text
stringlengths 105
4.17k
| source
stringclasses 883
values |
|---|---|
Another interesting recent development is research into models of just enough complexity through an estimation of the intrinsic complexity of the task being modelled. This approach has been successfully applied for multivariate time series prediction tasks such as traffic prediction. Finally, data can be augmented via methods such as cropping and rotating such that smaller training sets can be increased in size to reduce the chances of overfitting.
DNNs must consider many training parameters, such as the size (number of layers and number of units per layer), the learning rate, and initial weights. Sweeping through the parameter space for optimal parameters may not be feasible due to the cost in time and computational resources. Various tricks, such as batching (computing the gradient on several training examples at once rather than individual examples) speed up computation. Large processing capabilities of many-core architectures (such as GPUs or the Intel Xeon Phi) have produced significant speedups in training, because of the suitability of such processing architectures for the matrix and vector computations.
Alternatively, engineers may look for other types of neural networks with more straightforward and convergent training algorithms. CMAC (cerebellar model articulation controller) is one such kind of neural network. It doesn't require learning rates or randomized initial weights.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
CMAC (cerebellar model articulation controller) is one such kind of neural network. It doesn't require learning rates or randomized initial weights. The training process can be guaranteed to converge in one step with a new batch of data, and the computational complexity of the training algorithm is linear with respect to the number of neurons involved. Ting Qin, et al. "Continuous CMAC-QRLS and its systolic array". . Neural Processing Letters 22.1 (2005): 1-16.
## Hardware
Since the 2010s, advances in both machine learning algorithms and computer hardware have led to more efficient methods for training deep neural networks that contain many layers of non-linear hidden units and a very large output layer. By 2019, graphics processing units (GPUs), often with AI-specific enhancements, had displaced CPUs as the dominant method for training large-scale commercial cloud AI . OpenAI estimated the hardware computation used in the largest deep learning projects from AlexNet (2012) to AlphaZero (2017) and found a 300,000-fold increase in the amount of computation required, with a doubling-time trendline of 3.4 months.
Special electronic circuits called deep learning processors were designed to speed up deep learning algorithms. Deep learning processors include neural processing units (NPUs) in Huawei cellphones and cloud computing servers such as tensor processing units (TPU) in the Google Cloud Platform.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
Special electronic circuits called deep learning processors were designed to speed up deep learning algorithms. Deep learning processors include neural processing units (NPUs) in Huawei cellphones and cloud computing servers such as tensor processing units (TPU) in the Google Cloud Platform. Cerebras Systems has also built a dedicated system to handle large deep learning models, the CS-2, based on the largest processor in the industry, the second-generation Wafer Scale Engine (WSE-2).
Atomically thin semiconductors are considered promising for energy-efficient deep learning hardware where the same basic device structure is used for both logic operations and data storage.
In 2020, Marega et al. published experiments with a large-area active channel material for developing logic-in-memory devices and circuits based on floating-gate field-effect transistors (FGFETs).
In 2021, J. Feldmann et al. proposed an integrated photonic hardware accelerator for parallel convolutional processing. The authors identify two key advantages of integrated photonics over its electronic counterparts: (1) massively parallel data transfer through wavelength division multiplexing in conjunction with frequency combs, and (2) extremely high data modulation speeds. Their system can execute trillions of multiply-accumulate operations per second, indicating the potential of integrated photonics in data-heavy AI applications.
## Applications
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
Their system can execute trillions of multiply-accumulate operations per second, indicating the potential of integrated photonics in data-heavy AI applications.
## Applications
### Automatic speech recognition
Large-scale automatic speech recognition is the first and most convincing successful case of deep learning. LSTM RNNs can learn "Very Deep Learning" tasks that involve multi-second intervals containing speech events separated by thousands of discrete time steps, where one time step corresponds to about 10 ms. LSTM with forget gates is competitive with traditional speech recognizers on certain tasks.
The initial success in speech recognition was based on small-scale recognition tasks based on TIMIT. The data set contains 630 speakers from eight major dialects of American English, where each speaker reads 10 sentences. Its small size lets many configurations be tried. More importantly, the TIMIT task concerns phone-sequence recognition, which, unlike word-sequence recognition, allows weak phone bigram language models. This lets the strength of the acoustic modeling aspects of speech recognition be more easily analyzed. The error rates listed below, including these early results and measured as percent phone error rates (PER), have been summarized since 1991.
Method Percent phoneerror rate (PER) (%)
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
The error rates listed below, including these early results and measured as percent phone error rates (PER), have been summarized since 1991.
Method Percent phoneerror rate (PER) (%) Randomly Initialized RNN 26.1 Bayesian Triphone GMM-HMM 25.6 Hidden Trajectory (Generative) Model 24.8 Monophone Randomly Initialized DNN 23.4 Monophone DBN-DNN 22.4 Triphone GMM-HMM with BMMI Training 21.7 Monophone DBN-DNN on fbank 20.7 Convolutional DNN 20.0 Convolutional DNN w. Heterogeneous Pooling 18.7 Ensemble DNN/CNN/RNN 18.3 Bidirectional LSTM 17.8 Hierarchical Convolutional Deep Maxout Network 16.5
The debut of DNNs for speaker recognition in the late 1990s and speech recognition around 2009-2011 and of LSTM around 2003–2007, accelerated progress in eight major areas:
- Scale-up/out and accelerated DNN training and decoding
- Sequence discriminative training
- Feature processing by deep models with solid understanding of the underlying mechanisms
- Adaptation of DNNs and related deep models
- Multi-task and transfer learning by DNNs and related deep models
- CNNs and how to design them to best exploit domain knowledge of speech
- RNN and its rich LSTM variants
- Other types of deep models including tensor-based models and integrated deep generative/discriminative models.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
Method Percent phoneerror rate (PER) (%) Randomly Initialized RNN 26.1 Bayesian Triphone GMM-HMM 25.6 Hidden Trajectory (Generative) Model 24.8 Monophone Randomly Initialized DNN 23.4 Monophone DBN-DNN 22.4 Triphone GMM-HMM with BMMI Training 21.7 Monophone DBN-DNN on fbank 20.7 Convolutional DNN 20.0 Convolutional DNN w. Heterogeneous Pooling 18.7 Ensemble DNN/CNN/RNN 18.3 Bidirectional LSTM 17.8 Hierarchical Convolutional Deep Maxout Network 16.5
The debut of DNNs for speaker recognition in the late 1990s and speech recognition around 2009-2011 and of LSTM around 2003–2007, accelerated progress in eight major areas:
- Scale-up/out and accelerated DNN training and decoding
- Sequence discriminative training
- Feature processing by deep models with solid understanding of the underlying mechanisms
- Adaptation of DNNs and related deep models
- Multi-task and transfer learning by DNNs and related deep models
- CNNs and how to design them to best exploit domain knowledge of speech
- RNN and its rich LSTM variants
- Other types of deep models including tensor-based models and integrated deep generative/discriminative models.
All major commercial speech recognition systems (e.g., Microsoft Cortana, Xbox, Skype Translator, Amazon Alexa, Google Now, Apple Siri, Baidu and iFlyTek voice search, and a range of Nuance speech products, etc.) are based on deep learning.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
Randomly Initialized RNN 26.1 Bayesian Triphone GMM-HMM 25.6 Hidden Trajectory (Generative) Model 24.8 Monophone Randomly Initialized DNN 23.4 Monophone DBN-DNN 22.4 Triphone GMM-HMM with BMMI Training 21.7 Monophone DBN-DNN on fbank 20.7 Convolutional DNN 20.0 Convolutional DNN w. Heterogeneous Pooling 18.7 Ensemble DNN/CNN/RNN 18.3 Bidirectional LSTM 17.8 Hierarchical Convolutional Deep Maxout Network 16.5
The debut of DNNs for speaker recognition in the late 1990s and speech recognition around 2009-2011 and of LSTM around 2003–2007, accelerated progress in eight major areas:
- Scale-up/out and accelerated DNN training and decoding
- Sequence discriminative training
- Feature processing by deep models with solid understanding of the underlying mechanisms
- Adaptation of DNNs and related deep models
- Multi-task and transfer learning by DNNs and related deep models
- CNNs and how to design them to best exploit domain knowledge of speech
- RNN and its rich LSTM variants
- Other types of deep models including tensor-based models and integrated deep generative/discriminative models.
All major commercial speech recognition systems (e.g., Microsoft Cortana, Xbox, Skype Translator, Amazon Alexa, Google Now, Apple Siri, Baidu and iFlyTek voice search, and a range of Nuance speech products, etc.) are based on deep learning.
### Image recognition
A common evaluation set for image classification is the MNIST database data set.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
All major commercial speech recognition systems (e.g., Microsoft Cortana, Xbox, Skype Translator, Amazon Alexa, Google Now, Apple Siri, Baidu and iFlyTek voice search, and a range of Nuance speech products, etc.) are based on deep learning.
### Image recognition
A common evaluation set for image classification is the MNIST database data set. MNIST is composed of handwritten digits and includes 60,000 training examples and 10,000 test examples. As with TIMIT, its small size lets users test multiple configurations. A comprehensive list of results on this set is available.
Deep learning-based image recognition has become "superhuman", producing more accurate results than human contestants. This first occurred in 2011 in recognition of traffic signs, and in 2014, with recognition of human faces.
Deep learning-trained vehicles now interpret 360° camera views. Another example is Facial Dysmorphology Novel Analysis (FDNA) used to analyze cases of human malformation connected to a large database of genetic syndromes.
### Visual art processing
Closely related to the progress that has been made in image recognition is the increasing application of deep learning techniques to various visual art tasks.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
Another example is Facial Dysmorphology Novel Analysis (FDNA) used to analyze cases of human malformation connected to a large database of genetic syndromes.
### Visual art processing
Closely related to the progress that has been made in image recognition is the increasing application of deep learning techniques to various visual art tasks. DNNs have proven themselves capable, for example, of
- identifying the style period of a given painting
- Neural Style Transfer capturing the style of a given artwork and applying it in a visually pleasing manner to an arbitrary photograph or video
- generating striking imagery based on random visual input fields.
### Natural language processing
Neural networks have been used for implementing language models since the early 2000s. LSTM helped to improve machine translation and language modeling.
Other key techniques in this field are negative sampling and word embedding. Word embedding, such as word2vec, can be thought of as a representational layer in a deep learning architecture that transforms an atomic word into a positional representation of the word relative to other words in the dataset; the position is represented as a point in a vector space. Using word embedding as an RNN input layer allows the network to parse sentences and phrases using an effective compositional vector grammar. A compositional vector grammar can be thought of as probabilistic context free grammar (PCFG) implemented by an RNN.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
Using word embedding as an RNN input layer allows the network to parse sentences and phrases using an effective compositional vector grammar. A compositional vector grammar can be thought of as probabilistic context free grammar (PCFG) implemented by an RNN. Recursive auto-encoders built atop word embeddings can assess sentence similarity and detect paraphrasing. Deep neural architectures provide the best results for constituency parsing, sentiment analysis, information retrieval, spoken language understanding, machine translation, contextual entity linking, writing style recognition, named-entity recognition (token classification), text classification, and others.
Recent developments generalize word embedding to sentence embedding.
Google Translate (GT) uses a large end-to-end long short-term memory (LSTM) network. Google Neural Machine Translation (GNMT) uses an example-based machine translation method in which the system "learns from millions of examples". It translates "whole sentences at a time, rather than pieces". Google Translate supports over one hundred languages. The network encodes the "semantics of the sentence rather than simply memorizing phrase-to-phrase translations". GT uses English as an intermediate between most language pairs.
