Spreading in Disordered Lattices with Different Nonlinearities

We study the spreading of initially localized states in a nonlinear disordered lattice described by the nonlinear Schr\"odinger equation with random on-site potentials - a nonlinear generalization of the Anderson model of localization. We use a nonlinear diffusion equation to describe the subdiffusive spreading. To confirm the self-similar nature of the evolution we characterize the peak structure of the spreading states with help of R\'enyi entropies and in particular with the structural entropy. The latter is shown to remain constant over a wide range of time. Furthermore, we report on the dependence of the spreading exponents on the nonlinearity index in the generalized nonlinear Schr\"odinger disordered lattice, and show that these quantities are in accordance with previous theoretical estimates, based on assumptions of weak and very weak chaoticity of the dynamics.Comment: 5 pages, 6 figure

Phase synchronization in time-delay systems

Though the notion of phase synchronization has been well studied in chaotic dynamical systems without delay, it has not been realized yet in chaotic time-delay systems exhibiting non-phase coherent hyperchaotic attractors. In this article we report the first identification of phase synchronization in coupled time-delay systems exhibiting hyperchaotic attractor. We show that there is a transition from non-synchronized behavior to phase and then to generalized synchronization as a function of coupling strength. These transitions are characterized by recurrence quantification analysis, by phase differences based on a new transformation of the attractors and also by the changes in the Lyapunov exponents. We have found these transitions in coupled piece-wise linear and in Mackey-Glass time-delay systems.Comment: 4 pages, 3 Figures (To appear in Physical Review E Rapid Communication

Synchronization in driven versus autonomous coupled chaotic maps

The phenomenon of synchronization occurring in a locally coupled map lattice subject to an external drive is compared to the synchronization process in an autonomous coupled map system with similar local couplings plus a global interaction. It is shown that chaotic synchronized states in both systems are equivalent, but the collective states arising after the chaotic synchronized state becomes unstable can be different in these two systems. It is found that the external drive induces chaotic synchronization as well as synchronization of unstable periodic orbits of the local dynamics in the driven lattice. On the other hand, the addition of a global interaction in the autonomous system allows for chaotic synchronization that is not possible in a large coupled map system possessing only local couplings.Comment: 4 pages, 3 figs, accepted in Phys. Rev.

Optimal Phase Description of Chaotic Oscillators

We introduce an optimal phase description of chaotic oscillations by generalizing the concept of isochrones. On chaotic attractors possessing a general phase description, we define the optimal isophases as Poincar\'e surfaces showing return times as constant as possible. The dynamics of the resultant optimal phase is maximally decoupled of the amplitude dynamics, and provides a proper description of phase resetting of chaotic oscillations. The method is illustrated with the R\"ossler and Lorenz systems.Comment: 10 Pages, 14 Figure

Intermittent generalized synchronization in unidirectionally coupled chaotic oscillators

A new behavior type of unidirectionally coupled chaotic oscillators near the generalized synchronization transition has been detected. It has been shown that the generalized synchronization appearance is preceded by the intermitted behavior: close to threshold parameter value the coupled chaotic systems demonstrate the generalized synchronization most of the time, but there are time intervals during which the synchronized oscillations are interrupted by non-synchronous bursts. This type of the system behavior has been called intermitted generalized synchronization (IGS) by analogy with intermitted lag synchronization (ILS) [Phys. Rev. E \textbf{62}, 7497 (2000)].Comment: 8 pages, 5 figures, using epl.cls; published in Europhysics Letters. 70, 2 (2005) 169-17

Studying Attractor Symmetries by Means of Cross Correlation Sums

We use the cross correlation sum introduced recently by H. Kantz to study symmetry properties of chaotic attractors. In particular, we apply it to a system of six coupled nonlinear oscillators which was shown by Kroon et al. to have attractors with several different symmetries, and compare our results with those obtained by ``detectives" in the sense of Golubitsky et al.Comment: LaTeX file, 16 pages and 16 postscript figures; tarred, gzipped and uuencoded; submitted to 'Nonlinearity

Role of delay in the mechanism of cluster formation

We study the role of delay in phase synchronization and phenomena responsible for cluster formation in delayed coupled maps on various networks. Using numerical simulations, we demonstrate that the presence of delay may change the mechanism of unit to unit interaction. At weak coupling values, same parity delays are associated with the same phenomenon of cluster formation and exhibit similar dynamical evolution. Intermediate coupling values yield rich delay-induced driven cluster patterns. A Lyapunov function analysis sheds light on the robustness of the driven clusters observed for delayed bipartite networks. Our results reveal that delay may lead to a completely different relation, between dynamical and structural clusters, than observed for the undelayed case.Comment: 4+ pages, 4 figues, PRE Rapid Communication (in press