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For a (classically) integrable quantum mechanical system with two degrees of freedom, the functional dependence $\hat{H}=H_Q(\hat{J}_1,\hat{J}_2)$ of the Hamiltonian operator on the action operators is analyzed and compared with the corresponding functional relationship $H(p_1,q_1;p_2,q_2) = H_C(J_1,J_2)$ in the classical limit of that system. The former is shown to converge toward the latter in some asymptotic regime associated with the classical limit, but the convergence is, in general, non-uniform. The existence of the function $\hat{H}=H_Q(\hat{J}_1,\hat{J}_2)$ in the integrable regime of a parametric quantum system explains empirical results for the dimensionality of manifolds in parameter space on which at least two levels are degenerate. The comparative analysis is carried out for an integrable one-parameter two-spin model. Additional results presented for the (integrable) circular billiard model illuminate the same conclusions from a different angle.
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arxiv:nlin/0004039
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Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the periodic orbit sum and the uncertainty principle of the usual Fourier analysis, thus yielding results of high resolution and high precision. Based on the close analogy between periodic orbit trace formulae for regular and chaotic systems the technique is generalized in this paper for the semiclassical quantization of integrable systems. Thus, harmonic inversion is shown to be a universal tool which can be applied to a wide range of physical systems. The method is further generalized in two directions: Firstly, the periodic orbit quantization will be extended to include higher order hbar corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbits for semiclassical quantization, i.e., to improve the efficiency of the semiclassical method. As a representative of regular systems, we choose the circle billiard, whose periodic orbits and quantum eigenvalues can easily be obtained.
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arxiv:nlin/0005045
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We consider a one-dimensional persisent random walk viewed as a deterministic process with a form of time reversal symmetry. Particle reservoirs placed at both ends of the system induce a density current which drives the system out of equilibrium. The phase space distribution is singular in the stationary state and has a cumulative form expressed in terms of generalized Takagi functions. The entropy production rate is computed using the coarse-graining formalism of Gaspard, Gilbert and Dorfman. In the continuum limit, we show that the value of the entropy production rate is independent of the coarse-graining and agrees with the phenomenological entropy production rate of irreversible thermodynamics.
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arxiv:nlin/0005063
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In one-dimensional anharmonic lattices, we construct nonlinear standing waves (SWs) reducing to harmonic SWs at small amplitude. For SWs with spatial periodicity incommensurate with the lattice period, a transition by breaking of analyticity versus wave amplitude is observed. As a consequence of the discreteness, oscillatory linear instabilities, persisting for arbitrarily small amplitude in infinite lattices, appear for all wave numbers Q not equal to zero or \pi. Incommensurate analytic SWs with |Q|>\pi/2 may however appear as 'quasi-stable', as their instability growth rate is of higher order.
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arxiv:nlin/0005068
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A brief review is given of recent results devoted to the effects of large-scale anisotropy on the inertial-range statistics of the passive scalar quantity $\theta(t,{\bf x})$, advected by the synthetic turbulent velocity field with the covariance $\propto\delta(t-t')|{\bf x}-{\bf x'}|^{\eps}$. Inertial-range anomalous scaling behavior is established, and explicit asymptotic expressions for the structure functions $ S_n (\r) \equiv < [\theta(t,{\bf x}+\r)-\theta(t,{\bf x})]^{n}>$ are obtained; they are represented by superpositions of power laws with universal (independent of the anisotropy parameters) anomalous exponents, calculated to the first order in $\eps$ in any space dimension. The exponents are associated with tensor composite operators built of the scalar gradients, and exhibit a kind of hierarchy related to the degree of anisotropy: the less is the rank, the less is the dimension and, consequently, the more important is the contribution to the inertial-range behavior. The leading terms of the even (odd) structure functions are given by the scalar (vector) operators. The small-scale anisotropy reveals itself in odd correlation functions: for the incompressible velocity field, $S_{3}/S_{2}^{3/2}$ decreases going down towards to the depth of the inertial range, while the higher-order odd ratios increase; if the compressibility is strong enough, the skewness factor also becomes increasing.
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arxiv:nlin/0007015
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Hitchin's twistor treatment of Schlesinger's equations is extended to the general isomonodromic deformation problem. It is shown that a generic linear system of ordinary differential equations with gauge group SL(n,C) on a Riemann surface X can be obtained by embedding X in a twistor space Z on which sl(n,C) acts. When a certain obstruction vanishes, the isomonodromic deformations are given by deforming X in Z. This is related to a description of the deformations in terms of Hamiltonian flows on a symplectic manifold constructed from affine orbits in the dual Lie algebra of a loop group.
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arxiv:nlin/0007024
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I propose that stiffness may be defined and quantified for nonlinear systems using Lyapunov exponents, and demonstrate the relationship that exists between stiffness and the fractal dimension of a strange attractor: that stiff chaos is thin chaos.
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arxiv:nlin/0007031
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We study the effects of crosscorrelations of noises on the scaling properties of the correlation functions in a reduced model for Magnetohydrodynamic (MHD) turbulence [A. Basu, J.K Bhattacharjee and S. Ramaswamy [{\em Eur. Phys. J B} {\bf 9}, 725 (1999)]. We show that in {\em dimension d} crosscorrelations with sufficient long wavelength singularity become relevant and take the system to the long range noise fixed point. The crosscorrelations also affect the ratio of energies of the magnetic and velocity fields ($E_b/E_v$) in the strong coupling phase. In dimension $d=1$ the fluctuation-dissipation theorem (FDT) does not hold in presence of short range crosscorrelations. We discuss the possible effects of crosscorrelations on the scaling properties of fully developed MHD turbulence.
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arxiv:nlin/0009016
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We analyze the synchronization transition for a pair of coupled identical Kauffman networks in the chaotic phase. The annealed model for Kauffman networks shows that synchronization appears through a transcritical bifurcation, and provides an approximate description for the whole dynamics of the coupled networks. We show that these analytical predictions are in good agreement with numerical results for sufficiently large networks, and study finite-size effects in detail. Preliminary analytical and numerical results for partially disordered networks are also presented.
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arxiv:nlin/0010013
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Consider a linear autonomous Hamiltonian system with a time periodic bound state solution. In this paper we study the structural instability of this bound state ^M relative to time almost periodic perturbations which are small, localized and Hamiltonian. This class of perturbations includes those whose time dependence is periodic, but encompasses a large class of those with finite (quasiperiodic) or infinitely many non-commensurate frequencies. Problems of the type considered arise in many areas of application including ionization physics and the propagation of light in optical fibers in the presence of defects. The mechanism of instability is radiation damping due to resonant coupling of the bound state to the continuum modes by the time-dependent perturbation. This results in a transfer of energy from the discrete modes to the continuum. The rate of decay of solutions is slow and hence the decaying bound states can be viewed as metastable. These results generalize those of A. Soffer and M.I. Weinstein, who treated localized time-periodic perturbations of a particular form. In the present work, new analytical issues need to be addressed in view of (i) the presence of infinitely many frequencies which may resonate with the continuum as well as (ii) the possible accumulation of such resonances in the continuous spectrum. The theory is applied to a general class of Schr\"odinger operators.