### Drug discovery and toxicology
A large percentage of candidate drugs fail to win regulatory approval.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
GT uses English as an intermediate between most language pairs.
### Drug discovery and toxicology
A large percentage of candidate drugs fail to win regulatory approval. These failures are caused by insufficient efficacy (on-target effect), undesired interactions (off-target effects), or unanticipated toxic effects. Research has explored use of deep learning to predict the biomolecular targets, off-targets, and toxic effects of environmental chemicals in nutrients, household products and drugs.
AtomNet is a deep learning system for structure-based rational drug design. AtomNet was used to predict novel candidate biomolecules for disease targets such as the Ebola virus and multiple sclerosis.
In 2017 graph neural networks were used for the first time to predict various properties of molecules in a large toxicology data set. In 2019, generative neural networks were used to produce molecules that were validated experimentally all the way into mice.
### Customer relationship management
Deep reinforcement learning has been used to approximate the value of possible direct marketing actions, defined in terms of RFM variables. The estimated value function was shown to have a natural interpretation as customer lifetime value.
### Recommendation systems
Recommendation systems have used deep learning to extract meaningful features for a latent factor model for content-based music and journal recommendations.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
The estimated value function was shown to have a natural interpretation as customer lifetime value.
### Recommendation systems
Recommendation systems have used deep learning to extract meaningful features for a latent factor model for content-based music and journal recommendations. Multi-view deep learning has been applied for learning user preferences from multiple domains. The model uses a hybrid collaborative and content-based approach and enhances recommendations in multiple tasks.
### Bioinformatics
An autoencoder ANN was used in bioinformatics, to predict gene ontology annotations and gene-function relationships.
In medical informatics, deep learning was used to predict sleep quality based on data from wearables and predictions of health complications from electronic health record data.
Deep neural networks have shown unparalleled performance in predicting protein structure, according to the sequence of the amino acids that make it up. In 2020, AlphaFold, a deep-learning based system, achieved a level of accuracy significantly higher than all previous computational methods.
### Deep Neural Network Estimations
Deep neural networks can be used to estimate the entropy of a stochastic process and called Neural Joint Entropy Estimator (NJEE). Such an estimation provides insights on the effects of input random variables on an independent random variable.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
### Deep Neural Network Estimations
Deep neural networks can be used to estimate the entropy of a stochastic process and called Neural Joint Entropy Estimator (NJEE). Such an estimation provides insights on the effects of input random variables on an independent random variable. Practically, the DNN is trained as a classifier that maps an input vector or matrix X to an output probability distribution over the possible classes of random variable Y, given input X. For example, in image classification tasks, the NJEE maps a vector of pixels' color values to probabilities over possible image classes. In practice, the probability distribution of Y is obtained by a Softmax layer with number of nodes that is equal to the alphabet size of Y. NJEE uses continuously differentiable activation functions, such that the conditions for the universal approximation theorem holds. It is shown that this method provides a strongly consistent estimator and outperforms other methods in case of large alphabet sizes.
### Medical image analysis
Deep learning has been shown to produce competitive results in medical application such as cancer cell classification, lesion detection, organ segmentation and image enhancement. Modern deep learning tools demonstrate the high accuracy of detecting various diseases and the helpfulness of their use by specialists to improve the diagnosis efficiency.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
### Medical image analysis
Deep learning has been shown to produce competitive results in medical application such as cancer cell classification, lesion detection, organ segmentation and image enhancement. Modern deep learning tools demonstrate the high accuracy of detecting various diseases and the helpfulness of their use by specialists to improve the diagnosis efficiency.
### Mobile advertising
Finding the appropriate mobile audience for mobile advertising is always challenging, since many data points must be considered and analyzed before a target segment can be created and used in ad serving by any ad server. Deep learning has been used to interpret large, many-dimensioned advertising datasets. Many data points are collected during the request/serve/click internet advertising cycle. This information can form the basis of machine learning to improve ad selection.
### Image restoration
Deep learning has been successfully applied to inverse problems such as denoising, super-resolution, inpainting, and film colorization. These applications include learning methods such as "Shrinkage Fields for Effective Image Restoration" which trains on an image dataset, and Deep Image Prior, which trains on the image that needs restoration.
### Financial fraud detection
Deep learning is being successfully applied to financial fraud detection, tax evasion detection, and anti-money laundering.
### Materials science
In November 2023, researchers at Google DeepMind and Lawrence Berkeley National Laboratory announced that they had developed an AI system known as GNoME.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
### Financial fraud detection
Deep learning is being successfully applied to financial fraud detection, tax evasion detection, and anti-money laundering.
### Materials science
In November 2023, researchers at Google DeepMind and Lawrence Berkeley National Laboratory announced that they had developed an AI system known as GNoME. This system has contributed to materials science by discovering over 2 million new materials within a relatively short timeframe. GNoME employs deep learning techniques to efficiently explore potential material structures, achieving a significant increase in the identification of stable inorganic crystal structures. The system's predictions were validated through autonomous robotic experiments, demonstrating a noteworthy success rate of 71%. The data of newly discovered materials is publicly available through the Materials Project database, offering researchers the opportunity to identify materials with desired properties for various applications. This development has implications for the future of scientific discovery and the integration of AI in material science research, potentially expediting material innovation and reducing costs in product development. The use of AI and deep learning suggests the possibility of minimizing or eliminating manual lab experiments and allowing scientists to focus more on the design and analysis of unique compounds.
### Military
The United States Department of Defense applied deep learning to train robots in new tasks through observation.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
The use of AI and deep learning suggests the possibility of minimizing or eliminating manual lab experiments and allowing scientists to focus more on the design and analysis of unique compounds.
### Military
The United States Department of Defense applied deep learning to train robots in new tasks through observation.
### Partial differential equations
Physics informed neural networks have been used to solve partial differential equations in both forward and inverse problems in a data driven manner. One example is the reconstructing fluid flow governed by the Navier-Stokes equations. Using physics informed neural networks does not require the often expensive mesh generation that conventional CFD methods rely on.
### Deep backward stochastic differential equation method
Deep backward stochastic differential equation method is a numerical method that combines deep learning with Backward stochastic differential equation (BSDE). This method is particularly useful for solving high-dimensional problems in financial mathematics. By leveraging the powerful function approximation capabilities of deep neural networks, deep BSDE addresses the computational challenges faced by traditional numerical methods in high-dimensional settings. Specifically, traditional methods like finite difference methods or Monte Carlo simulations often struggle with the curse of dimensionality, where computational cost increases exponentially with the number of dimensions.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
By leveraging the powerful function approximation capabilities of deep neural networks, deep BSDE addresses the computational challenges faced by traditional numerical methods in high-dimensional settings. Specifically, traditional methods like finite difference methods or Monte Carlo simulations often struggle with the curse of dimensionality, where computational cost increases exponentially with the number of dimensions. Deep BSDE methods, however, employ deep neural networks to approximate solutions of high-dimensional partial differential equations (PDEs), effectively reducing the computational burden.
In addition, the integration of Physics-informed neural networks (PINNs) into the deep BSDE framework enhances its capability by embedding the underlying physical laws directly into the neural network architecture. This ensures that the solutions not only fit the data but also adhere to the governing stochastic differential equations. PINNs leverage the power of deep learning while respecting the constraints imposed by the physical models, resulting in more accurate and reliable solutions for financial mathematics problems.
### Image reconstruction
Image reconstruction is the reconstruction of the underlying images from the image-related measurements. Several works showed the better and superior performance of the deep learning methods compared to analytical methods for various applications, e.g., spectral imaging and ultrasound imaging.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
### Image reconstruction
Image reconstruction is the reconstruction of the underlying images from the image-related measurements. Several works showed the better and superior performance of the deep learning methods compared to analytical methods for various applications, e.g., spectral imaging and ultrasound imaging.
### Weather prediction
Traditional weather prediction systems solve a very complex system of partial differential equations. GraphCast is a deep learning based model, trained on a long history of weather data to predict how weather patterns change over time. It is able to predict weather conditions for up to 10 days globally, at a very detailed level, and in under a minute, with precision similar to state of the art systems.
### Epigenetic clock
An epigenetic clock is a biochemical test that can be used to measure age. Galkin et al. used deep neural networks to train an epigenetic aging clock of unprecedented accuracy using >6,000 blood samples. The clock uses information from 1000 CpG sites and predicts people with certain conditions older than healthy controls: IBD, frontotemporal dementia, ovarian cancer, obesity. The aging clock was planned to be released for public use in 2021 by an Insilico Medicine spinoff company Deep Longevity.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
The clock uses information from 1000 CpG sites and predicts people with certain conditions older than healthy controls: IBD, frontotemporal dementia, ovarian cancer, obesity. The aging clock was planned to be released for public use in 2021 by an Insilico Medicine spinoff company Deep Longevity.
## Relation to human cognitive and brain development
Deep learning is closely related to a class of theories of brain development (specifically, neocortical development) proposed by cognitive neuroscientists in the early 1990s. These developmental theories were instantiated in computational models, making them predecessors of deep learning systems. These developmental models share the property that various proposed learning dynamics in the brain (e.g., a wave of nerve growth factor) support the self-organization somewhat analogous to the neural networks utilized in deep learning models. Like the neocortex, neural networks employ a hierarchy of layered filters in which each layer considers information from a prior layer (or the operating environment), and then passes its output (and possibly the original input), to other layers. This process yields a self-organizing stack of transducers, well-tuned to their operating environment.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
Like the neocortex, neural networks employ a hierarchy of layered filters in which each layer considers information from a prior layer (or the operating environment), and then passes its output (and possibly the original input), to other layers. This process yields a self-organizing stack of transducers, well-tuned to their operating environment. A 1995 description stated, "...the infant's brain seems to organize itself under the influence of waves of so-called trophic-factors ... different regions of the brain become connected sequentially, with one layer of tissue maturing before another and so on until the whole brain is mature".
A variety of approaches have been used to investigate the plausibility of deep learning models from a neurobiological perspective. On the one hand, several variants of the backpropagation algorithm have been proposed in order to increase its processing realism. Other researchers have argued that unsupervised forms of deep learning, such as those based on hierarchical generative models and deep belief networks, may be closer to biological reality. In this respect, generative neural network models have been related to neurobiological evidence about sampling-based processing in the cerebral cortex.
Although a systematic comparison between the human brain organization and the neuronal encoding in deep networks has not yet been established, several analogies have been reported.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
In this respect, generative neural network models have been related to neurobiological evidence about sampling-based processing in the cerebral cortex.
Although a systematic comparison between the human brain organization and the neuronal encoding in deep networks has not yet been established, several analogies have been reported. For example, the computations performed by deep learning units could be similar to those of actual neurons and neural populations. Similarly, the representations developed by deep learning models are similar to those measured in the primate visual system both at the single-unit and at the population levels.
## Commercial activity
Facebook's AI lab performs tasks such as automatically tagging uploaded pictures with the names of the people in them.
Google's DeepMind Technologies developed a system capable of learning how to play Atari video games using only pixels as data input. In 2015 they demonstrated their AlphaGo system, which learned the game of Go well enough to beat a professional Go player. Google Translate uses a neural network to translate between more than 100 languages.
In 2017, Covariant.ai was launched, which focuses on integrating deep learning into factories.
