Lyapunov spectral analysis of a nonequilibrium Ising-like transition

By simulating a nonequilibrium coupled map lattice that undergoes an Ising-like phase transition, we show that the Lyapunov spectrum and related dynamical quantities such as the dimension correlation length~$\xi_\delta$ are insensitive to the onset of long-range ferromagnetic order. As a function of lattice coupling constant~$g$ and for certain lattice maps, the Lyapunov dimension density and other dynamical order parameters go through a minimum. The occurrence of this minimum as a function of~$g$ depends on the number of nearest neighbors of a lattice point but not on the lattice symmetry, on the lattice dimensionality or on the position of the Ising-like transition. In one-space dimension, the spatial correlation length associated with magnitude fluctuations and the length~$\xi_\delta$ are approximately equal, with both varying linearly with the radius of the lattice coupling.Comment: 29 pages of text plus 15 figures, uses REVTeX macros. Submitted to Phys. Rev. E

Testing for Chaos in Deterministic Systems with Noise

Recently, we introduced a new test for distinguishing regular from chaotic dynamics in deterministic dynamical systems and argued that the test had certain advantages over the traditional test for chaos using the maximal Lyapunov exponent. In this paper, we investigate the capability of the test to cope with moderate amounts of noisy data. Comparisons are made between an improved version of our test and both the ``tangent space'' and ``direct method'' for computing the maximal Lyapunov exponent. The evidence of numerical experiments, ranging from the logistic map to an eight-dimensional Lorenz system of differential equations (the Lorenz 96 system), suggests that our method is superior to tangent space methods and that it compares very favourably with direct methods

Analytical and Numerical Studies of Noise-induced Synchronization of Chaotic Systems

We study the effect that the injection of a common source of noise has on the trajectories of chaotic systems, addressing some contradictory results present in the literature. We present particular examples of 1-d maps and the Lorenz system, both in the chaotic region, and give numerical evidence showing that the addition of a common noise to different trajectories, which start from different initial conditions, leads eventually to their perfect synchronization. When synchronization occurs, the largest Lyapunov exponent becomes negative. For a simple map we are able to show this phenomenon analytically. Finally, we analyze the structural stability of the phenomenon.Comment: 10 pages including 12 postscript figures, revtex. Additional work in http://www.imedea.uib.es/Nonlinear . The paper with higher-resolution figures can be obtained from http://www.imedea.uib.es/PhysDept/publicationsDB/date.htm

Complex Dynamics and Synchronization of Delayed-Feedback Nonlinear Oscillators

We describe a flexible and modular delayed-feedback nonlinear oscillator that is capable of generating a wide range of dynamical behaviours, from periodic oscillations to high-dimensional chaos. The oscillator uses electrooptic modulation and fibre-optic transmission, with feedback and filtering implemented through real-time digital-signal processing. We consider two such oscillators that are coupled to one another, and we identify the conditions under which they will synchronize. By examining the rates of divergence or convergence between two coupled oscillators, we quantify the maximum Lyapunov exponents or transverse Lyapunov exponents of the system, and we present an experimental method to determine these rates that does not require a mathematical model of the system. Finally, we demonstrate a new adaptive control method that keeps two oscillators synchronized even when the coupling between them is changing unpredictably.Comment: 24 pages, 13 figures. To appear in Phil. Trans. R. Soc. A (special theme issue to accompany 2009 International Workshop on Delayed Complex Systems

Nonlinear time-series analysis revisited

In 1980 and 1981, two pioneering papers laid the foundation for what became known as nonlinear time-series analysis: the analysis of observed data---typically univariate---via dynamical systems theory. Based on the concept of state-space reconstruction, this set of methods allows us to compute characteristic quantities such as Lyapunov exponents and fractal dimensions, to predict the future course of the time series, and even to reconstruct the equations of motion in some cases. In practice, however, there are a number of issues that restrict the power of this approach: whether the signal accurately and thoroughly samples the dynamics, for instance, and whether it contains noise. Moreover, the numerical algorithms that we use to instantiate these ideas are not perfect; they involve approximations, scale parameters, and finite-precision arithmetic, among other things. Even so, nonlinear time-series analysis has been used to great advantage on thousands of real and synthetic data sets from a wide variety of systems ranging from roulette wheels to lasers to the human heart. Even in cases where the data do not meet the mathematical or algorithmic requirements to assure full topological conjugacy, the results of nonlinear time-series analysis can be helpful in understanding, characterizing, and predicting dynamical systems

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 kind 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.Comment: 142 Latex pgs. 41 included eps figures, submitted to Physics Reports. Related information at this http://axtnt2.phys.uniroma1.i