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arxiv:nlin/0012021
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It is shown using three series of Rayleigh number simulations of varying aspect ratio AR and Prandtl number Pr that the normalized dissipation at the wall, while significantly greater than 1, approaches a constant dependent upon AR and Pr. It is also found that the peak velocity, not the mean square velocity, obeys the experimental scaling of Ra^{0.5}. The scaling of the mean square velocity is closer to Ra^{0.46}, which is shown to be consistent with experimental measurements and the numerical results for the scaling of Nu and the temperature if there are strong correlations between the velocity and temperature.
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arxiv:nlin/0012061
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A discussion about dependences of the (fractal) basin boundary dimension with the definition of the basins and the size of the exits is presented for systems with one or more exits. In particular, it is shown that the dimension is largely independent of the choice of the basins, and decreases with the size of the exits. Considering the limit of small exits, a strong relation between fractals in exit systems and chaos in closed systems is found. The discussion is illustrated by simple examples of one-dimensional maps.
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arxiv:nlin/0101020
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Intermittency in fluid turbulence can be evidentiated through the analysis of Probability Distribution Functions (PDF) of velocity fluctuations, which display a strong non-gaussian behavior at small scales. In this paper we investigate the occurrence of intermittency in plasma turbulence by studying the departure from the gaussian distribution of PDF for both velocity and magnetic fluctuations. We use data coming from two different experiments, namely in situ satellite observations of the inner solar wind and turbulent fluctuations in a magnetically confined fusion plasma. Moreover we investigate also time intermittency observed in a simplified shell model which mimics 3D MHD equations. We found that the departure from a gaussian distribution is the main characteristic of all cases. The scaling behaviour of PDFs are then investigated by using two different models built up in the past years, in order to capture the essence of intermittency in turbulence.
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arxiv:nlin/0101053
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The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general solution of them is presented in explicit form.
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arxiv:nlin/0101059
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In assemblies of globally coupled dynamical units, weak noise perturbing independently the individual units can cause anomalous dispersion in the synchronized cloud of the units in the phase space. When the noise-free dynamics of the synchronized assembly is nonperiodic, various moments of the linear dimension of the cloud as a function of the noise strength exhibit multiscaling properties with parameter-dependent scaling exponents. Some numerical evidence of this peculiar behavior as well as its interpretation in terms of a multiplicative stochastic process with small additive noise is provided. Universality of the phenomenon is also discussed.
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arxiv:nlin/0102006
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The parity of a bit string of length $N$ is a global quantity that can be efficiently compute using a global counter in ${O} (N)$ time. But is it possible to find the parity using cellular automata with a set of local rule tables without using any global counter? Here, we report a way to solve this problem using a number of $r=1$ binary, uniform, parallel and deterministic cellular automata applied in succession for a total of ${O} (N^2)$ time.
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arxiv:nlin/0102026
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Given a continuous function $f(x)$, suppose that the sign of $f$ only has finitely many discontinuous points in the interval $[0,1]$. We show how to use a sequence of one dimensional deterministic binary cellular automata to determine the sign of $f(\rho)$ where $\rho$ is the (number) density of 1s in an arbitrarily given bit string of finite length provided that $f$ satisfies certain technical conditions.
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arxiv:nlin/0103057
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We study a class of 1+1 quadratically nonlinear water wave equations that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still preserves integrability via the inverse scattering transform (IST) method. This IST-integrable class of equations contains both the KdV equation and the CH equation as limiting cases. It arises as the compatibility condition for a second order isospectral eigenvalue problem and a first order equation for the evolution of its eigenfunctions. This integrable equation is shown to be a shallow water wave equation derived by asymptotic expansion at one order higher approximation than KdV. We compare its traveling wave solutions to KdV solitons.
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arxiv:nlin/0104004
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In helical turbulence a linear cascade of helicity accompanying the energy cascade has been suggested. Since energy and helicity have different dimensionality we suggest the existence of a characteristic inner scale, $\xi=k_H^{-1}$, for helicity dissipation in a regime of hydrodynamic fully developed turbulence and estimate it on dimensional grounds. This scale is always larger than the Kolmogorov scale, $\eta=k_E^{-1}$, and their ratio $\eta / \xi $ vanishes in the high Reynolds number limit, so the flow will always be helicity free in the small scales.
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arxiv:nlin/0104008
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It is shown experimentally that vertical pairing of two identical microspheres suspended in the sheath of a radio-frequency (rf) discharge at low gas pressures (a few Pa), appears at a well defined instability threshold of the rf power. The transition is reversible, but with significant hysteresis on the second stage. A simple model, which uses measured microsphere resonance frequencies and takes into account besides Coulomb interaction between negatively charged microspheres also their interaction with positive ion wake charges, seems to explain the instability threshold quite well.
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arxiv:nlin/0104048
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We study the two time correlation for the noise driven dynamics of the double-well oscillator in the Suzuki regime. It is seen that for very small noise strength the correaltion function shows a lack of translational invariance for very long times characterised by the Suzuki scaling variable. We see that in this strongly out-of-equilibrium situation, the conventional mode-coupling approximation is not a convenient tool.
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arxiv:nlin/0105059
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The quasi-classical limit of the scalar nonlocal dbar-problem is derived and a quasi-classical version of the dbar-dressing method is presented. Dispersionless KP, mKP and 2DTL hierarchies are discussed as illustrative examples. It is shown that the universal Whitham hierarchy it is nothing but the ring of symmetries for the quasi-classical dbar-problem. The reduction problem is discussed and, in particular, the d2DTL equation of B type is derived.
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arxiv:nlin/0105071
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When a medium composed of microscopic elements is subjected to a high intensity field, the individual behaviors of microscopic elements can become chaotic. In such cases it is important to consider the effects of this irregularity at microscopical level onto the macroscopic behavior of the medium. We show that the macroscopic field produced by a large group of chaotic scatterers can remain regular, due to the partial or complete phase coherence of the scattering elements and the incoherence of the chaotic components of their responses. Thus when only macroscopic fields are observed, one may be unaware of chaotic microscopical motion, as it appears to be hidden from the observer. The coupling among the elements may lead to partial chaos synchronization, which exposes the chaotic nature of the system making the oscillations of macroscopic fields more irregular.
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arxiv:nlin/0106008
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We investigate quantum integrals of motion in the sine-Gordon, Zhiber-Shabat and similar systems. When the coupling constants in these models take special values a new quantum symmetry appears. In those cases, correlation functions can be obtained, and they have a power law behavior.
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arxiv:nlin/0106020
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To study the convergence to equilibrium in random maps we developed the spectral theory of the corresponding transfer (Perron-Frobenius) operators acting in a certain Banach space of generalized functions. The random maps under study in a sense fill the gap between expanding and hyperbolic systems since among their (deterministic) components there are both expanding and contracting ones. We prove stochastic stability of the Perron-Frobenius spectrum and developed its finite rank operator approximations by means of a ``stochastically smoothed'' Ulam approximation scheme. A counterexample to the original Ulam conjecture about the approximation of the SBR measure and the discussion of the instability of spectral approximations by means of the original Ulam scheme are presented as well.
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arxiv:nlin/0106030
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We have found an oscillating instability of fast-running cracks in thin rubber sheets. A well-defined transition from straight to oscillating cracks occurs as the amount of biaxial strain increases. Measurements of the amplitude and wavelength of the oscillation near the onset of this instability indicate that the instability is a Hopf bifurcation.