As of 2008, researchers at The University of Texas at Austin (UT) developed a machine learning framework called Training an Agent Manually via Evaluative Reinforcement, or TAMER, which proposed new methods for robots or computer programs to learn how to perform tasks by interacting with a human instructor.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
In 2017, Covariant.ai was launched, which focuses on integrating deep learning into factories.
As of 2008, researchers at The University of Texas at Austin (UT) developed a machine learning framework called Training an Agent Manually via Evaluative Reinforcement, or TAMER, which proposed new methods for robots or computer programs to learn how to perform tasks by interacting with a human instructor. First developed as TAMER, a new algorithm called Deep TAMER was later introduced in 2018 during a collaboration between U.S. Army Research Laboratory (ARL) and UT researchers. Deep TAMER used deep learning to provide a robot with the ability to learn new tasks through observation. Using Deep TAMER, a robot learned a task with a human trainer, watching video streams or observing a human perform a task in-person. The robot later practiced the task with the help of some coaching from the trainer, who provided feedback such as "good job" and "bad job".
## Criticism and comment
Deep learning has attracted both criticism and comment, in some cases from outside the field of computer science.
### Theory
A main criticism concerns the lack of theory surrounding some methods. Learning in the most common deep architectures is implemented using well-understood gradient descent. However, the theory surrounding other algorithms, such as contrastive divergence is less clear. (e.g., Does it converge? If so, how fast?
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
(e.g., Does it converge? If so, how fast? What is it approximating?) Deep learning methods are often looked at as a black box, with most confirmations done empirically, rather than theoretically.
In further reference to the idea that artistic sensitivity might be inherent in relatively low levels of the cognitive hierarchy, a published series of graphic representations of the internal states of deep (20-30 layers) neural networks attempting to discern within essentially random data the images on which they were trained demonstrate a visual appeal: the original research notice received well over 1,000 comments, and was the subject of what was for a time the most frequently accessed article on The Guardian's website.
### Errors
Some deep learning architectures display problematic behaviors, such as confidently classifying unrecognizable images as belonging to a familiar category of ordinary images (2014) and misclassifying minuscule perturbations of correctly classified images (2013). Goertzel hypothesized that these behaviors are due to limitations in their internal representations and that these limitations would inhibit integration into heterogeneous multi-component artificial general intelligence (AGI) architectures. These issues may possibly be addressed by deep learning architectures that internally form states homologous to image-grammar decompositions of observed entities and events.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
Goertzel hypothesized that these behaviors are due to limitations in their internal representations and that these limitations would inhibit integration into heterogeneous multi-component artificial general intelligence (AGI) architectures. These issues may possibly be addressed by deep learning architectures that internally form states homologous to image-grammar decompositions of observed entities and events. Learning a grammar (visual or linguistic) from training data would be equivalent to restricting the system to commonsense reasoning that operates on concepts in terms of grammatical production rules and is a basic goal of both human language acquisition and artificial intelligence (AI).
### Cyber threat
As deep learning moves from the lab into the world, research and experience show that artificial neural networks are vulnerable to hacks and deception. By identifying patterns that these systems use to function, attackers can modify inputs to ANNs in such a way that the ANN finds a match that human observers would not recognize. For example, an attacker can make subtle changes to an image such that the ANN finds a match even though the image looks to a human nothing like the search target. Such manipulation is termed an "adversarial attack".
In 2016 researchers used one ANN to doctor images in trial and error fashion, identify another's focal points, and thereby generate images that deceived it. The modified images looked no different to human eyes.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
In 2016 researchers used one ANN to doctor images in trial and error fashion, identify another's focal points, and thereby generate images that deceived it. The modified images looked no different to human eyes. Another group showed that printouts of doctored images then photographed successfully tricked an image classification system. One defense is reverse image search, in which a possible fake image is submitted to a site such as TinEye that can then find other instances of it. A refinement is to search using only parts of the image, to identify images from which that piece may have been taken.
Another group showed that certain psychedelic spectacles could fool a facial recognition system into thinking ordinary people were celebrities, potentially allowing one person to impersonate another. In 2017 researchers added stickers to stop signs and caused an ANN to misclassify them.
ANNs can however be further trained to detect attempts at deception, potentially leading attackers and defenders into an arms race similar to the kind that already defines the malware defense industry. ANNs have been trained to defeat ANN-based anti-malware software by repeatedly attacking a defense with malware that was continually altered by a genetic algorithm until it tricked the anti-malware while retaining its ability to damage the target.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
ANNs can however be further trained to detect attempts at deception, potentially leading attackers and defenders into an arms race similar to the kind that already defines the malware defense industry. ANNs have been trained to defeat ANN-based anti-malware software by repeatedly attacking a defense with malware that was continually altered by a genetic algorithm until it tricked the anti-malware while retaining its ability to damage the target.
In 2016, another group demonstrated that certain sounds could make the Google Now voice command system open a particular web address, and hypothesized that this could "serve as a stepping stone for further attacks (e.g., opening a web page hosting drive-by malware)".
In "data poisoning", false data is continually smuggled into a machine learning system's training set to prevent it from achieving mastery.
### Data collection ethics
The deep learning systems that are trained using supervised learning often rely on data that is created and/or annotated by humans. It has been argued that not only low-paid clickwork (such as on Amazon Mechanical Turk) is regularly deployed for this purpose, but also implicit forms of human microwork that are often not recognized as such.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
### Data collection ethics
The deep learning systems that are trained using supervised learning often rely on data that is created and/or annotated by humans. It has been argued that not only low-paid clickwork (such as on Amazon Mechanical Turk) is regularly deployed for this purpose, but also implicit forms of human microwork that are often not recognized as such. The philosopher Rainer Mühlhoff distinguishes five types of "machinic capture" of human microwork to generate training data: (1) gamification (the embedding of annotation or computation tasks in the flow of a game), (2) "trapping and tracking" (e.g. CAPTCHAs for image recognition or click-tracking on Google search results pages), (3) exploitation of social motivations (e.g. tagging faces on Facebook to obtain labeled facial images), (4) information mining (e.g. by leveraging quantified-self devices such as activity trackers) and (5) clickwork.
|
https://en.wikipedia.org/wiki/Deep_learning%23Deep_neural_networks
|
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.
The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length or distance of each segment.
## Definition
The shortest path problem can be defined for graphs whether undirected, directed, or mixed. The definition for undirected graphs states that every edge can be traversed in either direction.
### Directed graph
s require that consecutive vertices be connected by an appropriate directed edge.
Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices
$$
P = ( v_1, v_2, \ldots, v_n ) \in V \times V \times \cdots \times V
$$
such that _ BLOCK1_ is adjacent to
$$
v_{i+1}
$$
for
$$
1 \leq i < n
$$
.
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
A path in an undirected graph is a sequence of vertices
$$
P = ( v_1, v_2, \ldots, v_n ) \in V \times V \times \cdots \times V
$$
such that _ BLOCK1_ is adjacent to
$$
v_{i+1}
$$
for
$$
1 \leq i < n
$$
. Such a path
$$
P
$$
is called a path of length
$$
n-1
$$
from
$$
v_1
$$
to
$$
v_n
$$
. (The
$$
v_i
$$
are variables; their numbering relates to their position in the sequence and need not relate to a canonical labeling.)
Let
$$
E = \{e_{i, j}\}
$$
where
$$
e_{i, j}
$$
is the edge incident to both
$$
v_i
$$
and
$$
v_j
$$
.
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
(The
$$
v_i
$$
are variables; their numbering relates to their position in the sequence and need not relate to a canonical labeling.)
Let
$$
E = \{e_{i, j}\}
$$
where
$$
e_{i, j}
$$
is the edge incident to both
$$
v_i
$$
and
$$
v_j
$$
. Given a real-valued weight function
$$
f: E \rightarrow \mathbb{R}
$$
, and an undirected (simple) graph
$$
G
$$
, the shortest path from
$$
v
$$
to
$$
v'
$$
is the path
$$
P = ( v_1, v_2, \ldots, v_n )
$$
(where
$$
v_1 = v
$$
and
$$
v_n = v'
$$
) that over all possible
$$
n
$$
minimizes the sum
$$
\sum_{i =1}^{n-1} f(e_{i, i+1}).
$$
When each edge in the graph has unit weight or
$$
f: E \rightarrow \{1\}
$$
, this is equivalent to finding the path with fewest edges.
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
minimizes the sum
$$
\sum_{i =1}^{n-1} f(e_{i, i+1}).
$$
When each edge in the graph has unit weight or
$$
f: E \rightarrow \{1\}
$$
, this is equivalent to finding the path with fewest edges.
The problem is also sometimes called the single-pair shortest path problem, to distinguish it from the following variations:
- The single-source shortest path problem, in which we have to find shortest paths from a source vertex v to all other vertices in the graph.
- The single-destination shortest path problem, in which we have to find shortest paths from all vertices in the directed graph to a single destination vertex v. This can be reduced to the single-source shortest path problem by reversing the arcs in the directed graph.
- The all-pairs shortest path problem, in which we have to find shortest paths between every pair of vertices v, v' in the graph.
These generalizations have significantly more efficient algorithms than the simplistic approach of running a single-pair shortest path algorithm on all relevant pairs of vertices.
## Algorithms
Several well-known algorithms exist for solving this problem and its variants.
- Dijkstra's algorithm solves the single-source shortest path problem with only non-negative edge weights.
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
## Algorithms
Several well-known algorithms exist for solving this problem and its variants.
- Dijkstra's algorithm solves the single-source shortest path problem with only non-negative edge weights.
- Bellman–Ford algorithm solves the single-source problem if edge weights may be negative.
- A* search algorithm solves for single-pair shortest path using heuristics to try to speed up the search.
- Floyd–Warshall algorithm solves all pairs shortest paths.
- Johnson's algorithm solves all pairs shortest paths, and may be faster than Floyd–Warshall on sparse graphs.
- Viterbi algorithm solves the shortest stochastic path problem with an additional probabilistic weight on each node.
Additional algorithms and associated evaluations may be found in .
## Single-source shortest paths
###
### Undirected graph
s
Weights Time complexity Author + O(V2) + O((E + V) log V) (binary heap) + O(E + V log V) (Fibonacci heap) O(E) (requires constant-time multiplication) +
### Unweighted graphs
Algorithm Time complexity Author Breadth-first search O(E + V)
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
### Undirected graph
s
Weights Time complexity Author + O(V2) + O((E + V) log V) (binary heap) + O(E + V log V) (Fibonacci heap) O(E) (requires constant-time multiplication) +
### Unweighted graphs
Algorithm Time complexity Author Breadth-first search O(E + V)
### Directed acyclic graphs
An algorithm using topological sorting can solve the single-source shortest path problem in time in arbitrarily-weighted directed acyclic graphs.
### Directed graphs with nonnegative weights
The following table is taken from , with some corrections and additions.
A green background indicates an asymptotically best bound in the table; L is the maximum length (or weight) among all edges, assuming integer edge weights.
