Coherent regimes of globally coupled dynamical systems

The paper presents a method by which the mean field dynamics of a population of dynamical systems with parameter diversity and global coupling can be described in terms of a few macroscopic degrees of freedom. The method applies to populations of any size and functional form in the region of coherence. It requires linear variation or a narrow distribution for the dispersed parameter. Although being an approximation, the method allows us to quantitatively study the collective regimes that arise as a result of diversity and coupling and to interpret the transitions among these regimes as bifurcations of the effective macroscopic degrees of freedom. To illustrate, the phenomenon of oscillator death and the route to full locking are examined for chaotic oscillators with time scale mismatch.Comment: 5 pages, 3 figure

Identical synchronization of time-continuous chaotic oscillators

Considering a system of two coupled identical chaotic oscillators, the paper first establishes the conditions of transverse stability for the fully synchronized chaotic state. Periodic orbit threshold theory is applied to determine the bifurcations through which low-periodic orbits embedded in the fully synchronized state lose their transverse stability, and the appearance of globally and locally riddled basins of attraction is discussed in terms of the sub-, respectively supercritical nature of the riddling bifurcations. We show how the introduction of a small parameter mismatch between the interacting chaotic oscillators causes a shift of the synchronization manifold. The presence of a coupling asymmetry is found to lead to further modifications of the destabilization process. Finally, the paper considers the problem of partial synchronization in a system of four coupled Rössler oscillators