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arxiv:nlin/0107030
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The validity of semiclassical expansions in the power of $\hbar$ for the quantum Green's function have been extensively tested for billiards systems, but in the case of chaotic dynamics with smooth potential, even if formula are existing, a quantitative comparison is still missing. In this paper, extending the theory developed by Gaspard et al., Adv. Chem. Phys. XC 105 (1995), based on the classical Green's functions, we present an efficient method allowing the calculation of $\hbar$ corrections for the propagator, the quantum Green's function, and their traces. Especially, we show that the previously published expressions for $\hbar$ corrections to the traces are incomplete.
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arxiv:nlin/0107031
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We report an infinite class of discrete hierarchies which naturally generalize familiar discrete KP one.
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arxiv:nlin/0107049
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A cycle expansion technique for discrete sums of several PF operators, similar to the one used in standard classical dynamical zeta-function formalism is constructed. It is shown that the corresponding expansion coefficients show an interesting universal behavior, which illustrates the details of the interference between the particlar mappings entering the sum.
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arxiv:nlin/0107060
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Anomalous correlation functions of the temperature field in two-dimensional turbulent convection are shown to be universal with respect to the choice of external sources. Moreover, they are equal to the anomalous correlations of the concentration field of a passive tracer advected by the convective flow itself. The statistics of velocity differences is found to be universal, self-similar and close to Gaussian. These results point to the conclusion that temperature intermittency in two-dimensional turbulent convection may be traced back to the existence of statistically preserved structures, as it is in passive scalar turbulence.
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arxiv:nlin/0110016
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The methodology of the Riemann-Hilbert (RH) factorisation approach for Lax-pair isospectral deformations is used to derive, in the solitonless sector, the leading-order asymptotics as $t \to \pm \infty$ $(x/t \sim \mathcal{O}(1))$ of solutions to the Cauchy problem for the defocusing non-linear Schr\"{o}dinger equation (D${}_{f}$NLSE), $\mi \partial_{t}u +\partial_{x}^{2}u-2(| u |^{2}-1)u=0$, with (finite-density) initial data $u(x,0)=_{x \to \pm \infty} \exp (\tfrac{\mi (1 \mp 1) \theta}{2})(1+ o(1))$, $\theta \in [0,2 \pi)$. A limiting case of these asymptotics related to the RH problem for the Painlev\'{e} II equation, or one of its special reductions, is also identified.
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arxiv:nlin/0110024
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A sequence of bifurcations is studied in a one-dimensional pattern forming system subject to the variation of two experimental control parameters: a dimensionless electrical forcing number ${\cal R}$ and a shear Reynolds number ${\rm Re}$. The pattern is an azimuthally periodic array of traveling vortices with integer mode number $m$. Varying ${\cal R}$ and ${\rm Re}$ permits the passage through several codimension-two points. We find that the coefficients of the nonlinear terms in a generic Landau equation for the primary bifurcation are discontinuous at the codimension-two points. Further, we map the stability boundaries in the space of the two parameters by studying the subcritical secondary bifurcations in which $m \to m+1$ when ${\cal R}$ is increased at constant ${\rm Re}$.
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arxiv:nlin/0111007
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We derive a simple and general relation between the fidelity of quantum motion, characterizing the stability of quantum dynamics with respect to arbitrary static perturbation of the unitary evolution propagator, and the integrated time auto-correlation function of the generator of perturbation. Surprisingly, this relation predicts the slower decay of fidelity the faster decay of correlations is. In particular, for non-ergodic and non-mixing dynamics, where asymptotic decay of correlations is absent, a qualitatively different and faster decay of fidelity is predicted on a time scale 1/delta as opposed to mixing dynamics where the fidelity is found to decay exponentially on a time-scale 1/delta^2, where delta is a strength of perturbation. A detailed discussion of a semi-classical regime of small effective values of Planck constant is given where classical correlation functions can be used to predict quantum fidelity decay. Note that the correct and intuitively expected classical stability behavior is recovered in the classical limit hbar->0, as the two limits delta->0 and hbar->0 do not commute. In addition we also discuss non-trivial dependence on the number of degrees of freedom. All the theoretical results are clearly demonstrated numerically on a celebrated example of a quantized kicked top.
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arxiv:nlin/0111014
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A polynomial deformation of the Kowalewski top is considered. This deformation includes as a degeneration a new integrable case for the Kirchhoff equations found recently by one of the authors. A $5\times 5$ matrix Lax pair for the deformed Kowalewski top is proposed. Also deformations of the two-field Kowalewski gyrostat and the $so(p,q)$ Kowalewski top are found. All our Lax pairs are deformations of the corresponding Lax representations found by Reyman and Semenov-Tian {S}hansky. In addition, a similar deformation of the Goryachev-Chaplygin top and its $3\times 3$ matrix Lax representation is constructed.
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arxiv:nlin/0111035
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New infinite number of one- and two-point B\"{a}cklund transformations (BTs) with explicit expressions are constructed for the high-order constrained flows of the AKNS hierarchy. It is shown that these BTs are canonical transformations including B\"{a}cklund parameter $\eta$ and a spectrality property holds with respect to $\eta$ and the 'conjugated' variable $\mu$ for which the point $(\eta, \mu)$ belongs to the spectral curve. Also the formulas of m-times repeated Darboux transformations for the high-order constrained flows of the AKNS hierarchy are presented.
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arxiv:nlin/0112017
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Asymptotic properties of the solution of two-dimensional randomly forced Navier-Stokes equation with long-range correlations of the driving force are analyzed in the two-loop order of perturbation theory with the use of renormalization group. Kolmogorov constant of the energy spectrum is calculated for both the inverse energy cascade and the direct enstrophy cascade in the second order of the $\epsilon$ expansion.
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arxiv:nlin/0201025
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We present a systematic way to analyze and model systems having many characteristic time-scales. The method we propose is employed for a test-case of a meandering jet model manifesting chaotic tracer dispersion with long time-correlations. We first choose a suitable state space partition and analyze the symbolic dynamics associated to the fluid particle position. In a second step we construct a stochastic process in terms of a multi-time Markovian model. This corresponds to a hierarchy of random travelers on a graph where each traveler moves at his own time scale. The results are compared on the basis of statistical measures such as entropies and correlation functions.
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arxiv:nlin/0201027
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The N=2 supersymmetric {\alpha}=1 KdV hierarchy in N=2 superspace is considered and its rich symmetry structure is uncovered. New nonpolynomial and nonlocal, bosonic and fermionic symmetries and Hamiltonians, bi-Hamiltonian structure as well as a recursion operator connecting all symmetries and Hamiltonian structures of the N=2 {\alpha}=1 KdV hierarchy are constructed in explicit form. It is observed that the algebra of symmetries of the N=2 supersymmetric {\alpha}=1 KdV hierarchy possesses two different subalgebras of N=2 supersymmetry.