Weights Algorithm Time complexity Author Bellman–Ford algorithm , , Dijkstra's algorithm with list , , Minty (see ), Dijkstra's algorithm with binary heap Dijkstra's algorithm with Fibonacci heap , Quantum Dijkstra algorithm with adjacency list Dürr et al. 2006 Dial's algorithm (Dijkstra's algorithm using a bucket queue with L buckets) , Gabow's algorithm , Thorup
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
Weights Algorithm Time complexity Author Bellman–Ford algorithm , , Dijkstra's algorithm with list , , Minty (see ), Dijkstra's algorithm with binary heap Dijkstra's algorithm with Fibonacci heap , Quantum Dijkstra algorithm with adjacency list Dürr et al. 2006 Dial's algorithm (Dijkstra's algorithm using a bucket queue with L buckets) , Gabow's algorithm , Thorup
### Directed graphs with arbitrary weights without negative cycles
Weights Algorithm Time complexity Author Bellman–Ford algorithm , , Johnson-Dijkstra with binary heap Johnson-Dijkstra with Fibonacci heap , , adapted after Johnson's technique applied to Dial's algorithm , adapted after Interior-point method with Laplacian solverInterior-point method with flow solverRobust interior-point method with sketching interior-point method with dynamic min-ratio cycle data structureBased on low-diameter decompositionHop-limited shortest paths
### Directed graphs with arbitrary weights with negative cycles
Finds a negative cycle or calculates distances to all vertices.
Weights Algorithm Time complexity Author Andrew V. Goldberg
### Planar graphs with nonnegative weights
Weights Algorithm Time complexity Author
##
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
Author Andrew V. Goldberg
### Planar graphs with nonnegative weights
Weights Algorithm Time complexity Author
##
## Applications
Network flows are a fundamental concept in graph theory and operations research, often used to model problems involving the transportation of goods, liquids, or information through a network. A network flow problem typically involves a directed graph where each edge represents a pipe, wire, or road, and each edge has a capacity, which is the maximum amount that can flow through it. The goal is to find a feasible flow that maximizes the flow from a source node to a sink node.
Shortest Path Problems can be used to solve certain network flow problems, particularly when dealing with single-source, single-sink networks. In these scenarios, we can transform the network flow problem into a series of shortest path problems.
### Transformation Steps
1. Create a Residual Graph:
1. For each edge (u, v) in the original graph, create two edges in the residual graph:
1. (u, v) with capacity c(u, v)
1. (v, u) with capacity 0
1. The residual graph represents the remaining capacity available in the network.
1. Find the Shortest Path:
1.
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
The residual graph represents the remaining capacity available in the network.
1. Find the Shortest Path:
1. Use a shortest path algorithm (e.g., Dijkstra's algorithm, Bellman-Ford algorithm) to find the shortest path from the source node to the sink node in the residual graph.
1. Augment the Flow:
1. Find the minimum capacity along the shortest path.
1. Increase the flow on the edges of the shortest path by this minimum capacity.
1. Decrease the capacity of the edges in the forward direction and increase the capacity of the edges in the backward direction.
1. Update the Residual Graph:
1. Update the residual graph based on the augmented flow.
1. Repeat:
1. Repeat steps 2-4 until no more paths can be found from the source to the sink.
## All-pairs shortest paths
The all-pairs shortest path problem finds the shortest paths between every pair of vertices , in the graph. The all-pairs shortest paths problem for unweighted directed graphs was introduced by , who observed that it could be solved by a linear number of matrix multiplications that takes a total time of .
Undirected graph
Weights Time complexity Algorithm + Floyd–
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
The all-pairs shortest paths problem for unweighted directed graphs was introduced by , who observed that it could be solved by a linear number of matrix multiplications that takes a total time of .
Undirected graph
Weights Time complexity Algorithm + Floyd– Warshall algorithm Seidel's algorithm (expected running time) + applied to every vertex (requires constant-time multiplication).
Directed graph
Weights Time complexity Algorithm (no negative cycles) Floyd–Warshall algorithm (no negative cycles) Quantum search (no negative cycles) Johnson–Dijkstra (no negative cycles)
Applications
Shortest path algorithms are applied to automatically find directions between physical locations, such as driving directions on web mapping websites like MapQuest or Google Maps. For this application fast specialized algorithms are available.
If one represents a nondeterministic abstract machine as a graph where vertices describe states and edges describe possible transitions, shortest path algorithms can be used to find an optimal sequence of choices to reach a certain goal state, or to establish lower bounds on the time needed to reach a given state. For example, if vertices represent the states of a puzzle like a Rubik's Cube and each directed edge corresponds to a single move or turn, shortest path algorithms can be used to find a solution that uses the minimum possible number of moves.
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
If one represents a nondeterministic abstract machine as a graph where vertices describe states and edges describe possible transitions, shortest path algorithms can be used to find an optimal sequence of choices to reach a certain goal state, or to establish lower bounds on the time needed to reach a given state. For example, if vertices represent the states of a puzzle like a Rubik's Cube and each directed edge corresponds to a single move or turn, shortest path algorithms can be used to find a solution that uses the minimum possible number of moves.
In a networking or telecommunications mindset, this shortest path problem is sometimes called the min-delay path problem and usually tied with a widest path problem. For example, the algorithm may seek the shortest (min-delay) widest path, or widest shortest (min-delay) path.
A more lighthearted application is the games of "six degrees of separation" that try to find the shortest path in graphs like movie stars appearing in the same film.
Other applications, often studied in operations research, include plant and facility layout, robotics, transportation, and VLSI design.
### Road networks
A road network can be considered as a graph with positive weights. The nodes represent road junctions and each edge of the graph is associated with a road segment between two junctions.
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
### Road networks
A road network can be considered as a graph with positive weights. The nodes represent road junctions and each edge of the graph is associated with a road segment between two junctions. The weight of an edge may correspond to the length of the associated road segment, the time needed to traverse the segment, or the cost of traversing the segment. Using directed edges it is also possible to model one-way streets. Such graphs are special in the sense that some edges are more important than others for long-distance travel (e.g. highways). This property has been formalized using the notion of highway dimension. There are a great number of algorithms that exploit this property and are therefore able to compute the shortest path a lot quicker than would be possible on general graphs.
All of these algorithms work in two phases. In the first phase, the graph is preprocessed without knowing the source or target node. The second phase is the query phase. In this phase, source and target node are known. The idea is that the road network is static, so the preprocessing phase can be done once and used for a large number of queries on the same road network.
The algorithm with the fastest known query time is called hub labeling and is able to compute shortest path on the road networks of Europe or the US in a fraction of a microsecond.
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
The idea is that the road network is static, so the preprocessing phase can be done once and used for a large number of queries on the same road network.
The algorithm with the fastest known query time is called hub labeling and is able to compute shortest path on the road networks of Europe or the US in a fraction of a microsecond. Other techniques that have been used are:
- ALT (A* search, landmarks, and triangle inequality)
- Arc flags
- Contraction hierarchies
- Transit node routing
- Reach-based pruning
- Labeling
- Hub labels
## Related problems
For shortest path problems in computational geometry, see Euclidean shortest path.
The shortest multiple disconnected path is a representation of the primitive path network within the framework of Reptation theory. The widest path problem seeks a path so that the minimum label of any edge is as large as possible.
Other related problems may be classified into the following categories.
### Paths with constraints
Unlike the shortest path problem, which can be solved in polynomial time in graphs without negative cycles, shortest path problems which include additional constraints on the desired solution path are called Constrained Shortest Path First, and are harder to solve. One example is the constrained shortest path problem, which attempts to minimize the total cost of the path while at the same time maintaining another metric below a given threshold.
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
### Paths with constraints
Unlike the shortest path problem, which can be solved in polynomial time in graphs without negative cycles, shortest path problems which include additional constraints on the desired solution path are called Constrained Shortest Path First, and are harder to solve. One example is the constrained shortest path problem, which attempts to minimize the total cost of the path while at the same time maintaining another metric below a given threshold. This makes the problem NP-complete (such problems are not believed to be efficiently solvable for large sets of data, see P = NP problem). Another NP-complete example requires a specific set of vertices to be included in the path, which makes the problem similar to the Traveling Salesman Problem (TSP). The TSP is the problem of finding the shortest path that goes through every vertex exactly once, and returns to the start. The problem of finding the longest path in a graph is also NP-complete.
### Partial observability
The Canadian traveller problem and the stochastic shortest path problem are generalizations where either the graph is not completely known to the mover, changes over time, or where actions (traversals) are probabilistic.
### Strategic shortest paths
Sometimes, the edges in a graph have personalities: each edge has its own selfish interest.
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
### Partial observability
The Canadian traveller problem and the stochastic shortest path problem are generalizations where either the graph is not completely known to the mover, changes over time, or where actions (traversals) are probabilistic.
### Strategic shortest paths
Sometimes, the edges in a graph have personalities: each edge has its own selfish interest. An example is a communication network, in which each edge is a computer that possibly belongs to a different person. Different computers have different transmission speeds, so every edge in the network has a numeric weight equal to the number of milliseconds it takes to transmit a message. Our goal is to send a message between two points in the network in the shortest time possible. If we know the transmission-time of each computer (the weight of each edge), then we can use a standard shortest-paths algorithm. If we do not know the transmission times, then we have to ask each computer to tell us its transmission-time. But, the computers may be selfish: a computer might tell us that its transmission time is very long, so that we will not bother it with our messages. A possible solution to this problem is to use a variant of the VCG mechanism, which gives the computers an incentive to reveal their true weights.
### Negative cycle detection
In some cases, the main goal is not to find the shortest path, but only to detect if the graph contains a negative cycle.
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
A possible solution to this problem is to use a variant of the VCG mechanism, which gives the computers an incentive to reveal their true weights.
### Negative cycle detection
In some cases, the main goal is not to find the shortest path, but only to detect if the graph contains a negative cycle. Some shortest-paths algorithms can be used for this purpose:
- The Bellman–Ford algorithm can be used to detect a negative cycle in time
$$
O(|V||E|)
$$
.
- Cherkassky and Goldberg survey several other algorithms for negative cycle detection.
## General algebraic framework on semirings: the algebraic path problem
Many problems can be framed as a form of the shortest path for some suitably substituted notions of addition along a path and taking the minimum. The general approach to these is to consider the two operations to be those of a semiring. Semiring multiplication is done along the path, and the addition is between paths. This general framework is known as the algebraic path problem.
Most of the classic shortest-path algorithms (and new ones) can be formulated as solving linear systems over such algebraic structures.
More recently, an even more general framework for solving these (and much less obviously related problems) has been developed under the banner of valuation algebras.
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
Most of the classic shortest-path algorithms (and new ones) can be formulated as solving linear systems over such algebraic structures.
More recently, an even more general framework for solving these (and much less obviously related problems) has been developed under the banner of valuation algebras.
## Shortest path in stochastic time-dependent networks
In real-life, a transportation network is usually stochastic and time-dependent. The travel duration on a road segment depends on many factors such as the amount of traffic (origin-destination matrix), road work, weather, accidents and vehicle breakdowns. A more realistic model of such a road network is a stochastic time-dependent (STD) network.
There is no accepted definition of optimal path under uncertainty (that is, in stochastic road networks). It is a controversial subject, despite considerable progress during the past decade. One common definition is a path with the minimum expected travel time. The main advantage of this approach is that it can make use of efficient shortest path algorithms for deterministic networks. However, the resulting optimal path may not be reliable, because this approach fails to address travel time variability.
To tackle this issue, some researchers use travel duration distribution instead of its expected value. So, they find the probability distribution of total travel duration using different optimization methods such as dynamic programming and Dijkstra's algorithm .
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
To tackle this issue, some researchers use travel duration distribution instead of its expected value. So, they find the probability distribution of total travel duration using different optimization methods such as dynamic programming and Dijkstra's algorithm . These methods use stochastic optimization, specifically stochastic dynamic programming to find the shortest path in networks with probabilistic arc length. The terms travel time reliability and travel time variability are used as opposites in the transportation research literature: the higher the variability, the lower the reliability of predictions.
