Lyapunov spectrum and synchronization of piecewise linear map lattices with power-law coupling

Spatiotemporal dynamics of a digital phase-locked loop based coupled map lattice system

We explore the spatiotemporal dynamics of a coupled map lattice (CML) system, which is realized with a one dimensional array of locally coupled digital phase-locked loops (DPLLs). DPLL is a nonlinear feedback-controlled system widely used as an important building block of electronic communication systems. We derive the phase-error equation of the spatially extended system of coupled DPLLs, which resembles a form of the equation of a CML system. We carry out stability analysis for the synchronized homogeneous solutions using the circulant matrix formalism. It is shown through extensive numerical simulations that with the variation of nonlinearity parameter and coupling strength the system shows transitions among several generic features of spatiotemporal dynamics, viz., synchronized fixed point solution, frozen random pattern, pattern selection, spatiotemporal intermittency, and fully developed spatiotemporal chaos. We quantify the spatiotemporal dynamics using quantitative measures like average quadratic deviation and spatial correlation function. We emphasize that instead of using an idealized model of CML, which is usually employed to observe the spatiotemporal behaviors, we consider a real world physical system and establish the existence of spatiotemporal chaos and other patterns in this system. We also discuss the importance of the present study in engineering application like removal of clock-skew in parallel processors.

Using recurrences to characterize the hyperchaos-chaos transition

Hyperchaos occurs in a dynamical system with more than one positive Lyapunov exponent. When the equations governing the time evolution of the dynamical system are known, the transition from chaos to hyperchaos can be readily obtained when the second largest Lyapunov exponent crosses zero. If the only information available on the system is a time series, however, such method is difficult to apply. We propose the use of recurrence quantification analysis of a time series to characterize the chaos-hyperchaos transition. We present results obtained from recurrence plots of coupled chaotic piecewise-linear maps and Chua-Matsumoto circuits, but the method can be applied as well to other systems, even when one does not know their dynamical equations.

Paths to globally generalized synchronization in scale-free networks

We apply the auxiliary-system approach to study paths to globally generalized synchronization in scale-free networks of identical chaotic oscillators, including Hénon maps, logistic maps, and Lorentz oscillators. As the coupling strength epsilon between nodes of the network is increased, transitions from partially to globally generalized synchronization and intermittent behaviors near the synchronization thresholds, are found. The generalized synchronization starts from the hubs of the network and then spreads throughout the whole network with the increase of epsilon . Our result is useful for understanding the synchronization process in complex networks.

Spatial recurrence plots

We propose an extension of the recurrence plot concept to perform quantitative analyzes of roughness and disorder of spatial patterns at a fixed time. We introduce spatial recurrence plots (SRPs) as a graphical representation of the pointwise correlation matrix, in terms of a two-dimensional spatial return plot. This technique is applied to the study of complex patterns generated by coupled map lattices, which are characterized by measures of complexity based on SRPs. We show that the complexity measures we propose for SRPs provide a systematic way of investigating the distribution of spatially coherent structures, such as synchronization domains, in lattice profiles. This approach has potential for many more applications, e.g., in surface roughness analyzes.

Statistics of finite-time Lyapunov exponents in the Ulam map

The statistical properties of finite-time Lyapunov exponents at the Ulam point of the logistic map are investigated. The exact analytical expression for the autocorrelation function of one-step Lyapunov exponents is obtained, allowing the calculation of the variance of exponents computed over time intervals of length n. The variance anomalously decays as 1/n(2). The probability density of finite-time exponents noticeably deviates from the Gaussian shape, decaying with exponential tails and presenting 2(n-1) spikes that narrow and accumulate close to the mean value with increasing n. The asymptotic expression for this probability distribution function is derived. It provides an adequate smooth approximation to describe numerical histograms built for not too small n, where the finiteness of bin size trims the sharp peaks.

Validity of numerical trajectories in the synchronization transition of complex systems

We investigate the relationship between the loss of synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems. In the neighborhood of the critical transition to strongly nonhyperbolic behavior, the system undergoes on-off intermittency with respect to the synchronization state. There are potentially severe consequences of these facts on the validity of the computer-generated trajectories obtained from dynamical systems whose synchronization manifolds share the same nonhyperbolic properties.

Analytical results for coupled-map lattices with long-range interactions

We obtain exact analytical results for lattices of maps with couplings that decay with distance as r(-alpha). We analyze the effect of the coupling range on the system dynamics through the Lyapunov spectrum. For lattices whose elements are piecewise linear maps, we get an algebraic expression for the Lyapunov spectrum. When the local dynamics is given by a nonlinear map, the Lyapunov spectrum for a completely synchronized state is analytically obtained. The critical line characterizing the synchronization transition is determined from the expression for the largest transversal Lyapunov exponent. In particular, it is shown that in the thermodynamical limit, such transition is only possible for sufficiently long-range interactions, namely, for alpha<alpha(c)=d, where d is the lattice dimension.

Chaotic bursting at the onset of unstable dimension variability

Dynamical systems possessing symmetries have invariant manifolds. According to the transversal stability properties of this invariant manifold, nearby trajectories may spend long stretches of time in its vicinity before being repelled from it as a chaotic burst, after which the trajectories return to their original laminar behavior. The onset of chaotic bursting is determined by the loss of transversal stability of low-period periodic orbits embedded in the invariant manifold, in such a way that the shadowability of chaotic orbits is broken due to unstable dimension variability, characterized by finite-time Lyapunov exponents fluctuating about zero. We use a two-dimensional map with an invariant subspace to estimate shadowing distances and times from the statistical properties of the bursts in the transversal direction. A stochastic model (biased random walk with reflecting barrier) is used to relate the shadowability properties to the distribution of the finite-time Lyapunov exponents.