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arxiv:nlin/0201061
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Numerous experimental and theoretical studies have established that intramolecular vibrational energy redistribution (IVR) in isolated molecules has a heirarchical tier structure. The tier structure implies strong correlations between the energy level motions of a quantum system and its intensity-weighted spectrum. A measure, which explicitly accounts for this correaltion, was first introduced by one of us as a sensitive probe of phase space localization. It correlates eigenlevel velocities with the overlap intensities between the eigenstates and some localized state of interest. A semiclassical theory for the correlation is developed for systems that are classically integrable and complements earlier work focusing exclusively on the chaotic case. Application to a model two dimensional effective spectroscopic Hamiltonian shows that the correlation measure can provide information about the terms in the molecular Hamiltonian which play an important role in an energy range of interest and the character of the dynamics. Moreover, the correlation function is capable of highlighting relevant phase space structures including the local resonance features associated with a specific bright state. In addition to being ideally suited for multidimensional systems with a large density of states, the measure can also be used to gain insights into the phase space transport and localization. It is argued that the overlap intensity-level velocity correlation function provides a novel way of studying vibrational energy redistribution in isolated molecules. The correlation function is ideally suited to analyzing the parametric spectra of molecules in external fields.
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arxiv:nlin/0202005
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Preliminary results of extensive numerical experiments on a simple model specified by the smooth canonical strongly chaotic 2D-map with global virtual invariant curves (VICs) are presented and discussed. We focus on the statistics of the diffusion rate of individual trajectories in dependence on the model parameters. Our previous conjecture about the fractal statistics determined by the critical structure of both the phase space and the motion is confirmed and studied in some detail. Particularly, we have found specific characteristics of what we termed the VIC diffusion suppression which is related to a new type of the critical structure. An example of ergodic motion with a surprising "hidden" critical structure strongly affecting the diffusion rate was also encountered and discussed.
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arxiv:nlin/0202017
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As a first step toward realizing a dynamical system that evolves while spontaneously determining its own rule for time evolution, function dynamics (FD) is analyzed. FD consists of a functional equation with a self-referential term, given as a dynamical system of a 1-dimensional map. Through the time evolution of this system, a dynamical graph (a network) emerges. This graph has three interesting properties: i) vertices appear as stable elements, ii) the terminals of directed edges change in time, and iii) some vertices determine the dynamics of edges, and edges determine the stability of the vertices, complementarily. Two aspects of FD are studied, the generation of a graph (network) structure and the dynamics of this graph (network) in the system.
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arxiv:nlin/0202033
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A neural network with fixed topology can be regarded as a parametrization of functions, which decides on the correlations between functional variations when parameters are adapted. We propose an analysis, based on a differential geometry point of view, that allows to calculate these correlations. In practise, this describes how one response is unlearned while another is trained. Concerning conventional feed-forward neural networks we find that they generically introduce strong correlations, are predisposed to forgetting, and inappropriate for task decomposition. Perspectives to solve these problems are discussed.
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arxiv:nlin/0202038
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Low-dimensional periodic arrays of scatterers with a moving point particle are ideal models for studying deterministic diffusion. For such systems the diffusion coefficient is typically an irregular function under variation of a control parameter. Here we propose a systematic scheme of how to approximate deterministic diffusion coefficients of this kind in terms of correlated random walks. We apply this approach to two simple examples which are a one-dimensional map on the line and the periodic Lorentz gas. Starting from suitable Green-Kubo formulas we evaluate hierarchies of approximations for their parameter-dependent diffusion coefficients. These approximations converge exactly yielding a straightforward interpretation of the structure of these irregular diffusion coeficients in terms of dynamical correlations.
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arxiv:nlin/0202040
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In the first half of the paper, some recent advances in coupled dynamical systems, in particular, a globally coupled map are surveyed. First, dominance of Milnor attractors in partially ordered phase is demonstrated. Second, chaotic itinerancy in high-dimensional dynamical systems is briefly reviewed, with discussion on a possible connection with a Milnor attractor network. Third, infinite-dimensional collective dynamics is studied, in the thermodynamic limit of the globally coupled map, where bifurcation to lower-dimensional attractors by the addition of noise is briefly reviewed. Following the study of coupled dynamical systems, a scenario for developmental process of cell society is proposed, based on numerical studies of a system with interacting units with internal dynamics and reproduction. Differentiation of cell types is found as a natural consequence of such a system. "Stem cells" that either proliferate or differentiate to different types generally appear in the system, where irreversible loss of multipotency is demonstrated. Robustness of the developmental process against microscopic and macroscopic perturbations is found and explained, while irreversibility in developmental process is analyzed in terms of the gain of stability, loss of diversity and chaotic instability. Construction of a phenomenology theory for development is discussed in comparison with the thermodynamics.
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arxiv:nlin/0203040
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We study the evolution of the energy distribution for a stadium with moving walls. We consider one period driving cycle, which is characterized by an amplitude $A$ and wall velocity $V$. This evolving energy distribution has both "parametric" and "stochastic" components. The latter are important for the theory of quantum irreversibility and dissipation in driven mesoscopic devices (eg in the context of quantum computation). For extremely slow wall velocity $V$ the spreading mechanism is dominated by transitions between neighboring levels, while for larger (non-adiabatic) velocities the spreading mechanism has both perturbative and non-perturbative features. We present, for the first time, a numerical study which is aimed in identifying the latter features. A procedure is developed for the determination of the various $V$ regimes. The possible implications on linear response theory are discussed.
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arxiv:nlin/0204025
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Biskamp and Schwarz [Phys. Plasmas, 8, 3282 (2001)] have reported that the energy spectrum of two-dimensional magnetohydrodynamic turbulence is proportional to $k^{-3/2}$, which is a prediction of Iroshnikov-Kraichnan phenomenology. In this comment we report some earlier results which conclusively show that for two-dimensional magnetohydrodynamic turbulence, Kolmogorov-like phenomenology (spectral index 5/3) is better model than Iroshnikov-Kraichnan phenomenology; these results are based on energy flux analysis.
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arxiv:nlin/0204026
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The renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by the Gaussian self-similar velocity field with finite, and not small, correlation time. The inertial-range energy spectrum of the velocity is chosen in the form $E(k)\propto k^{1-2\eps}$, and the correlation time at the wavenumber $k$ scales as $k^{-2+\eta}$. Inertial-range anomalous scaling for the structure functions and other correlation functions emerges as a consequence of the existence in the model of composite operators with negative scaling dimensions, identified with anomalous exponents. For $\eta>\eps$, these exponents are the same as in the rapid-change limit of the model; for $\eta<\eps$, they are the same as in the limit of a time-independent (quenched) velocity field. For $\eps=\eta$ (local turnover exponent), the anomalous exponents are nonuniversal through the dependence on a dimensionless parameter, the ratio of the velocity correlation time and the scalar turnover time. The universality reveals itself, however, only in the second order of the $\eps$ expansion, and the exponents are derived to order $O(\eps^{2})$, including anisotropic contributions. It is shown that, for moderate $n$, the order of the structure function, and $d$, the space dimensionality, finite correlation time enhances the intermittency in comparison with the both limits: the rapid-change and quenched ones. The situation changes when $n$ and/or $d$ become large enough: the correction to the rapid-change limit due to the finite correlation time is positive (that is, the anomalous scaling is suppressed), it is maximal for the quenched limit and monotonically decreases as the correlation time tends to zero.