To account for variability, researchers have suggested two alternative definitions for an optimal path under uncertainty. The most reliable path is one that maximizes the probability of arriving on time given a travel time budget. An α-reliable path is one that minimizes the travel time budget required to arrive on time with a given probability.
|
https://en.wikipedia.org/wiki/Shortest_path_problem
|
In lab experiments that study chaos theory, approaches designed to control chaos are based on certain observed system behaviors. Any chaotic attractor contains an infinite number of unstable, periodic orbits. Chaotic dynamics, then, consists of a motion where the system state moves in the neighborhood of one of these orbits for a while, then falls close to a different unstable, periodic orbit where it remains for a limited time and so forth. This results in a complicated and unpredictable wandering over longer periods of time.
Control of chaos is the stabilization, by means of small system perturbations, of one of these unstable periodic orbits. The result is to render an otherwise chaotic motion more stable and predictable, which is often an advantage. The perturbation must be tiny compared to the overall size of the attractor of the system to avoid significant modification of the system's natural dynamics.
Several techniques have been devised for chaos control, but most are developments of two basic approaches: the Ott–Grebogi–Yorke (OGY) method and Pyragas continuous control. Both methods require a previous determination of the unstable periodic orbits of the chaotic system before the controlling algorithm can be designed.
|
https://en.wikipedia.org/wiki/Control_of_chaos
|
Several techniques have been devised for chaos control, but most are developments of two basic approaches: the Ott–Grebogi–Yorke (OGY) method and Pyragas continuous control. Both methods require a previous determination of the unstable periodic orbits of the chaotic system before the controlling algorithm can be designed.
## OGY method
Edward Ott, Celso Grebogi and James A. Yorke were the first to make the key observation that the infinite number of unstable periodic orbits typically embedded in a chaotic attractor could be taken advantage of for the purpose of achieving control by means of applying only very small perturbations. After making this general point, they illustrated it with a specific method, since called the Ott–Grebogi–Yorke (OGY) method of achieving stabilization of a chosen unstable periodic orbit. In the OGY method, small, wisely chosen, kicks are applied to the system once per cycle, to maintain it near the desired unstable periodic orbit.
To start, one obtains information about the chaotic system by analyzing a slice of the chaotic attractor. This slice is a Poincaré section. After the information about the section has been gathered, one allows the system to run and waits until it comes near a desired periodic orbit in the section. Next, the system is encouraged to remain on that orbit by perturbing the appropriate parameter.
|
https://en.wikipedia.org/wiki/Control_of_chaos
|
After the information about the section has been gathered, one allows the system to run and waits until it comes near a desired periodic orbit in the section. Next, the system is encouraged to remain on that orbit by perturbing the appropriate parameter. When the control parameter is actually changed, the chaotic attractor is shifted and distorted somewhat. If all goes according to plan, the new attractor encourages the system to continue on the desired trajectory. One strength of this method is that it does not require a detailed model of the chaotic system but only some information about the Poincaré section. It is for this reason that the method has been so successful in controlling a wide variety of chaotic systems.
The weaknesses of this method are in isolating the Poincaré section and in calculating the precise perturbations necessary to attain stability.
## Pyragas method
In the Pyragas method of stabilizing a periodic orbit, an appropriate continuous controlling signal is injected into the system, whose intensity is practically zero as the system evolves close to the desired periodic orbit but increases when it drifts away from the desired orbit. Both the Pyragas and OGY methods are part of a general class of methods called "closed loop" or "feedback" methods which can be applied based on knowledge of the system obtained through solely observing the behavior of the system as a whole over a suitable period of time.
|
https://en.wikipedia.org/wiki/Control_of_chaos
|
## Pyragas method
In the Pyragas method of stabilizing a periodic orbit, an appropriate continuous controlling signal is injected into the system, whose intensity is practically zero as the system evolves close to the desired periodic orbit but increases when it drifts away from the desired orbit. Both the Pyragas and OGY methods are part of a general class of methods called "closed loop" or "feedback" methods which can be applied based on knowledge of the system obtained through solely observing the behavior of the system as a whole over a suitable period of time. The method was proposed by Lithuanian physicist .
## Applications
Experimental control of chaos by one or both of these methods has been achieved in a variety of systems, including turbulent fluids, oscillating chemical reactions, magneto-mechanical oscillators and cardiac tissues. attempt the control of chaotic bubbling with the OGY method and using electrostatic potential as the primary control variable.
Forcing two systems into the same state is not the only way to achieve synchronization of chaos. Both control of chaos and synchronization constitute parts of cybernetical physics, a research area on the border between physics and control theory.
## References
## External links
- Chaos control bibliography (1997–2000)
Category:Chaos theory
Category:Nonlinear systems
|
https://en.wikipedia.org/wiki/Control_of_chaos
|
In physics, Newtonian dynamics (also known as Newtonian mechanics) is the study of the dynamics of a particle or a small body according to Newton's laws of motion.
## Mathematical generalizations
Typically, the Newtonian dynamics occurs in a three-dimensional Euclidean space, which is flat. However, in mathematics Newton's laws of motion can be generalized to multidimensional and curved spaces. Often the term Newtonian dynamics is narrowed to Newton's second law
$$
\displaystyle m\,\mathbf a=\mathbf F
$$
.
## Newton's second law in a multidimensional space
Consider
$$
\displaystyle N
$$
particles with masses
$$
\displaystyle m_1,\,\ldots,\,m_N
$$
in the regular three-dimensional Euclidean space. Let
$$
\displaystyle \mathbf r_1,\,\ldots,\,\mathbf r_N
$$
be their radius-vectors in some inertial coordinate system. Then the motion of these particles is governed by Newton's second law applied to each of them
The three-dimensional radius-vectors
$$
\displaystyle\mathbf r_1,\,\ldots,\,\mathbf r_N
$$
can be built into a single
$$
\displaystyle n=3N
$$
-dimensional radius-vector.
|
https://en.wikipedia.org/wiki/Newtonian_dynamics
|
Let
$$
\displaystyle \mathbf r_1,\,\ldots,\,\mathbf r_N
$$
be their radius-vectors in some inertial coordinate system. Then the motion of these particles is governed by Newton's second law applied to each of them
The three-dimensional radius-vectors
$$
\displaystyle\mathbf r_1,\,\ldots,\,\mathbf r_N
$$
can be built into a single
$$
\displaystyle n=3N
$$
-dimensional radius-vector. Similarly, three-dimensional velocity vectors
$$
\displaystyle\mathbf v_1,\,\ldots,\,\mathbf v_N
$$
can be built into a single
$$
\displaystyle n=3N
$$
-dimensional velocity vector:
In terms of the multidimensional vectors () the equations () are written as
i.e. they take the form of Newton's second law applied to a single particle with the unit mass
$$
\displaystyle m=1
$$
.
Definition. The equations () are called the
equations of a Newtonian dynamical system in a flat multidimensional Euclidean space, which is called the configuration space of this system. Its points are marked by the radius-vector
$$
\displaystyle\mathbf r
$$
.
|
https://en.wikipedia.org/wiki/Newtonian_dynamics
|
The equations () are called the
equations of a Newtonian dynamical system in a flat multidimensional Euclidean space, which is called the configuration space of this system. Its points are marked by the radius-vector
$$
\displaystyle\mathbf r
$$
. The space whose points are marked by the pair of vectors
$$
\displaystyle(\mathbf r,\mathbf v)
$$
is called the phase space of the dynamical system ().
## Euclidean structure
The configuration space and the phase space of the dynamical system () both are Euclidean spaces, i. e. they are equipped with a Euclidean structure. The Euclidean structure of them is defined so that the kinetic energy of the single multidimensional particle with the unit mass
$$
\displaystyle m=1
$$
is equal to the sum of kinetic energies of the three-dimensional particles with the masses
$$
\displaystyle m_1,\,\ldots,\,m_N
$$
:
## Constraints and internal coordinates
In some cases the motion of the particles with the masses
$$
\displaystyle m_1,\,\ldots,\,m_N
$$
can be constrained. Typical constraints look like scalar equations of the form
Constraints of the form () are called holonomic and scleronomic.
|
https://en.wikipedia.org/wiki/Newtonian_dynamics
|
In some cases the motion of the particles with the masses
$$
\displaystyle m_1,\,\ldots,\,m_N
$$
can be constrained. Typical constraints look like scalar equations of the form
Constraints of the form () are called holonomic and scleronomic. In terms of the radius-vector
$$
\displaystyle\mathbf r
$$
of the Newtonian dynamical system () they are written as
Each such constraint reduces by one the number of degrees of freedom of the Newtonian dynamical system (). Therefore, the constrained system has
$$
\displaystyle n=3\,N-K
$$
degrees of freedom.
Definition. The constraint equations () define an
$$
\displaystyle n
$$
-dimensional manifold
$$
\displaystyle M
$$
within the configuration space of the Newtonian dynamical system (). This manifold
$$
\displaystyle M
$$
is called the configuration space of the constrained system. Its tangent bundle _ BLOCK6_ is called the phase space of the constrained system.
Let
$$
\displaystyle q^1,\,\ldots,\,q^n
$$
be the internal coordinates of a point of
$$
\displaystyle M
$$
. Their usage is typical for the Lagrangian mechanics.
|
https://en.wikipedia.org/wiki/Newtonian_dynamics
|
Let
$$
\displaystyle q^1,\,\ldots,\,q^n
$$
be the internal coordinates of a point of
$$
\displaystyle M
$$
. Their usage is typical for the Lagrangian mechanics. The radius-vector
$$
\displaystyle\mathbf r
$$
is expressed as some definite function of
$$
\displaystyle q^1,\,\ldots,\,q^n
$$
:
The vector-function () resolves the constraint equations () in the sense that upon substituting () into () the equations () are fulfilled identically in
$$
\displaystyle q^1,\,\ldots,\,q^n
$$
.
## Internal presentation of the velocity vector
The velocity vector of the constrained Newtonian dynamical system is expressed in terms of the partial derivatives of the vector-function
():
The quantities
$$
\displaystyle\dot q^1,\,\ldots,\,\dot q^n
$$
are called internal components of the velocity vector. Sometimes they are denoted with the use of a separate symbol
and then treated as independent variables. The quantities
are used as internal coordinates of a point of the phase space
$$
\displaystyle TM
$$
of the constrained Newtonian dynamical system.
|
https://en.wikipedia.org/wiki/Newtonian_dynamics
|
Sometimes they are denoted with the use of a separate symbol
and then treated as independent variables. The quantities
are used as internal coordinates of a point of the phase space
$$
\displaystyle TM
$$
of the constrained Newtonian dynamical system.
## Embedding and the induced Riemannian metric
Geometrically, the vector-function () implements an embedding of the configuration space
$$
\displaystyle M
$$
of the constrained Newtonian dynamical system into the
$$
\displaystyle 3\,N
$$
-dimensional flat configuration space of the unconstrained
Newtonian dynamical system (). Due to this embedding the Euclidean structure of the ambient space induces the Riemannian metric onto the manifold
$$
\displaystyle M
$$
. The components of the metric tensor of this induced metric are given by the formula
where
$$
\displaystyle(\ ,\ )
$$
is the scalar product associated with the Euclidean structure ().
|
https://en.wikipedia.org/wiki/Newtonian_dynamics
|
Due to this embedding the Euclidean structure of the ambient space induces the Riemannian metric onto the manifold
$$
\displaystyle M
$$
. The components of the metric tensor of this induced metric are given by the formula
where
$$
\displaystyle(\ ,\ )
$$
is the scalar product associated with the Euclidean structure ().
