Spatial Lyapunov Exponents Research Articles

Heavy particles in a persistent random flow with traps.

We study a one-dimensional model for heavy particles in a compressible fluid. The fluid-velocity field is modeled by a persistent Gaussian random function, and the particles are assumed to be weakly inertial. Since one-dimensional fluid-velocity fields are always compressible, the model exhibits spatial trapping regions where particles tend to accumulate. We determine the statistics of fluid-velocity gradients in the vicinity of these traps and show how this allows one to determine the spatial Lyapunov exponent and the rate of caustic formation. We compare our analytical results with numerical simulations of the model and explore the limits of validity of the theory. Finally, we discuss implications for higher-dimensional systems.

Pinning and depinning: From periodic to chaotic and random media.

We study the propagation of dissipative structures in inhomogeneous media with a focus on pinning and depinning transitions. We model spatial complexity in the medium as generated by dynamical systems. We are thus able to capture transitions from periodic to quasiperiodic, to homoclinic and heteroclinic, and to chaotic media. Depinning bifurcations exhibit universal laws depending on extreme value statistics that are encoded in the dimension of ergodic measures, only. A key condition limiting this approach bounds spatial Lyapunov exponents in terms of interface localization, and we explore the breakdown of smoothness and universality when this condition is violated and fluctuations in the medium occur on length scales shorter than a typical interface width.

Spatial chaos on surface and its associated bifurcation and Feigenbaum problem

The qualitative theory of nonlinear spatial dynamical systems has attracted increasing attention recently. In particular, we have studied construction of spatial periodic orbits, dynamical behaviors of spatial chaos in the sense of Li–Yorke–Marotto, spatial Lyapunov exponents, control and generalized synchronization of spatial chaotic systems. In this paper, we apply a special mathematical transform to obtain spatial chaos on surface and its associated bifurcation and Feigenbaum problem. The 2D Logistic system is used for illustration. In addition, we also illustrate the difference in essence about chaos on surface as high-dimensional spatial system and dimension in phase space and we also analyze the parallel and different characters of theory system between 1D and 2D Logistic system.

Computation of the spectrum of spatial Lyapunov exponents for the spatially extended beam-plasma systems and electron-wave devices

The spectrum of Lyapunov exponents is powerful tool for the analysis of the complex system dynamics. In the general framework of nonlinear dynamics, a number of the numerical techniques have been developed to obtain the spectrum of Lyapunov exponents for the complex temporal behavior of the systems with a few degree of freedom. Unfortunately, these methods cannot be applied directly to analysis of complex spatio-temporal dynamics of plasma devices which are characterized by the infinite phase space, since they are the spatially extended active media. In the present paper, we propose the method for the calculation of the spectrum of the spatial Lyapunov exponents (SLEs) for the spatially extended beam-plasma systems. The calculation technique is applied to the analysis of chaotic spatio-temporal oscillations in three different beam-plasma model: (1) simple plasma Pierce diode, (2) coupled Pierce diodes, and (3) electron-wave system with backward electromagnetic wave. We find an excellent agreement between the system dynamics and the behavior of the spectrum of the spatial Lyapunov exponents. Along with the proposed method, the possible problems of SLEs calculation are also discussed. It is shown that for the wide class of the spatially extended systems, the set of quantities included in the system state for SLEs calculation can be reduced using the appropriate feature of the plasma systems.

On the spectrum of spatial Lyapunov exponents for a nonlinear active medium described by a complex Ginzburg-Landau equation

A new method is proposed for calculating the spectrum of Lyapunov exponents for spatially distributed systems. The proposed method is applied to a distributed nonlinear active medium described by a complex Ginzburg-Landau equation with periodic boundary conditions.

Generalized chaotic synchronization in coupled Ginzburg-Landau equations

Generalized synchronization is analyzed in unidirectionally coupled oscillatory systems exhibiting spatiotemporal chaotic behavior described by Ginzburg-Landau equations. Several types of coupling between the systems are analyzed. The largest spatial Lyapunov exponent is proposed as a new characteristic of the state of a distributed system, and its calculation is described for a distributed oscillatory system. Partial generalized synchronization is introduced as a new type of chaotic synchronization in spatially nonuniform distributed systems. The physical mechanisms responsible for the onset of generalized chaotic synchronization in spatially distributed oscillatory systems are elucidated. It is shown that the onset of generalized chaotic synchronization is described by a modified Ginzburg-Landau equation with additional dissipation irrespective of the type of coupling. The effect of noise on the onset of a generalized synchronization regime in coupled distributed systems is analyzed.

ON SPATIAL LYAPUNOV EXPONENTS AND SPATIAL CHAOS

In this paper, we investigate the periodic orbits, spatial Lyapunov exponents, and stability of spatially periodic orbits of the general 2D Logistic system [Formula: see text] where a is a real constant and μ is a parameter. The existence of spatial chaos in the sense of Li and Yorke is proved using the Marotto theorem. These results extend the corresponding results in the 1D Logistic system [Formula: see text] where n0is a fixed integer. These results also improve some existing results of the 2D coupled map lattice (CML) model [Formula: see text] where ε > 0 is the coupling constant.