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arxiv:nlin/0204044
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A model is introduced to describe guided propagation of a linear or nonlinear pulse which encounters a localized nonlinear defect, that may be either static or breather-like one (the model with the static defect applies to an optical pulse in a long fiber link with an inserted additional section of a nonlinear fiber). In the case when the host waveguide is linear, the pulse has a Gaussian shape. In that case, an immediate result of its interaction with the nonlinear defect can be found in an exact analytical form, amounting to transformation of the incoming Gaussian into an infinite array of overlapping Gaussian pulses. Further evolution of the array in the linear host medium is found numerically by means of the Fourier transform. An important ingredient of the linear medium is the third-order dispersion, that eventually splits the array into individual pulses. If the host medium is nonlinear, the input pulse is taken as a fundamental soliton. The soliton is found to be much more resistant to the action of the nonlinear defect than the Gaussian pulse in the linear host medium. In this case, the third-order-dispersion splits the soliton proper and wavepackets generated by the action of the defect.
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arxiv:nlin/0204065
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A new class of 1D discrete nonlinear Schr${\ddot{\rm{o}}}$dinger Hamiltonians with tunable nonlinerities is introduced, which includes the integrable Ablowitz-Ladik system as a limit. A new subset of equations, which are derived from these Hamiltonians using a generalized definition of Poisson brackets, and collectively refered to as the N-AL equation, is studied. The symmetry properties of the equation are discussed. These equations are shown to possess propagating localized solutions, having the continuous translational symmetry of the one-soliton solution of the Ablowitz-Ladik nonlinear Schr${\ddot{\rm{o}}}$dinger equation. The N-AL systems are shown to be suitable to study the combined effect of the dynamical imbalance of nonlinearity and dispersion and the Peierls-Nabarro potential, arising from the lattice discreteness, on the propagating solitary wave like profiles. A perturbative analysis shows that the N-AL systems can have discrete breather solutions, due to the presence of saddle center bifurcations in phase portraits. The unstaggered localized states are shown to have positive effective mass. On the other hand, large width but small amplitude staggered localized states have negative effective mass. The collison dynamics of two colliding solitary wave profiles are studied numerically. Notwithstanding colliding solitary wave profiles are seen to exhibit nontrivial nonsolitonic interactions, certain universal features are observed in the collison dynamics. Future scopes of this work and possible applications of the N-AL systems are discussed.
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arxiv:nlin/0205002
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We study the topology of the set of singular points (blow-ups) in the solution of the nonperiodic Toda lattice defined on real split semisimple Lie algebra $\mathfrak g$. The set of blow-ups is called the Painlev\'e divisor. The isospectral manifold of the Toda lattice is compactified through the companion embedding which maps themanifold to the flag manifold associated with the underlying Lie algebra $\mathfrak g$. The Painlev\'e divisor is then given by the intersections of the compactified manifold with the Bruhat cells in the flag manifold. In this paper, we give explicit description of the topology of the Painlev\'e divisor for the cases of all the rank two Lie algebra, $A_2,B_2, C_2, G_2$, and $A_3$ type. The results are obtained by using the Mumford system and the limit matrices introduced originally for the periodic Toda lattice. We also give a Lie theoretic description of the Painlev\'e divisor of codimension one case, and propose several conjecturesfor the general case.
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arxiv:nlin/0205013
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We investigate the mKdV hierarchy with integral type of source (mKdVHWS), which consist of the reduced AKNS eigenvalue problem with $r=q$ and the mKdV hierarchy with extra term of the integration of square eigenfunction. First we propose a method to find the explicit evolution equation for eigenfunction of the auxiliary linear problems of the mKdVHWS. Then we determine the evolution equations of scattering data corresponding to the mKdVHWS which allow us to solve the equation in the mKdVHWS by inverse scattering transformation.
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arxiv:nlin/0205024
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We study the periodic forced response of a system of two limit cycle oscillators that interact with each other via a time delayed coupling. Detailed bifurcation diagrams in the parameter space of the forcing amplitude and forcing frequency are obtained for various interesting limits using numerical and analytical means. In particular, the effects of the coupling strength, the natural frequency spread of the two oscillators and the time delay parameter on the size and nature of the entrainment domain are delineated. The system is found to display a nonlinear response on certain critical contours in the space of the coupling strength and time delay. Time delay offers a novel tuning knob for controlling the system response over a wide range of frequencies and this may have important practical applications.
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arxiv:nlin/0205035
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The field theoretic renormalization group is applied to Kraichnan's model of a passive scalar quantity advected by the Gaussian velocity field with the pair correlation function $\propto\delta(t-t')/k^{d+\epsilon}$. Inertial-range anomalous scaling for the structure functions and various pair correlators is established as a consequence of the existence in the corresponding operator product expansions of ``dangerous'' composite operators (powers of the local dissipation rate), whose {\it negative} critical dimensions determine anomalous exponents. The latter are calculated to order $\epsilon^3$ of the $\epsilon$ expansion (three-loop approximation).
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arxiv:nlin/0205052
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Soliton interactions in systems modelled by coupled nonlinear Schroedinger (CNLS) equations and encountered in phenomena such as wave propagation in optical fibers and photorefractive media possess unusual features : shape changing intensity redistributions, amplitude dependent phase shifts and relative separation distances. We demonstrate these properties in the case of integrable 2-CNLS equations. As a simple example, we consider the stationary two-soliton solution which is equivalent to the so-called partially coherent soliton (PCS) solution discussed much in the recent literature.
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arxiv:nlin/0205057
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Lax pairs with operator valued coefficients, which are explicitly connected by means of an additional condition, are considered. This condition is proved to be covariant with respect to the Darboux transformation of a general form. Nonlinear equations arising from the compatibility condition of the Lax pairs in the matrix case include, in particular, Nahm equations, Volterra, Bogoyavlenskii and Toda lattices. The examples of another one-, two- and multi-field lattice equations are also presented.
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arxiv:nlin/0205061
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Employing symbolic dynamics for geodesic motion on the tesselated pseudosphere, the so-called Hadamard-Gutzwiller model, we construct extremely long periodic orbits without compromising accuracy. We establish criteria for such long orbits to behave ergodically and to yield reliable statistics for self-crossings and avoided crossings. Self-encounters of periodic orbits are reflected in certain patterns within symbol sequences, and these allow for analytic treatment of the crossing statistics. In particular, the distributions of crossing angles and avoided-crossing widths thus come out as related by analytic continuation. Moreover, the action difference for Sieber-Richter pairs of orbits (one orbit has a self-crossing which the other narrowly avoids and otherwise the orbits look very nearly the same) results to all orders in the crossing angle. These findings may be helpful for extending the work of Sieber and Richter towards a fuller understanding of the classical basis of quantum spectral fluctuations.
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arxiv:nlin/0206023
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We have measured resonance spectra in a superconducting microwave cavity with the shape of a three-dimensional generalized Bunimovich stadium billiard and analyzed their spectral fluctuation properties. The experimental length spectrum exhibits contributions from periodic orbits of non-generic modes and from unstable periodic orbit of the underlying classical system. It is well reproduced by our theoretical calculations based on the trace formula derived by Balian and Duplantier for chaotic electromagnetic cavities.
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arxiv:nlin/0206028
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We discuss the impact of recent developments in the theory of chaotic dynamical systems, particularly the results of Sinai and Ruelle, on microwave experiments designed to study quantum chaos. The properties of closed Sinai billiard microwave cavities are discussed in terms of universal predictions from random matrix theory, as well as periodic orbit contributions which manifest as `scars' in eigenfunctions. The semiclassical and classical Ruelle zeta-functions lead to quantum and classical resonances, both of which are observed in microwave experiments on n-disk hyperbolic billiards.