## Kinetic energy of a constrained Newtonian dynamical system
Since the Euclidean structure of an unconstrained system of
$$
\displaystyle N
$$
particles is introduced through their kinetic energy, the induced Riemannian structure on the configuration space
$$
\displaystyle N
$$
of a constrained system preserves this relation to the kinetic energy:
The formula () is derived by substituting () into () and taking into account ().
## Constraint forces
For a constrained Newtonian dynamical system the constraints described by the equations () are usually implemented by some mechanical framework. This framework produces some auxiliary forces including the force that maintains the system within its configuration manifold
$$
\displaystyle M
$$
. Such a maintaining force is perpendicular to
$$
\displaystyle M
$$
. It is called the normal force.
|
https://en.wikipedia.org/wiki/Newtonian_dynamics
|
Such a maintaining force is perpendicular to
$$
\displaystyle M
$$
. It is called the normal force. The force
$$
\displaystyle\mathbf F
$$
from () is subdivided into two components
The first component in () is tangent to the configuration manifold
$$
\displaystyle M
$$
. The second component is perpendicular to
$$
\displaystyle M
$$
. In coincides with the normal force
$$
\displaystyle\mathbf N
$$
.
Like the velocity vector (), the tangent force
$$
\displaystyle\mathbf F_\parallel
$$
has its internal presentation
The quantities
$$
F^1,\,\ldots,\,F^n
$$
in () are called the internal components of the force vector.
## Newton's second law in a curved space
The Newtonian dynamical system () constrained to the configuration manifold
$$
\displaystyle M
$$
by the constraint equations () is described by the differential equations
where _ BLOCK1_ are Christoffel symbols of the metric connection produced by the Riemannian metric ().
|
https://en.wikipedia.org/wiki/Newtonian_dynamics
|
## Newton's second law in a curved space
The Newtonian dynamical system () constrained to the configuration manifold
$$
\displaystyle M
$$
by the constraint equations () is described by the differential equations
where _ BLOCK1_ are Christoffel symbols of the metric connection produced by the Riemannian metric ().
## Relation to Lagrange equations
Mechanical systems with constraints are usually described by Lagrange equations:
where
$$
T=T(q^1,\ldots,q^n,w^1,\ldots,w^n)
$$
is the kinetic energy the constrained dynamical system given by the formula (). The quantities
$$
Q_1,\,\ldots,\,Q_n
$$
in
() are the inner covariant components of the tangent force vector
$$
\mathbf F_\parallel
$$
(see () and ()). They are produced from the inner contravariant components
$$
F^1,\,\ldots,\,F^n
$$
of the vector
$$
\mathbf F_\parallel
$$
by means of the standard index lowering procedure using the metric ():
The equations () are equivalent to the equations (). However, the metric () and
other geometric features of the configuration manifold
$$
\displaystyle M
$$
are not explicit in ().
|
https://en.wikipedia.org/wiki/Newtonian_dynamics
|
They are produced from the inner contravariant components
$$
F^1,\,\ldots,\,F^n
$$
of the vector
$$
\mathbf F_\parallel
$$
by means of the standard index lowering procedure using the metric ():
The equations () are equivalent to the equations (). However, the metric () and
other geometric features of the configuration manifold
$$
\displaystyle M
$$
are not explicit in (). The metric () can be recovered from the kinetic energy
$$
\displaystyle T
$$
by means of the formula
|
https://en.wikipedia.org/wiki/Newtonian_dynamics
|
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve.
## Examples
include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.
At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
Often the function is deterministic, that is, for a given time interval only one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables.
The study of dynamical systems is the focus of dynamical systems theory, which has applications to a wide variety of fields such as mathematics, physics, biology, chemistry, engineering, economics, history, and medicine. Dynamical systems are a fundamental part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly and self-organization processes, and the edge of chaos concept.
## Overview
The concept of a dynamical system has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is an implicit relation that gives the state of the system for only a short time into the future. (The relation is either a differential equation, difference equation or other time scale.) To determine the state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system. If the system can be solved, then, given an initial point, it is possible to determine all its future positions, a collection of points known as a trajectory or orbit.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
The iteration procedure is referred to as solving the system or integrating the system. If the system can be solved, then, given an initial point, it is possible to determine all its future positions, a collection of points known as a trajectory or orbit.
Before the advent of computers, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system.
For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because:
- The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions. To address these questions several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability. The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their equivalence changes with the different notions of stability.
- The type of trajectory may be more important than one particular trajectory.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their equivalence changes with the different notions of stability.
- The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class. Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes.
## Linear dynamical systems
and systems that have two numbers describing a state are examples of dynamical systems where the possible classes of orbits are understood.
- The behavior of trajectories as a function of a parameter may be what is needed for an application. As a parameter is varied, the dynamical systems may have bifurcation points where the qualitative behavior of the dynamical system changes. For example, it may go from having only periodic motions to apparently erratic behavior, as in the transition to turbulence of a fluid.
- The trajectories of the system may appear erratic, as if random. In these cases it may be necessary to compute averages using one very long trajectory or many different trajectories. The averages are well defined for ergodic systems and a more detailed understanding has been worked out for hyperbolic systems.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
In these cases it may be necessary to compute averages using one very long trajectory or many different trajectories. The averages are well defined for ergodic systems and a more detailed understanding has been worked out for hyperbolic systems. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of statistical mechanics and of chaos.
## History
Many people regard French mathematician Henri Poincaré as the founder of dynamical systems. Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included the Poincaré recurrence theorem, which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state.
Aleksandr Lyapunov developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamical system.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamical system.
In 1913, George David Birkhoff proved Poincaré's "Last Geometric Theorem", a special case of the three-body problem, a result that made him world-famous. In 1927, he published his Dynamical Systems. Birkhoff's most durable result has been his 1931 discovery of what is now called the ergodic theorem. Combining insights from physics on the ergodic hypothesis with measure theory, this theorem solved, at least in principle, a fundamental problem of statistical mechanics. The ergodic theorem has also had repercussions for dynamics.
Stephen Smale made significant advances as well. His first contribution was the Smale horseshoe that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others.
Oleksandr Mykolaiovych Sharkovsky developed Sharkovsky's theorem on the periods of discrete dynamical systems in 1964. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.
In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.
In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity. Palestinian mechanical engineer Ali H. Nayfeh applied nonlinear dynamics in mechanical and engineering systems. His pioneering work in applied nonlinear dynamics has been influential in the construction and maintenance of machines and structures that are common in daily life, such as ships, cranes, bridges, buildings, skyscrapers, jet engines, rocket engines, aircraft and spacecraft.
## Formal definition
In the most general sense,Mazzola C. and Giunti M. (2012), "Reversible dynamics and the directionality of time". In Minati G., Abram M., Pessa E. (eds.), Methods, models, simulations and approaches towards a general theory of change, pp. 161–171, Singapore: World Scientific. .
|
https://en.wikipedia.org/wiki/Dynamical_system
|
In Minati G., Abram M., Pessa E. (eds.), Methods, models, simulations and approaches towards a general theory of change, pp. 161–171, Singapore: World Scientific. .
a dynamical system is a tuple (T, X, Φ) where T is a monoid, written additively, X is a non-empty set and Φ is a function
$$
\Phi: U \subseteq (T \times X) \to X
$$
with
$$
\mathrm{proj}_{2}(U) = X
$$
(where
$$
\mathrm{proj}_{2}
$$
is the 2nd projection map)
and for any x in X:
$$
\Phi(0,x) = x
$$
$$
\Phi(t_2,\Phi(t_1,x)) = \Phi(t_2 + t_1, x),
$$
for
$$
\, t_1,\, t_2 + t_1 \in I(x)
$$
and
$$
\ t_2 \in I(\Phi(t_1, x))
$$
, where we have defined the set
$$
I(x) := \{ t \in T : (t,x) \in U \}
$$
for any x in X.
In particular, in the case that _ BLOCK8_
|
https://en.wikipedia.org/wiki/Dynamical_system
|
a dynamical system is a tuple (T, X, Φ) where T is a monoid, written additively, X is a non-empty set and Φ is a function
$$
\Phi: U \subseteq (T \times X) \to X
$$
with
$$
\mathrm{proj}_{2}(U) = X
$$
(where
$$
\mathrm{proj}_{2}
$$
is the 2nd projection map)
and for any x in X:
$$
\Phi(0,x) = x
$$
$$
\Phi(t_2,\Phi(t_1,x)) = \Phi(t_2 + t_1, x),
$$
for
$$
\, t_1,\, t_2 + t_1 \in I(x)
$$
and
$$
\ t_2 \in I(\Phi(t_1, x))
$$
, where we have defined the set
$$
I(x) := \{ t \in T : (t,x) \in U \}
$$
for any x in X.
In particular, in the case that _ BLOCK8_ we have for every x in X that _ BLOCK9_
|
https://en.wikipedia.org/wiki/Dynamical_system
|
we have for every x in X that _ BLOCK9_ and thus that Φ defines a monoid action of T on X.
The function Φ(t,x) is called the evolution function of the dynamical system: it associates to every point x in the set X a unique image, depending on the variable t, called the evolution parameter. X is called phase space or state space, while the variable x represents an initial state of the system.
We often write
$$
\Phi_x(t) \equiv \Phi(t,x)
$$
$$
\Phi^t(x) \equiv \Phi(t,x)
$$
if we take one of the variables as constant. The function
$$
\Phi_x:I(x) \to X
$$
is called the flow through x and its graph is called the trajectory through x. The set
$$
\gamma_x \equiv\{\Phi(t,x) : t \in I(x)\}
$$
is called the orbit through x.
The orbit through x is the image of the flow through x.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
The set
$$
\gamma_x \equiv\{\Phi(t,x) : t \in I(x)\}
$$
is called the orbit through x.
The orbit through x is the image of the flow through x.
A subset S of the state space X is called Φ-invariant if for all x in S and all t in T
$$
\Phi(t,x) \in S.
$$
Thus, in particular, if S is Φ-invariant,
$$
I(x) = T
$$
for all x in S. That is, the flow through x must be defined for all time for every element of S.
More commonly there are two classes of definitions for a dynamical system: one is motivated by ordinary differential equations and is geometrical in flavor; and the other is motivated by ergodic theory and is measure theoretical in flavor.
### Geometrical definition
In the geometrical definition , a dynamical system is the tuple
$$
\langle \mathcal{T}, \mathcal{M}, f\rangle
$$
. _ BLOCK1_ is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. _ BLOCK2_ is a manifold, i.e. locally a Banach space or Euclidean space, or in the discrete case a graph.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
_ BLOCK2_ is a manifold, i.e. locally a Banach space or Euclidean space, or in the discrete case a graph. f is an evolution rule t → f t (with
$$
t\in\mathcal{T}
$$
) such that f t is a diffeomorphism of the manifold to itself. So, f is a "smooth" mapping of the time-domain
$$
\mathcal{T}
$$
into the space of diffeomorphisms of the manifold to itself. In other terms, f(t) is a diffeomorphism, for every time t in the domain
$$
\mathcal{T}
$$
.
