Persistent clusters in lattices of coupled nonidentical chaotic systems

Abstract

Two-dimensional ͑2D͒ lattices of diffusively coupled chaotic oscillators are studied. In previous work, it was shown that various cluster synchronization regimes exist when the oscillators are identical. Here, analytical and numerical studies allow us to conclude that these cluster synchronization regimes persist when the chaotic oscillators have slightly different parameters. In the analytical approach, the stability of almost-perfect synchronization regimes is proved via the Lyapunov function method for a wide class of systems, and the synchronization error is estimated. Examples include a 2D lattice of nonidentical Lorenz systems with scalar diffusive coupling. In the numerical study, it is shown that in lattices of Lorenz and Rössler systems the cluster synchronization regimes are stable and robust against up to 10%-15% parameter mismatch and against small noise. © 2003 American Institute of Physics. ͓DOI: 10.1063/1.1514202͔Lattices of coupled chaotic oscillators model many systems of interest in physics, biology, and engineering. In particular, the phenomenon of cluster synchronization, i.e., the synchronization of groups of oscillators, has received much attention. This phenomenon depends heavily on the coupling configuration between the oscillators. While in a network of globally coupled identical oscillators in principle any subset of the oscillators may synchronize, in diffusively coupled oscillators only very few of the decompositions into subsets of synchronized oscillators are possible. They have been characterized in recent papers in detail "see Refs. 14-16…. In that work, the oscillators were assumed to be identical and the symmetries of the resulting system of coupled oscillators were exploited. In more realistic models of physical systems, however, the individual oscillators have slightly different parameters and therefore the perfect symmetries in the coupled systems no longer exist. Similarly, perfect cluster synchronization cannot exist anymore, but approximate synchronization is still possible. The question then arises whether the cluster synchronization regimes that are observed in systems with identical oscillators persist approximatively under small parameter mismatch and the addition of small noise. This paper gives a positive answer and explores the limit of approximate synchronization both analytically and numerically.

Cluster synchronization in networks of coupled nonidentical dynamical systems

“…[22] and non-identical dynamical systems in Ref. [28], where the oscillators are coupled via interor/and intra-cluster manners. Ref.…”

Obtaining certificates for complete synchronisation of coupled oscillators

“…Sufficient conditions that guarantee global complete stability of the synchronised state of a system of coupled identical oscillators and that can be checked analytically -albeit not easily -were developed independently of each other by Wu [6,21] and Belykh and colleagues [5,22]. The approaches are different but related and based on graph theory and Lyapunov stability theory.…”

“…It can exhibit periodic and chaotic behaviour. In several papers by Belykh et al (for instance, see [5,22]), a network of coupled identical Lorenz oscillators was analysed. We now use the results presented so far in this paper to obtain lower bounds for the coupling strength that guarantees global complete synchronisation and compare our findings with those of [5].…”

“…This property leads to exponential stability of equilibria. In [28], Lewis studied autonomous dynamical systems which fulfill the contraction property presented in this paper as condition (22). Slotine [13] modernised Lewis's work and made it known to the wider audience under the name of contraction theory by applying it to problems in engineering.…”

Synchrony in tritrophic food chain metacommunities

“…Most methods for global synchronization of periodic and chaotic oscillators are based on the eigenvalues of the connectivity matrix and on the dynamics of the single oscillator [3,8,19,43,62,63]. An alternative approach, called connection graph method (CGM) [4], combines the Liapunov function approach with graph theoretical arguments.…”