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arxiv:nlin/0206031
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It is argued that the integrable modified nonlinear Schroedinger equation with the nonlinearity dispersion term is the true starting point to analytically describe subpicosecond pulse dynamics in monomode fibers. Contrary to the known assertions, solitons of this equation are free of self-steepining and the breather formation is possible.
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arxiv:nlin/0206041
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We both show experimentally and numerically that the time scales separation introduced by long range activation can induce oscillations and excitability in nonequilibrium reaction-diffusion systems that would otherwise only exhibit bistability. Namely, we show that the Chlorite-Tetrathionate reaction, where autocatalytic species diffuses faster than the substrates, the spatial bistability domain in the nonequilibrium phase diagram is extended with oscillatory and excitability domains. A simple model and a more realistic model qualitatively account for the observed behavior. The latter model provides quantitative agreement with the experiments.
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arxiv:nlin/0207031
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Turbulent shear flows, such as those occurring in the wall region of turbulent boundary layers, manifest a substantial increase of intermittency with respect to isotropic conditions. This suggests a close link between anisotropy and intermittency. However, a rigorous statistical description of anisotropic flows is, in most cases, hampered by the inhomogeneity of the field. This difficulty is absent for the homogeneous shear flow. For this flow the scale by scale budget is discussed here by using the appropriate form of the Karman-Howarth equation, to determine the range of scales where the shear is dominant. The issuing generalization of the four-fifths law is then used as the guideline to extend to shear dominated flows the Kolmogorov-Obhukhov theory of intermittency. The procedure leads naturally to the formulation of generalized structure functions and the description of intermittency thus obtained reduces to the K62 theory for vanishing shear. Also here the intermittency corrections to the scaling exponents are carried by the moments of the coarse grained energy dissipation field. Numerical experiments give indications that the dissipation field is statistically unaffected by the shear, thereby supporting the conjecture that the intermittency corrections are universal. This observation together with the present reformulation of the theory gives reason for the increased intermittency observed in the classical longitudinal velocity increments.
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arxiv:nlin/0207053
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Hierarchy of one-parameter families of chaotic maps with an invariant measure have been introduced, where their appropriate coupling has lead to the generation of some coupled chaotic maps with an invariant measure. It is shown that these chaotic maps (also the coupled maps) do not undergo any period doubling or period-n-tupling cascade bifurcation to chaos, but they have either single fixed point attractor at certain values of the parameters or they are ergodic in the complementary region. Using the invariant measure or Sinai-Rulle-Bowen measure the Kolmogrov-Sinai entropy of the chaotic maps (coupled maps) have been calculated analytically, where the numerical simulations support the results
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arxiv:nlin/0208011
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Molecular dynamic simulations are performed to investigate a long-time evolution of different type initial signals in nonlinear acoustic chains with realistic Exp-6 potential and with power ones. Finite number of long-lifetime kink-shaped excitations is observed in the system in thermodynamic equilibrium. Dynamical equilibrium between the processes of their growth and decay is found.
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arxiv:nlin/0209004
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We consider a discrete classical integrable model on the 3-dimensional cubic lattice. The solutions of this model can be used to parameterize the Boltzmann weights of the different 3-dimensional spin models. We have found the general solution of this model constructed in terms of the theta-functions defined on an arbitrary compact algebraic curve. The imposing of the periodic boundary conditions fixes the algebraic curve. We have shown that in this case the curve coincides with the spectral one of the auxiliary linear problem. In the case when the curve is a rational one, the soliton solutions have been constructed.
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arxiv:nlin/0209019
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We study the multi-species replicator model with linear fitness and random fitness matrices of various classes. By means of numerical resolution of the replicator equations, we determine the survival probability of a species in terms of its average interaction with the rest of the system. The role of the interaction pattern of the ecosystem in defining survival and extinction probabilities is emphasized.
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arxiv:nlin/0210018
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Cooperative effects of periodic force and noise in globally Cooperative effects of periodic force and noise in globally coupled systems are studied using a nonlinear diffusion equation for the number density. The amplitude of the order parameter oscillation is enhanced in an intermediate range of noise strength for a globally coupled bistable system, and the order parameter oscillation is entrained to the external periodic force in an intermediate range of noise strength. These enhancement phenomena of the response of the order parameter in the deterministic equations are interpreted as stochastic resonance and stochastic synchronization in globally coupled systems.
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arxiv:nlin/0210056
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Spatial solitons can exist in various kinds of nonlinear optical resonators with and without amplification. In the past years different types of these localized structures such as vortices, bright, dark solitons and phase solitons have been experimentally shown to exist. Many links appear to exist to fields different from optics, such as fluids, phase transitions or particle physics. These spatial resonator solitons are bistable and due to their mobility suggest schemes of information processing not possible with the fixed bistable elements forming the basic ingredient of traditional electronic processing. The recent demonstration of existence and manipulation of spatial solitons in emiconductor microresonators represents a step in the direction of such optical parallel processing applications. We review pattern formation and solitons in a general context, show some proof of principle soliton experiments on slow systems, and describe in more detail the experiments on semiconductor resonator solitons which are aimed at applications.
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arxiv:nlin/0210073
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We present numerical and experimental results for the development of islands of stability in atom-optics billiards with soft walls. As the walls are soften, stable regions appear near singular periodic trajectories in converging (focusing) and dispersing billiards, and are surrounded by areas of "stickiness" in phase-space. The size of these islands depends on the softness of the potential in a very sensitive way.
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arxiv:nlin/0210075
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We study the transport of charge due to polarons in a model of DNA which takes in account its 3D structure and the coupling of the electron wave function with the H--bond distortions and the twist motions of the base pairs. Perturbations of the ground states lead to moving polarons which travel long distances. The influence of parametric and structural disorder, due to the impact of the ambient, is considered, showing that the moving polarons survive to a certain degree of disorder. Comparison of the linear and tail analysis and the numerical results makes possible to obtain further information on the moving polaron properties.
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arxiv:nlin/0211026
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We provide the first statistical analysis of the decay rates of strongly driven 3D atomic Rydberg states. The distribution of the rates exhibits universal features due to Anderson localization, while universality of the time dependent decay requires particular initial conditions.
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arxiv:nlin/0212022
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A model of the heart tissue as a conductive system with two interacting pacemakers and a refractory time, is proposed. In the parametric space of the model the phase locking areas are investigated in detail. Obtained results allow us to predict the behaviour of excitable systems with two pacemakers depending on the type and intensity of their interaction and the initial phase. Comparison of the described phenomena with intrinsic pathologies of cardiac rhythms is presented.
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arxiv:nlin/0212025
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We present a numerical study of Riemann's formula for the oscillating part of the density of the primes and their powers. The formula is comprised of an infinite series of oscillatory terms, one for each zero of the zeta function on the critical line and was derived by Riemann in his paper on primes assuming the Riemann hypothesis. We show that high resolution spectral lines can be generated by the truncated series at all powers of primes and demonstrate explicitly that the relative line intensities are correct. We then derive a Gaussian sum rule for Riemann's formula. This is used to analyze the numerical convergence of the truncated series. The connections to quantum chaos and semiclassical physics are discussed.