#### Real dynamical system
A real dynamical system, real-time dynamical system, continuous time dynamical system, or flow is a tuple (T, M, Φ) with T an open interval in the real numbers R, M a manifold locally diffeomorphic to a Banach space, and Φ a continuous function. If Φ is continuously differentiable we say the system is a differentiable dynamical system. If the manifold M is locally diffeomorphic to Rn, the dynamical system is finite-dimensional; if not, the dynamical system is infinite-dimensional. This does not assume a symplectic structure.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
If the manifold M is locally diffeomorphic to Rn, the dynamical system is finite-dimensional; if not, the dynamical system is infinite-dimensional. This does not assume a symplectic structure. When T is taken to be the reals, the dynamical system is called global or a flow; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow.
#### Discrete dynamical system
A discrete dynamical system, discrete-time dynamical system is a tuple (T, M, Φ), where M is a manifold locally diffeomorphic to a Banach space, and Φ is a function. When T is taken to be the integers, it is a cascade or a map. If T is restricted to the non-negative integers we call the system a semi-cascade.
#### Cellular automaton
A cellular automaton is a tuple (T, M, Φ), with T a lattice such as the integers or a higher-dimensional integer grid, M is a set of functions from an integer lattice (again, with one or more dimensions) to a finite set, and Φ a (locally defined) evolution function. As such cellular automata are dynamical systems. The lattice in M represents the "space" lattice, while the one in T represents the "time" lattice.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
As such cellular automata are dynamical systems. The lattice in M represents the "space" lattice, while the one in T represents the "time" lattice.
#### Multidimensional generalization
Dynamical systems are usually defined over a single independent variable, thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called multidimensional systems. Such systems are useful for modeling, for example, image processing.
#### Compactification of a dynamical system
Given a global dynamical system (R, X, Φ) on a locally compact and Hausdorff topological space X, it is often useful to study the continuous extension Φ* of Φ to the one-point compactification X* of X. Although we lose the differential structure of the original system we can now use compactness arguments to analyze the new system (R, X*, Φ*).
In compact dynamical systems the limit set of any orbit is non-empty, compact and simply connected.
### Measure theoretical definition
A dynamical system may be defined formally as a measure-preserving transformation of a measure space, the triplet (T, (X, Σ, μ), Φ).
|
https://en.wikipedia.org/wiki/Dynamical_system
|
In compact dynamical systems the limit set of any orbit is non-empty, compact and simply connected.
### Measure theoretical definition
A dynamical system may be defined formally as a measure-preserving transformation of a measure space, the triplet (T, (X, Σ, μ), Φ). Here, T is a monoid (usually the non-negative integers), X is a set, and (X, Σ, μ) is a probability space, meaning that Σ is a sigma-algebra on X and μ is a finite measure on (X, Σ). A map Φ: X → X is said to be Σ-measurable if and only if, for every σ in Σ, one has
$$
\Phi^{-1}\sigma \in \Sigma
$$
. A map Φ is said to preserve the measure if and only if, for every σ in Σ, one has
$$
\mu(\Phi^{-1}\sigma ) = \mu(\sigma)
$$
. Combining the above, a map Φ is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The triplet (T, (X, Σ, μ), Φ), for such a Φ, is then defined to be a dynamical system.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
Combining the above, a map Φ is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The triplet (T, (X, Σ, μ), Φ), for such a Φ, is then defined to be a dynamical system.
The map Φ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates
$$
\Phi^n = \Phi \circ \Phi \circ \dots \circ \Phi
$$
for every integer n are studied. For continuous dynamical systems, the map Φ is understood to be a finite time evolution map and the construction is more complicated.
#### Relation to geometric definition
The measure theoretical definition assumes the existence of a measure-preserving transformation. Many different invariant measures can be associated to any one evolution rule. If the dynamical system is given by a system of differential equations the appropriate measure must be determined. This makes it difficult to develop ergodic theory starting from differential equations, so it becomes convenient to have a dynamical systems-motivated definition within ergodic theory that side-steps the choice of measure and assumes the choice has been made.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
If the dynamical system is given by a system of differential equations the appropriate measure must be determined. This makes it difficult to develop ergodic theory starting from differential equations, so it becomes convenient to have a dynamical systems-motivated definition within ergodic theory that side-steps the choice of measure and assumes the choice has been made. A simple construction (sometimes called the Krylov–Bogolyubov theorem) shows that for a large class of systems it is always possible to construct a measure so as to make the evolution rule of the dynamical system a measure-preserving transformation. In the construction a given measure of the state space is summed for all future points of a trajectory, assuring the invariance.
Some systems have a natural measure, such as the Liouville measure in Hamiltonian systems, chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system. For chaotic dissipative systems the choice of invariant measure is technically more challenging. The measure needs to be supported on the attractor, but attractors have zero Lebesgue measure and the invariant measures must be singular with respect to the Lebesgue measure. A small region of phase space shrinks under time evolution.
For hyperbolic dynamical systems, the Sinai–Ruelle–Bowen measures appear to be the natural choice.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
A small region of phase space shrinks under time evolution.
For hyperbolic dynamical systems, the Sinai–Ruelle–Bowen measures appear to be the natural choice. They are constructed on the geometrical structure of stable and unstable manifolds of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems.
## Construction of dynamical systems
The concept of evolution in time is central to the theory of dynamical systems as seen in the previous sections: the basic reason for this fact is that the starting motivation of the theory was the study of time behavior of classical mechanical systems. But a system of ordinary differential equations must be solved before it becomes a dynamic system. For example, consider an initial value problem such as the following:
$$
\dot{\boldsymbol{x}}=\boldsymbol{v}(t,\boldsymbol{x})
$$
$$
\boldsymbol{x}|_=\boldsymbol{x}_0
$$
where
-
$$
\dot{\boldsymbol{x}}
$$
represents the velocity of the material point x
- M is a finite dimensional manifold
- v: T × M → TM is a vector field in Rn or Cn and represents the change of velocity induced by the known forces acting on the given material point in the phase space M.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
But a system of ordinary differential equations must be solved before it becomes a dynamic system. For example, consider an initial value problem such as the following:
$$
\dot{\boldsymbol{x}}=\boldsymbol{v}(t,\boldsymbol{x})
$$
$$
\boldsymbol{x}|_=\boldsymbol{x}_0
$$
where
-
$$
\dot{\boldsymbol{x}}
$$
represents the velocity of the material point x
- M is a finite dimensional manifold
- v: T × M → TM is a vector field in Rn or Cn and represents the change of velocity induced by the known forces acting on the given material point in the phase space M. The change is not a vector in the phase space M, but is instead in the tangent space TM.
There is no need for higher order derivatives in the equation, nor for the parameter t in v(t,x), because these can be eliminated by considering systems of higher dimensions.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
The change is not a vector in the phase space M, but is instead in the tangent space TM.
There is no need for higher order derivatives in the equation, nor for the parameter t in v(t,x), because these can be eliminated by considering systems of higher dimensions.
Depending on the properties of this vector field, the mechanical system is called
- autonomous, when v(t, x) = v(x)
- homogeneous when v(t, 0) = 0 for all t
The solution can be found using standard ODE techniques and is denoted as the evolution function already introduced above
$$
\boldsymbol(t)=\Phi(t,\boldsymbol_0)
$$
The dynamical system is then (T, M, Φ).
Some formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy
$$
\dot{\boldsymbol{x}}-\boldsymbol{v}(t,\boldsymbol{x})=0 \qquad\Leftrightarrow\qquad \mathfrak\left(t,\Phi(t,\boldsymbol_0)\right)=0
$$
where
$$
\mathfrak{G}:{{(T\times M)}^M}\to\mathbf{C}
$$
is a functional from the set of evolution functions to the field of the complex numbers.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
Depending on the properties of this vector field, the mechanical system is called
- autonomous, when v(t, x) = v(x)
- homogeneous when v(t, 0) = 0 for all t
The solution can be found using standard ODE techniques and is denoted as the evolution function already introduced above
$$
\boldsymbol(t)=\Phi(t,\boldsymbol_0)
$$
The dynamical system is then (T, M, Φ).
Some formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy
$$
\dot{\boldsymbol{x}}-\boldsymbol{v}(t,\boldsymbol{x})=0 \qquad\Leftrightarrow\qquad \mathfrak\left(t,\Phi(t,\boldsymbol_0)\right)=0
$$
where
$$
\mathfrak{G}:{{(T\times M)}^M}\to\mathbf{C}
$$
is a functional from the set of evolution functions to the field of the complex numbers.
This equation is useful when modeling mechanical systems with complicated constraints.
Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally Banach spaces—in which case the differential equations are partial differential equations.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
This equation is useful when modeling mechanical systems with complicated constraints.
Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally Banach spaces—in which case the differential equations are partial differential equations.
Examples
- Arnold's cat map
- Baker's map is an example of a chaotic piecewise linear map
- Billiards and outer billiards
- Bouncing ball dynamics
- Circle map
- Complex quadratic polynomial
- Double pendulum
- Dyadic transformation
- Dynamical system simulation
- Hénon map
- Irrational rotation
- Kaplan–Yorke map
- List of chaotic maps
- Lorenz system
- Quadratic map simulation system
- Rössler map
- Swinging Atwood's machine
- Tent map
Linear dynamical systems
Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the N-dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers. The analysis of linear systems is possible because they satisfy a superposition principle: if u(t) and w(t) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will u(t) + w(t).
|
https://en.wikipedia.org/wiki/Dynamical_system
|
In a linear system the phase space is the N-dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers. The analysis of linear systems is possible because they satisfy a superposition principle: if u(t) and w(t) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will u(t) + w(t).
### Flows
For a flow, the vector field v(x) is an affine function of the position in the phase space, that is,
$$
\dot{x} = v(x) = A x + b,
$$
with A a matrix, b a vector of numbers and x the position vector. The solution to this system can be found by using the superposition principle (linearity).
The case b ≠ 0 with A = 0 is just a straight line in the direction of b:
$$
\Phi^t(x_1) = x_1 + b t.
$$
When b is zero and A ≠ 0 the origin is an equilibrium (or singular) point of the flow, that is, if x0 = 0, then the orbit remains there.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
The case b ≠ 0 with A = 0 is just a straight line in the direction of b:
$$
\Phi^t(x_1) = x_1 + b t.
$$
When b is zero and A ≠ 0 the origin is an equilibrium (or singular) point of the flow, that is, if x0 = 0, then the orbit remains there.
For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point x0,
$$
\Phi^t(x_0) = e^{t A} x_0.
$$
When b = 0, the eigenvalues of A determine the structure of the phase space. From the eigenvalues and the eigenvectors of A it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin.
The distance between two different initial conditions in the case A ≠ 0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for chaotic behavior.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for chaotic behavior.
### Maps
A discrete-time, affine dynamical system has the form of a matrix difference equation:
$$
x_{n+1} = A x_n + b,
$$
with A a matrix and b a vector. As in the continuous case, the change of coordinates x → x + (1 − A) –1b removes the term b from the equation. In the new coordinate system, the origin is a fixed point of the map and the solutions are of the linear system A nx0.
The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map.
As in the continuous case, the eigenvalues and eigenvectors of A determine the structure of phase space. For example, if u1 is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u1, with α ∈ R, is an invariant curve of the map. Points in this straight line run into the fixed point.
There are also many other discrete dynamical systems.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
Points in this straight line run into the fixed point.
There are also many other discrete dynamical systems.
## Local dynamics
The qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative): a singular point of the vector field (a point where v(x) = 0) will remain a singular point under smooth transformations; a periodic orbit is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible.
