April 15, 2019

A defining characteristic of synchronization in coupled systems is that the resulting dynamics occurs on a lower dimensional invariant submanifold S of the phase space P. The confinement of the collective dynamics to S can be viewed as a constraint, with the coupling playing the role of forces of constraint. The process can also be inverted: for many cases coupling functions can be designed in order to lead the dynamics in coupled systems to a specific target submanifold. The resulting dynamical state is one of generalized synchrony.

For a system of globally coupled chaotic oscillators, when the submanifold S is attracting in all transverse directions this state of generalized synchrony is stable. Several distinct coupling functions can achieve the same state of generalised synchrony, although with different stability properties. The existence of a subset of unstable directions corresponds to unusual dynamical patterns such as the so-called chimera states. For systems coupled in a bipartite topology, states with distinctive temporal patterns can be observed.