Statistical-thermodynamic approach to a chaotic dynamical system: Exactly solvable examples

The static and dynamic properties of a chaotic attractor of a two-dimensional map are studied, which belongs to a particular class of piecewise continuous invertible maps. Coverings of a natural size to cover the attractor are introduced, so that the microscopic information of the attractor is written on each box composing the cover. The statistical thermodynamics of the scaling indices and the size indices of the boxes is formulated. Analytic forms of the free energy functions of the scaling indices and the size indices of the boxes are obtained for examples of a hyperbolic and a nonhyperbolic chaotic attractor. The statistical thermodynamics of local Lyapunov exponents is also studied and a relation between the thermodynamics of scaling indices and of local Lyapunov exponents is invetigated. For the nonhyperbolic example, the free energy and entropy functions of local Lyapunov exponents are obtained in analytic forms. These results display the existence of phase transitions. A phase transition is seen in the thermodynamics of scaling indices also.     

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.     

Measures with infinite Lyapunov exponents for the periodic Lorentz gas

We study invariant measures for the periodic Lorentz gas which are supported on the set of points with infinite Lyapunov exponents. We construct examples of such measures which are measures of maximal entropy and ones which are not.     

Lyapunov exponents and kolmogorov-sinai entropy for a high-dimensional convex billiard

We compute the Lyapunov exponents and the Kolmogorov-Sinai (KS) entropy for a self-bound N-body system that is realized as a convex billiard. This system exhibits truly high-dimensional chaos, and 2N-4 Lyapunov exponents are found to be positive. The KS entropy increases linearly with the numbers of particles. We examine the chaos generating defocusing mechanism and investigate how high-dimensional chaos develops in this system with no dispersing elements.     

基于图像区域Lyapunov指数的海面舰船目标检测

为了检测海面背景中的舰船目标,分析了目标存在时背景信号混沌特征的变化,提出了一种基于图像区域Lyapunov指数的目标检测新方法. 新方法定义了图像灰度距离的概念,基于改进的Wolf方法将一维信号Lyapunov指数提取方法扩展到图像信号,利用图像区域最大灰度距离Lyapunov指数的变化检测淹没在混沌背景信号中的目标信号. 实验结果表明海面背景图像信号具有一定的混沌特征,利用新方法能有效检测出海面背景下的舰船目标,检测结果优于基于统计分析的方法. 关键词： Lyapunov指数 灰度距离 混沌特征 目标检测     

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measure-theoretic entropies and topological pressures in this general situation. Most of our results are obtained without the assumption of the existence of unique equilibrium measures or the differentiability of pressure functions. Some examples are constructed to illustrate the irregularity and the complexity of multifractal behaviors in the sub-additive case and in the case that the entropy map is not upper-semi continuous.     

Spatiotemporal chaos and noise

Low-dimensional chaotic dynamical systems can exhibit many characteristic properties of stochastic systems, such as broad Fourier spectra. They are distinguishable from stochastic processes through finite values for their dimension, Lyapunov exponents, and Kolmogorov-Sinai entropy. We discuss how these characteristic observables are modified in spatiotemporal chaotic systems like. coupled map lattices. We analyze with the help of Lyapunov concepts how the stochastic limit is approached and how these properties can be observed directly through local dimension measurements from reconstructed time series. Finally, we discuss the interaction of spatiotemporal attractors with external noise and possible connections to problems of pattern selection and stability.     

The conjugate-pairing rule for non-Hamiltonian systems

In systems that satisfy the Conjugate Pairing Rule (CPR), the spectrum of Lyapunov exponents is symmetric. The sum of each conjugate pair of exponents is identical. Since in dissipative systems the sum of all the exponents is the entropy production divided by Boltzmann's constant, the calculation of transport coefficients from the Lyapunov exponents is greatly simplified in systems that satisfy CPR. Sufficient conditions for CPR are well known: the underlying adiabatic dynamics should be symplectic. However, the necessary conditions for CPR are not known. In this paper we report on the results of computer simulations which shed light on the necessary conditions for the CPR to hold. We provide, for the first time, convincing evidence that the standard molecular dynamics algorithm for calculating shear viscosity violates the CPR, even in the thermodynamic limit. In spite of this it appears that the sum of the maximal exponents is equal to the entropy production per degree of freedom. Thus it appears that the shear viscosity can still be calculated using the standard viscosity algorithm by summing the maximal pair of exponents.(c) 1998 American Institute of Physics.     