Simple Dynamical Model with History Dependence

A lattice dynamics model exhibiting history dependence is proposed, inspired by recent sandpile experiments. The stress distribution depends on the preparation process, which is explained as the dependence of certain attractors on the preparation conditions of the system. The model has three phases, but the history dependence is shown to exist only in the phase where a perturbation is amplified selectively rather than globally when propagating in the downflow direction. The condition for this history dependence is given in terms of the spatial Lyapunov exponent.

Entropy potential and Lyapunov exponents.

According to a previous conjecture, spatial and temporal Lyapunov exponents of chaotic extended systems can be obtained from derivatives of a suitable function, the entropy potential. The validity and the consequences of this hypothesis are explored in detail. The numerical investigation of a continuous-time model provides a further confirmation to the existence of the entropy potential. Furthermore, it is shown that the knowledge of the entropy potential allows determining also Lyapunov spectra in general reference frames where the time-like and space-like axes point along generic directions in the space-time plane. Finally, the existence of an entropy potential implies that the integrated density of positive exponents (Kolmogorov-Sinai entropy) is independent of the chosen reference frame. (c) 1997 American Institute of Physics.

Localization of Mono‐ and Multichannel Waves in Disordered Periodic Systems

Analytical solution is obtained for the localization factor, namely, the exponential spatial decay rate, in the propagation of a disturbance through a chain of nearly periodic cell units. The localization is attributable to the departure from perfect spatial periodicity in the chain, known as disorder. In the case where multiple waves are permissible in the chain, each wave is characterized by a spatial Lyapunov exponent. These Lyapunov exponents constitute a spectrum of plus-minus pairs, and localization is governed by the largest negative Lyapunov exponent, which characterizes the least attenuated wave. This is then obtained by invoking Oseledec's multiplicative ergodic theorem, assuming that random disorders in different cell units are independent and identically distributed.

Chronotopic Lyapunov analysis. I. A detailed characterization of 1D systems

Instabilities in 1D spatially extended systems are studied with the aid of both temporal and spatial Lyapunov exponents. A suitable representation of the spectra allows a compact description of all the possible disturbances in tangent space. The analysis is carried out for chaotic and periodic spatiotemporal patterns. Singularities of the spectra and localization properties of the associated Lyapunov vectors are discussed.

Effect of delay in a bistable optical system with distributed nonlinear elements

We consider an optical ring cavity with distributed nonlinear elements and investigate the effect of delay using linear stability analysis. The lowest threshold for stability of the system is found and described by the spatial Lyapunov exponent α as α=0. When the system is lowest stability threshold, the delay has no effect on the system's stability. We also discuss the short delay approximation. In this regime, the stability threshold prediction is larger than that the system should have, and this also verifies that delay will induce instability.

Cooperative dynamics and functions in a collective nonlinear optical element system.

An optical ring cavity containing distributed nonlinear elements is proposed as a simple metaphorical model for investigating the dynamic properties of spatial chaos in a system far from thermal equilibrium. If the coupling between the elements is unidirectional, the stability of the disordered structure can be determined by the spatial Lyapunov exponent. This fact implies that spatial chaos is almost dynamically unstable and is replaced by spatiotemporal chaos. However, in the case of bidirectional coupling, the spatial chaos is self-induced over a wide range of the control parameter, which means that a memory function is formed cooperatively in the system. We predict several cooperative phenomena and describe their physical origin in terms of nonlinear dynamics. In particular, characteristics of spatial chaos applied to information storage are studied in detail. Applicability of predicted phenomena such as a cooperative all-optical switching, multivibrator operations, flip-flop operations, as well as spatial chaos memory, is discussed.

Spatial disorder in a nonlinear optical system: (1) The number of unstable modes in the short delay limit

The spatial disorder in an optical bistable system with N distributed nonlinear elements in the limit of a short delay time is studied. After using a continued fraction method, the spatial disordered structure is alternatively classified by the number of unstable modes of the system and the corresponding regimes are determined with the help of the spatial Lyapunov exponent. This procedure suggests that the asymptotic behavior of the system develop on a manifold whose dimension is N − [ N 2 ] at most. The presence of the decay mode, independent of control parameter, is due to the absence of delay time. The onset of the spatiotemporal instability originates from the “distributed” nature of the nonlinear elements, i.e. N > 1.

Self-induced spatial disorder in a nonlinear optical system.

An optical ring cavity containing distributed nonlinear elements is proposed as a promising candidate for investigation of the dynamic stability of spatial disorder in a system far from thermal equilibrium. If the interaction between the elements is unidirectional, the stability of disordered structure can be determined by the spatial Lyapunov exponent. This fact implies that spatial disorder is frozen under quite restricted conditions, and most of the spatially disordered structure is replaced by spatiotemporal chaos. However, in the case of a bidirectional interaction, the spatial disorder is self-induced over a wide range of the control parameter.