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 modes of continuous time oscillators that are diffusively coupled in a three-dimensional (3-D) lattice are studied in the paper via the corresponding linear invariant manifolds. Depending in an essential way on the number of oscillators composing the lattice in three volume directions, the set of possible regimes of spatiotemporal synchronization is examined. Sufficient conditions of the stability of cluster synchronization are obtained analytically for a wide class of coupled dynamical systems with complicated individual behavior. Dependence of the necessary coupling strengths for the onset of global synchronization on the number of oscillators in each lattice direction is discussed and an approximative formula is proposed. The appearance and order of stabilization of the cluster synchronization modes with increasing coupling between the oscillators are revealed for 2-D and 3-D lattices of coupled Lur'e systems and of coupled Rössler oscillators.

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