Computing the pressure for Axiom-A attractors by time series and large deviations for the Lyapunov exponent

For the Axiom-A attractors a relation is given between the topological pressure and the spectrum of the generalized Lyapunov exponents. As a consequence, a simple formula is found to compute the topological entropy of the attractor by means of a time series. The results are used to compute the large deviations for positive Lyapunov exponents.     

Cycles homoclinic to chaotic sets; robustness and resonance

For dynamical systems possessing invariant subspaces one can have a robust homoclinic cycle to a chaotic set. If such a cycle is stable, it manifests itself as long periods of quiescent chaotic behaviour interrupted by sudden transient 'bursts'. The time between the transients increases as the trajectory approaches the cycle. This behavior for a cycle connecting symmetrically related chaotic sets has been called 'cycling chaos' by Dellnitz et al. [IEEE Trans. Circ. Sys. I 42, 821-823 (1995)]. We characterise such cycles and their stability by means of normal Lyapunov exponents. We find persistence of states that are not Lyapunov stable but still attracting, and also states that are approximately periodic. For systems possessing a skew-product structure (such as naturally arises in chaotically forced systems) we show that the asymptotic stability and the attractivity of the cycle depends in a crucial way on what we call the footprint of the cycle. This is the spectrum of Lyapunov exponents of the chaotic invariant set in the expanding and contracting directions of the cycle. Numerical simulations and calculations for an example system of a homoclinic cycle parametrically forced by a Rossler attractor are presented; here we observe the creation of nearby chaotic attractors at resonance of transverse Lyapunov exponents. (c) 1997 American Institute of Physics.     

Chronotopic Lyapunov analysis: II. Toward a unified approach

From the analyticity properties of the equation governing infinitesimal perturbations, it is conjectured that all types of Lyapunov exponents introduced in spatially extended 1D systems can be derived from a single function that we call the entropy potential. The general consequences of its very existence on the Kolmogorov-Sinai entropy of generic spatiotemporal patterns are discussed.     

The Metric Entropy of Endomorphisms

We prove that, for a C ² non-invertible but non-degenerate map on a compact Riemannian manifold without boundary, an invariant measure satisfies an equality relating entropy, folding entropy and negative Lyapunov exponents. This generalizes Ledrappier-Young’s entropy formula [5] (for negative Lyapunov exponents of diffeomorphisms) to the case of endomorphisms. This work is supported by National Basic Research Program of China (973 Program) (2007 CB 814800).     

Lyapunov spectrum of a random Lorentz gas in a magnetic field

The Lyapunov exponents and the Kolmogorov Sinai entropy for 2- and 3-dimensional, dilute, random Lorentz gases in a magnetic field are calculated. The results are obtained by combining simple kinetic theory with geometric methods from dynamical systems theory. The Lyapunov exponents are explicitly calculated up to second order in the magnetic field.     

On estimates of Lyapunov exponents of synchronized coupled systems

Lyapunov exponents of a synchronized coupled system consist of those of the underlying individual systems and the transverse systems, based on a mode decomposition along the synchronization manifold. Estimates of bounds on the Lyapunov exponents (including transverse Lyapunov exponents) are derived. Several examples are used to validate the theoretical estimates.     

A Noisy SAR Image Fusion Method Based on NLM and GAN

The unavoidable noise often present in synthetic aperture radar (SAR) images, such as speckle noise, negatively impacts the subsequent processing of SAR images. Further, it is not easy to find an appropriate application for SAR images, given that the human visual system is sensitive to color and SAR images are gray. As a result, a noisy SAR image fusion method based on nonlocal matching and generative adversarial networks is presented in this paper. A nonlocal matching method is applied to processing source images into similar block groups in the pre-processing step. Then, adversarial networks are employed to generate a final noise-free fused SAR image block, where the generator aims to generate a noise-free SAR image block with color information, and the discriminator tries to increase the spatial resolution of the generated image block. This step ensures that the fused image block contains high resolution and color information at the same time. Finally, a fused image can be obtained by aggregating all the image blocks. By extensive comparative experiments on the SEN1–2 datasets and source images, it can be found that the proposed method not only has better fusion results but is also robust to image noise, indicating the superiority of the proposed noisy SAR image fusion method over the state-of-the-art methods.     