### Rectification
A flow in most small patches of the phase space can be made very simple. If y is a point where the vector field v(y) ≠ 0, then there is a change of coordinates for a region around y where the vector field becomes a series of parallel vectors of the same magnitude. This is known as the rectification theorem.
The rectification theorem says that away from singular points the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space M the dynamical system is integrable.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
The rectification theorem says that away from singular points the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space M the dynamical system is integrable. In most cases the patch cannot be extended to the entire phase space. There may be singular points in the vector field (where v(x) = 0); or the patches may become smaller and smaller as some point is approached. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one. If the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches.
### Near periodic orbits
In general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. Poincaré developed an approach that transforms the analysis near a periodic orbit to the analysis of a map. Pick a point x0 in the orbit γ and consider the points in phase space in that neighborhood that are perpendicular to v(x0). These points are a Poincaré section S(γ, x0), of the orbit.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
Pick a point x0 in the orbit γ and consider the points in phase space in that neighborhood that are perpendicular to v(x0). These points are a Poincaré section S(γ, x0), of the orbit. The flow now defines a map, the Poincaré map F : S → S, for points starting in S and returning to S. Not all these points will take the same amount of time to come back, but the times will be close to the time it takes x0.
The intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map F. By a translation, the point can be assumed to be at x = 0. The Taylor series of the map is F(x) = J · x + O(x2), so a change of coordinates h can only be expected to simplify F to its linear part
$$
h^{-1} \circ F \circ h(x) = J \cdot x.
$$
This is known as the conjugation equation. Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. Poincaré first approached it assuming all functions to be analytic and in the process discovered the non-resonant condition. If λ1, ..., λν are the eigenvalues of J they will be resonant if one eigenvalue is an integer linear combination of two or more of the others.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
Poincaré first approached it assuming all functions to be analytic and in the process discovered the non-resonant condition. If λ1, ..., λν are the eigenvalues of J they will be resonant if one eigenvalue is an integer linear combination of two or more of the others. As terms of the form λi – Σ (multiples of other eigenvalues) occurs in the denominator of the terms for the function h, the non-resonant condition is also known as the small divisor problem.
### Conjugation results
The results on the existence of a solution to the conjugation equation depend on the eigenvalues of J and the degree of smoothness required from h. As J does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of J are not in the unit circle, the dynamics near the fixed point x0 of F is called hyperbolic and when the eigenvalues are on the unit circle and complex, the dynamics is called elliptic.
In the hyperbolic case, the Hartman–Grobman theorem gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map J · x. The hyperbolic case is also structurally stable.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
In the hyperbolic case, the Hartman–Grobman theorem gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map J · x. The hyperbolic case is also structurally stable. Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of J in the complex plane, implying that the map is still hyperbolic.
The Kolmogorov–Arnold–Moser (KAM) theorem gives the behavior near an elliptic point.
## Bifurcation theory
When the evolution map Φt (or the vector field it is derived from) depends on a parameter μ, the structure of the phase space will also depend on this parameter. Small changes may produce no qualitative changes in the phase space until a special value μ0 is reached. At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation.
Bifurcation theory considers a structure in phase space (typically a fixed point, a periodic orbit, or an invariant torus) and studies its behavior as a function of the parameter μ. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
Bifurcation theory considers a structure in phase space (typically a fixed point, a periodic orbit, or an invariant torus) and studies its behavior as a function of the parameter μ. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems.
The bifurcations of a hyperbolic fixed point x0 of a system family Fμ can be characterized by the eigenvalues of the first derivative of the system DFμ(x0) computed at the bifurcation point. For a map, the bifurcation will occur when there are eigenvalues of DFμ on the unit circle. For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on Bifurcation theory.
Some bifurcations can lead to very complicated structures in phase space. For example, the Ruelle–Takens scenario describes how a periodic orbit bifurcates into a torus and the torus into a strange attractor. In another example, Feigenbaum period-doubling describes how a stable periodic orbit goes through a series of period-doubling bifurcations.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
For example, the Ruelle–Takens scenario describes how a periodic orbit bifurcates into a torus and the torus into a strange attractor. In another example, Feigenbaum period-doubling describes how a stable periodic orbit goes through a series of period-doubling bifurcations.
## Ergodic systems
In many dynamical systems, it is possible to choose the coordinates of the system so that the volume (really a ν-dimensional volume) in phase space is invariant. This happens for mechanical systems derived from Newton's laws as long as the coordinates are the position and the momentum and the volume is measured in units of (position) × (momentum). The flow takes points of a subset A into the points Φ t(A) and invariance of the phase space means that
$$
\mathrm{vol} (A) = \mathrm{vol} ( \Phi^t(A) ).
$$
In the Hamiltonian formalism, given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the Liouville measure.
In a Hamiltonian system, not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
In a Hamiltonian system, not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible. The states with the same energy form an energy shell Ω, a sub-manifold of the phase space. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution.
For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space. Then almost every point of A returns to A infinitely often. The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann's derivation of the increase in entropy in a dynamical system of colliding atoms.
One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the ergodic hypothesis. The hypothesis states that the length of time a typical trajectory spends in a region A is vol(A)/vol(Ω).
The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
The hypothesis states that the length of time a typical trajectory spends in a region A is vol(A)/vol(Ω).
The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. Koopman approached the study of ergodic systems by the use of functional analysis. An observable a is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height). The value of an observable can be computed at another time by using the evolution function φ t. This introduces an operator U t, the transfer operator,
$$
(U^t a)(x) = a(\Phi^{-t}(x)).
$$
By studying the spectral properties of the linear operator U it becomes possible to classify the ergodic properties of Φ t. In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φ t gets mapped into an infinite-dimensional linear problem involving U.
The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in equilibrium statistical mechanics. An average in time along a trajectory is equivalent to an average in space computed with the Boltzmann factor exp(−βH).
|
https://en.wikipedia.org/wiki/Dynamical_system
|
In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φ t gets mapped into an infinite-dimensional linear problem involving U.
The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in equilibrium statistical mechanics. An average in time along a trajectory is equivalent to an average in space computed with the Boltzmann factor exp(−βH). This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems. SRB measures replace the Boltzmann factor and they are defined on attractors of chaotic systems.
## Nonlinear dynamical systems and chaos
Simple nonlinear dynamical systems, including piecewise linear systems, can exhibit strongly unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This unpredictable behavior has been called chaos. Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent spaces perpendicular to an orbit can be decomposed into a combination of two parts: one with the points that converge towards the orbit (the stable manifold) and another of the points that diverge from the orbit (the unstable manifold).
|
https://en.wikipedia.org/wiki/Dynamical_system
|
Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent spaces perpendicular to an orbit can be decomposed into a combination of two parts: one with the points that converge towards the orbit (the stable manifold) and another of the points that diverge from the orbit (the unstable manifold).
This branch of mathematics deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible attractors?" or "Does the long-term behavior of the system depend on its initial condition?"
The chaotic behavior of complex systems is not the issue. Meteorology has been known for years to involve complex—even chaotic—behavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The Pomeau–Manneville scenario of the logistic map and the Fermi–Pasta–Ulam–Tsingou problem arose with just second-degree polynomials; the horseshoe map is piecewise linear.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
Chaos theory has been so surprising because chaos can be found within almost trivial systems. The Pomeau–Manneville scenario of the logistic map and the Fermi–Pasta–Ulam–Tsingou problem arose with just second-degree polynomials; the horseshoe map is piecewise linear.
### Solutions of finite duration
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, meaning here that in these solutions the system will reach the value zero at some time, called an ending time, and then stay there forever after. This can occur only when system trajectories are not uniquely determined forwards and backwards in time by the dynamics, thus solutions of finite duration imply a form of "backwards-in-time unpredictability" closely related to the forwards-in-time unpredictability of chaos. This behavior cannot happen for Lipschitz continuous differential equations according to the proof of the Picard-Lindelof theorem. These solutions are non-Lipschitz functions at their ending times and cannot be analytical functions on the whole real line.
|
https://en.wikipedia.org/wiki/Dynamical_system
|
This behavior cannot happen for Lipschitz continuous differential equations according to the proof of the Picard-Lindelof theorem. These solutions are non-Lipschitz functions at their ending times and cannot be analytical functions on the whole real line.
As example, the equation:
$$
y'= -\text{sgn}(y)\sqrt{|y|},\,\,y(0)=1
$$
Admits the finite duration solution:
$$
y(t)=\frac{1}{4}\left(1-\frac{t}{2}+\left|1-\frac{t}{2}\right|\right)^2
$$
that is zero for
$$
t \geq 2
$$
and is not Lipschitz continuous at its ending time
$$
t = 2.
$$
|
https://en.wikipedia.org/wiki/Dynamical_system
|
In mathematics, a basic algebraic operation is any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). These operations may be performed on numbers, in which case they are often called arithmetic operations. They may also be performed, in a similar way, on variables, algebraic expressions, and more generally, on elements of algebraic structures, such as groups and fields. An algebraic operation may also be defined more generally as a function from a Cartesian power of a given set to the same set.
The term algebraic operation may also be used for operations that may be defined by compounding basic algebraic operations, such as the dot product. In calculus and mathematical analysis, algebraic operation is also used for the operations that may be defined by purely algebraic methods. For example, exponentiation with an integer or rational exponent is an algebraic operation, but not the general exponentiation with a real or complex exponent. Also, the derivative is an operation on numerical functions and algebraic expressions that is not algebraic.
## Notation
Multiplication symbols are usually omitted, and implied, when there is no operator between two variables or terms, or when a coefficient is used.
|
https://en.wikipedia.org/wiki/Algebraic_operation
|
Also, the derivative is an operation on numerical functions and algebraic expressions that is not algebraic.
## Notation
Multiplication symbols are usually omitted, and implied, when there is no operator between two variables or terms, or when a coefficient is used. For example, 3 × x2 is written as 3x2, and 2 × x × y is written as 2xy. Sometimes, multiplication symbols are replaced with either a dot or center-dot, so that x × y is written as either x . y or x · y. Plain text, programming languages, and calculators also use a single asterisk to represent the multiplication symbol, and it must be explicitly used; for example, 3x is written as 3 * x.
Rather than using the ambiguous division sign (÷), division is usually represented with a vinculum, a horizontal line, as in . In plain text and programming languages, a slash (also called a solidus) is used, e.g. 3 / (x + 1).
Exponents are usually formatted using superscripts, as in x2.
|
https://en.wikipedia.org/wiki/Algebraic_operation
|
In plain text and programming languages, a slash (also called a solidus) is used, e.g. 3 / (x + 1).
Exponents are usually formatted using superscripts, as in x2. In plain text, the TeX mark-up language, and some programming languages such as MATLAB and Julia, the caret symbol, ^, represents exponents, so x2 is written as x ^ 2.George Grätzer, First Steps in LaTeX, Publisher Springer, 1999, , 9780817641320, page 17 In programming languages such as Ada, Fortran, Perl, Python and Ruby, a double asterisk is used, so x2 is written as x ** 2.
The plus–minus sign, ±, is used as a shorthand notation for two expressions written as one, representing one expression with a plus sign, the other with a minus sign. For example, y = x ± 1 represents the two equations y = x + 1 and y = x − 1. Sometimes, it is used for denoting a positive-or-negative term such as ±x.
## Arithmetic vs algebraic operations
Algebraic operations work in the same way as arithmetic operations, as can be seen in the table below.
|
https://en.wikipedia.org/wiki/Algebraic_operation
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.