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arxiv:nlin/0212042
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A family of maps or flows depending on a parameter $\nu$ which varies in an interval, spans a certain property if along the interval this property depends continuously on the parameter and achieves some asymptotic values along it. We consider families of periodically forced Hamiltonian systems for which the appropriately scaled frequency $\bar{\omega}(\nu)$ is spanned, namely it covers the semi-infinite line $[0,\infty).$ Under some natural assumptions on the family of flows and its adiabatic limit, we construct a convenient labelling scheme for the primary homoclinic orbits which may undergo a countable number of bifurcations along this interval. Using this scheme we prove that a properly defined flux function is $C^{1}$ in $\nu.$ Combining this proof with previous results of RK and Poje, immediately establishes that the flux function and the size of the chaotic zone depend on the frequency in a non-monotone fashion for a large class of Hamiltonian flows.
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arxiv:nlin/0301006
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Even if it is nonintegrable, a differential equation may nevertheless admit particular solutions which are globally analytic. On the example of the dynamical system of Kuramoto and Sivashinsky, which is generically chaotic and presents a high physical interest, we review various methods, all based on the structure of singularities, allowing us to characterize the analytic solution which depends on the largest possible number of constants of integration.
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arxiv:nlin/0302056
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It has been shown that for a certain special type of quantum graphs the random-matrix form factor can be recovered to at least third order in the scaled time \tau using periodic-orbit theory. Two types of contributing pairs of orbits were identified, those which require time-reversal symmetry and those which do not. We present a new technique of dealing with contribution from the former type of orbits. The technique allows us to derive the third order term of the expansion for general graphs. Although the derivation is rather technical, the advantages of the technique are obvious: it makes the derivation tractable, it identifies explicitly the orbit configurations which give the correct contribution, it is more algorithmical and more system-independent, making possible future applications of the technique to systems other than quantum graphs.
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arxiv:nlin/0305009
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We report progress in constructing Boltzmann weights for integrable 3-dimensional lattice spin models. We show that a large class of vertex solutions to the modified tetrahedron equation can be conveniently parameterized in terms of N-th roots of theta-functions on the Jacobian of a compact algebraic curve. Fay's identity guarantees the Fermat relations and the classical equations of motion for the parameters determining the Boltzmann weights. Our parameterization allows to write a simple formula for fused Boltzmann weights R which describe the partition function of an arbitrary open box and which also obey the modified tetrahedron equation. Imposing periodic boundary conditions we observe that the R satisfy the normal tetrahedron equation. The scheme described contains the Zamolodchikov-Baxter-Bazhanov model and the Chessboard model as special cases.
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arxiv:nlin/0305031
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We discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. The method is then applied to the Schr\"{o}dinger equation in order to obtain a realization of nonrelativistic quantum mechanics in discrete space-time. In this approach a quantum system on a lattice has a symmetry algebra isomorphic to that of the continuous case. Moreover, systems that are integrable, superintegrable or exactly solvable preserve these properties in the discrete case.
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arxiv:nlin/0305047
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Based on the heuristics that maintaining presumptions can be beneficial in uncertain environments, we propose a set of basic axioms for learning systems to incorporate the concept of prejudice. The simplest, memoryless model of a deterministic learning rule obeying the axioms is constructed, and shown to be equivalent to the logistic map. The system's performance is analysed in an environment in which it is subject to external randomness, weighing learning defectiveness against stability gained. The corresponding random dynamical system with inhomogeneous, additive noise is studied, and shown to exhibit the phenomena of noise induced stability and stochastic bifurcations. The overall results allow for the interpretation that prejudice in uncertain environments entails a considerable portion of stubbornness as a secondary phenomenon.
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arxiv:nlin/0306055
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With this paper we begin an investigation of difference schemes that possess Darboux transformations and can be regarded as natural discretizations of elliptic partial differential equations. We construct, in particular, the Darboux transformations for the general self adjoint schemes with five and seven neighbouring points. We also introduce a distinguished discretization of the two-dimensional stationary Schrodinger equation, described by a 5-point difference scheme involving two potentials, which admits a Darboux transformation.
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arxiv:nlin/0307045
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We derive simple conditions for the stability or instability of the synchronized oscillation of a class of networks of coupled phase-oscillators, which includes many of the systems used in neural modelling.
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arxiv:nlin/0308031
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We study the noise-induced escape process in a prototype dissipative nonequilibrium system, the Ikeda map. In the presence of a chaotic saddle embedded in the basin of attraction of the metastable state, we find the novel phenomenon of a strong enhancement of noise-induced escape. This result is established by employing the theory of quasipotentials. Our finding is of general validity and should be experimentally observable.
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arxiv:nlin/0309016
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The connection between symmetries and linearizations of discrete-time dynamical systems is being inverstigated. It is shown, that existence of semigroup structures related to the vector field and having linear representations enables reduction of linearization problem to a system of first order partial differential equations. By means of inverse of the Poincare map one can relate symmetries in such linearizable systems to continuous and discrete ones of the corresponding differential equations.
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arxiv:nlin/0309055
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The spectral statistics of the circular billiard with a point-scatterer is investigated. In the semiclassical limit, the spectrum is demonstrated to be composed of two uncorrelated level sequences. The first corresponds to states for which the scatterer is located in the classically forbidden region and its energy levels are not affected by the scatterer in the semiclassical limit while the second sequence contains the levels which are affected by the point-scatterer. The nearest neighbor spacing distribution which results from the superposition of these sequences is calculated analytically within some approximation and good agreement with the distribution that was computed numerically is found.
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arxiv:nlin/0309061
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Motivated by the rich variety of complex periodic and quasi-periodic patterns found in systems such as two-frequency forced Faraday waves, we study the interaction of two spatially periodic modes that are nearly resonant. Within the framework of two coupled one-dimensional Ginzburg-Landau equations we investigate analytically the stability of the periodic solutions to general perturbations, including perturbations that do not respect the periodicity of the pattern, and which may lead to quasi-periodic solutions. We study the impact of the deviation from exact resonance on the destabilizing modes and on the final states. In regimes in which the mode interaction leads to traveling waves our numerical simulations reveal localized waves in which the wavenumbers are resonant and which drift through a steady background pattern that has an off-resonant wavenumber ratio.
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arxiv:nlin/0309070
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Recently, we have proposed a {\em Euler-Lagrange transformation} for cellular automata(CA) by developing new transformation formulas. Applying this method to the Burgers CA(BCA), we have succeeded in obtaining the Lagrange representation of the BCA. In this paper, we apply this method to multi-value generalized Burgers CA(GBCA) which include the Fukui-Ishibashi model and the quick-start model associated with traffic flow. As a result, we have succeeded in clarifying the Euler-Lagrange correspondence of these models. It turns out, moreover that the GBCA can naturally be considered as a simple model of a multi-lane traffic flow.
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arxiv:nlin/0311057
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Coupled map lattices have been widely used as models in several fields of physics, such as chaotic strings, turbulence, and phase transitions, as well as in other disciplines, such as biology (ecology, evolution) and information processing. This paper investigates properties of periodic orbits in two coupled Tchebyscheff maps. The zeta function cycle expansions are used to compute dynamical averages appearing in Beck's theory of chaotic strings. The results show close agreement with direct simulation for most values of the coupling parameter, and yield information about the system complementary to that of direct simulation.