Anomalous sensitivity to initial conditions and entropy production in standard maps: Nonextensive approach

We perform a throughout numerical study of the average sensitivity to initial conditions and entropy production for two symplectically coupled standard maps focusing on the control-parameter region close to regularity. Although the system is ultimately strongly chaotic (positive Lyapunov exponents), it first stays lengthily in weak-chaotic regions (zero Lyapunov exponents). We argue that the nonextensive generalization of the classical formalism is an adequate tool in order to get nontrivial information about the first stage of this crossover phenomenon. Within this context we analyze the relation between the power-law sensitivity to initial conditions and the entropy production.     

Phase resetting effects for robust cycles between chaotic sets

In the presence of symmetries or invariant subspaces, attractors in dynamical systems can become very complicated, owing to the interaction with the invariant subspaces. This gives rise to a number of new phenomena, including that of robust attractors showing chaotic itinerancy. At the simplest level this is an attracting heteroclinic cycle between equilibria, but cycles between more general invariant sets are also possible. In this paper we introduce and discuss an instructive example of an ordinary differential equation where one can observe and analyze robust cycling behavior. By design, we can show that there is a robust cycle between invariant sets that may be chaotic saddles (whose internal dynamics correspond to a R?ssler system), and/or saddle equilibria. For this model, we distinguish between cycling that includes phase resetting connections (where there is only one connecting trajectory) and more general non(phase) resetting cases, where there may be an infinite number (even a continuum) of connections. In the nonresetting case there is a question of connection selection: which connections are observed for typical attracted trajectories? We discuss the instability of this cycling to resonances of Lyapunov exponents and relate this to a conjecture that phase resetting cycles typically lead to stable periodic orbits at instability, whereas more general cases may give rise to "stuck on" cycling. Finally, we discuss how the presence of positive Lyapunov exponents of the chaotic saddle mean that we need to be very careful in interpreting numerical simulations where the return times become long; this can critically influence the simulation of phase resetting and connection selection.     

Lyapunov Instability for a Hard-Disk Fluid in Equilibrium and Nonequilibrium Thermostated by Deterministic Scattering

We compute the full Lyapunov spectra for a hard-disk fluid under temperature gradient and under shear. The Lyapunov exponents are calculated using a recently developed formalism for systems with elastic hard collisions. The system is thermalized by deterministic and time-reversible scattering at the boundary, whereas the bulk dynamics remains Hamiltonian. This thermostating mechanism allows for energy fluctuations around a mean value which is reflected by only two vanishing Lyapunov exponents in equilibrium and nonequilibrium. In nonequilibrium steady states the phase-space volume is contracted on average, leading to a negative sum of the Lyapunov exponents. Since the system is driven inhomogeneously we do not expect the conjugate pairing rule to hold, which is indeed shown to be the case. Finally, the Kaplan–Yorke dimension and the Kolmogorov–Sinai entropy are calculated from the Lyapunov spectra.     

Predictability: a way to characterize complexity

Different aspects of the predictability problem in dynamical systems are reviewed. The deep relation among Lyapunov exponents, Kolmogorov–Sinai entropy, Shannon entropy and algorithmic complexity is discussed. In particular, we emphasize how a characterization of the unpredictability of a system gives a measure of its complexity. Adopting this point of view, we review some developments in the characterization of the predictability of systems showing different kinds of complexity: from low-dimensional systems to high-dimensional ones with spatio-temporal chaos and to fully developed turbulence. A special attention is devoted to finite-time and finite-resolution effects on predictability, which can be accounted with suitable generalization of the standard indicators. The problems involved in systems with intrinsic randomness is discussed, with emphasis on the important problems of distinguishing chaos from noise and of modeling the system. The characterization of irregular behavior in systems with discrete phase space is also considered.