Coupled nonlinear dynamical systems can exhibit a very rich range of synchronous behaviors. In the past couple of decades, numerous efforts have been made in understanding the various concepts such as generalized synchrony, phase synchrony, and lag synchrony in nonlinear systems. Our previous efforts have been focused on the process of desyncrhonization in coupled heterogeneous systems. We have shown that the desynchronization of two nonidentical nonlinear chaotic systems can be analyzed in terms of the evolution of the unstable periodic orbit structures transverse to the synchronization set. One can identify an observable critical transition, which we termed the Topological Decoherence, to describe the exponential growth of complexity of the coupled system with respect to the inherent structure originally embedded within the synchronous state (movies). For coupled nonidentical systems, complex geometrical structures, such as wrinkling, smearing, and cusps, can also resulted even when the coupled systems are in the strongly coupled “synchronous” state (pictures). For these wrinkled and or smeared "synchronous" systems, the detection of the correlation among the coupled components might be hampered.
The previous description is well suited for a small number of coupled elements. On the other extreme, one is interested in the development of synchronous behaviors in systems with a large number of coupled elements. For understanding the emergence of natural rhythm in biological systems, the Kuramoto model with a globally coupled network of phase oscillators has shown to be a valuable tool. Recently, we have extended this result to a network of nonlinear oscillators that are capable of more complex intrinsic dynamics(pictures). Furthermore, motivated by the diversity of neuronal subtypes, we have also derived analytical results for coherence transitions in a multi-population Kuramoto-type system. This should provide a usefulplatform in understanding the emergence of neural synchrony and rhythms.
EXTRA: Movies of Nonlinear Synchrony