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arxiv:nlin/0312024
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Sound scattering by a finite width beam on a single rigid body rotation vortex flow is detected by a linear array of transducers (both smaller than a flow cell), and analyzed using a revised scattering theory. Both the phase and amplitude of the scattered signal are obtained on 64 elements of the detector array and used for the analysis of velocity and vorticity fields. Due to averaging on many pulses the signal-to-noise ratio of the phases difference in the scattered sound signal can be amplified drastically, and the resolution of the method in the detection of circulation, vortex radius, vorticity, and vortex location becomes comparable with that obtained earlier by time-reversal mirror (TRM) method (P. Roux, J. de Rosny, M. Tanter, and M. Fink, {\sl Phys. Rev. Lett.} {\bf 79}, 3170 (1997)). The revised scattering theory includes two crucial steps, which allow overcoming limitations of the existing theories. First, the Huygens construction of a far field scattering signal is carried out from a signal obtained at any intermediate plane. Second, a beam function that describes a finite width beam is introduced, which allows using a theory developed for an infinite width beam for the relation between a scattering amplitude and the vorticity structure function. Structure functions of the velocity and vorticity fields deduced from the sound scattering signal are compared with those obtained from simultaneous particle image velocimetry (PIV) measurements. Good quantitative agreement is found.
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arxiv:nlin/0401004
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Following our first report (A. Groisman and V. Steinberg, $\sl Nature$ $\bf 405$, 53 (2000)) we present an extended account of experimental observations of elasticity induced turbulence in three different systems: a swirling flow between two plates, a Couette-Taylor (CT) flow between two cylinders, and a flow in a curvilinear channel (Dean flow). All three set-ups had high ratio of width of the region available for flow to radius of curvature of the streamlines. The experiments were carried out with dilute solutions of high molecular weight polyacrylamide in concentrated sugar syrups. High polymer relaxation time and solution viscosity ensured prevalence of non-linear elastic effects over inertial non-linearity, and development of purely elastic instabilities at low Reynolds number (Re) in all three flows. Above the elastic instability threshold, flows in all three systems exhibit features of developed turbulence. Those include: (i)randomly fluctuating fluid motion excited in a broad range of spatial and temporal scales; (ii) significant increase in the rates of momentum and mass transfer (compared to those expected for a steady flow with a smooth velocity profile). Phenomenology, driving mechanisms, and parameter dependence of the elastic turbulence are compared with those of the conventional high Re hydrodynamic turbulence in Newtonian fluids.
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arxiv:nlin/0401006
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Direct and inverse recursion operator is derived for the vacuum Einstein equations for metrics with two commuting Killing vectors that are orthogonal to a foliation by 2-dimensional leaves.
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arxiv:nlin/0401014
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The statistical properties of fluid particles transported by a fully developed turbulent flow are investigated by means of high resolution direct numerical simulations. Single trajectory statistics is investigated in a time range spanning more than three decades, from less than a tenth of the Kolmogorov timescale up to one large-eddy turnover time. Acceleration and velocity statistics show a neat quantitative agreement with recent experimental results. Trapping effects in vortex filaments give rise to enhanced small-scale intermittency on Lagrangian observables.
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arxiv:nlin/0402032
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We study resonance patterns of a spiral-shaped dielectric microcavity with chaotic ray dynamics. Many resonance patterns of this microcavity, with refractive indices $n=2$ and 3, exhibit strong localization of simple geometric shape, and we call them {\em quasi-scarred resonances} in the sense that there is, unlike the conventional scarring, no underlying periodic orbits. It is shown that the formation of quasi-scarred pattern can be understood in ter ms of ray dynamical probability distributions and wave properties like uncertainty and interference.
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arxiv:nlin/0403025
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We show that the generalized diffusion coefficient of a subdiffusive intermittent map is a fractal function of control parameters. A modified continuous time random walk theory yields its coarse functional form and correctly describes a dynamical phase transition from normal to anomalous diffusion marked by strong suppression of diffusion. Similarly, the probability density of moving particles is governed by a time-fractional diffusion equation on coarse scales while exhibiting a specific fine structure. Approximations beyond stochastic theory are derived from a generalized Taylor-Green-Kubo formula.
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arxiv:nlin/0403039
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We discuss a simple analytical model of the turbulent boundary layer (TBL) over flat plane. The model offers an analytical description of the profiles of mean velocity and turbulent activity in the entire boundary region, from the viscous sub-layer, through the buffer layer further into the log-law turbulent region. In contrast to various existing interpolation formulas the model allows one to generalize the description of simple TBL of a Newtonian fluid for more complicated flows of turbulent suspensions laden with heavy particles, bubbles, long-chain polymers, to include the gravity acceleration, etc.
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arxiv:nlin/0404010
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The concept of cross diffusion is applied to some biological systems. The conditions for persistence and Turing instability in the presence of cross diffusion are derived. Many examples including: predator-prey, epidemics (with and without delay), hawk-dove-retaliate and prisoner's dilemma games are given.
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arxiv:nlin/0404021
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Nash equilibria are defined using uncorrelated behavioural or mixed joint probability distributions effectively assuming that players of bounded rationality must discard information to locate equilibria. We propose instead that rational players will use all the information available in correlated distributions to constrain payoff function topologies and gradients to generate novel "constrained" equilibria, each one a backwards induction pathway optimizing payoffs in the constrained space. In the finite iterated prisoner's dilemma, we locate constrained equilibria maximizing payoffs via cooperation additional to the unconstrained (Nash) equilibrium maximizing payoffs via defection. Our approach clarifies the usual assumptions hidden in backwards induction.
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arxiv:nlin/0404023
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We find normal and seminormal forms for a sl(3)-valued zero curvature representation (ZCR). We prove a theorem about reducibility of ZCR's, which says that if one of the matrix in a ZCR (A,B) falls to a proper subalgebra of sl(3), then the second matrix either falls to the same subalgebra or the ZCR is almost trivial. In the end of this paper we show examples of ZCR's and their normal forms.
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arxiv:nlin/0404036
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In this paper a proposal is made of an adaptive coupling function for achieving synchronization between two lasers subject to optical feedback. Such a control scheme requires knowledge of the systems' parameters. For the first time we demonstrate that when these parameters are not available on-line parameter estimation can be applied.Generalisation of the approach to the multi-feedback systems is also presented.
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arxiv:nlin/0404053
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We give a representation--theoretic interpretation of recent discovered coupled soliton equations using vertex operators construction of affinization of not simple but quadratic Lie algebras. In this setup we are able to obtain new integrable hierarchies coupled to each Drinfeld--Sokolov of $A$, $B$, $C$, $D$ hierarchies and to construct their soliton solutions.
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arxiv:nlin/0405040
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We investigate the propagation of partially coherent beams in spatially nonlocal nonlinear media with a logarithmic type of nonlinearity. We derive analytical formulas for the evolution of the beam parameters and conditions for the formation of nonlocal incoherent solitons.
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arxiv:nlin/0405043
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An experimental analysis of the asynchronous version of the "Game of Life" is performed to estimate how topology perturbations modify its evolution. We focus on the study of a phase transition from an "inactive-sparse phase" to a "labyrinth phase" and produce experimental data to quantify these changes as a function of the density of the initial configuration, the value of the synchrony rate, and the topology missing-link rate. An interpretation of the experimental results is given using the hypothesis that initial "germs" colonize the whole lattice and the validity of this hypothesis is tested.
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arxiv:nlin/0405061
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