Synchronization conditions and desynchronizing patterns in coupled limit-cycle and chaotic systems

Many coupling schemes for both limit-cycle and chaotic systems involve adding linear combinations of dynamical variables from various oscillators in an array of identical oscillators to each oscillator node of the array. Examples of such couplings are (nearest neighbor) diffusive coupling, all-to-all coupling, star coupling, and random linear couplings. We show that for a given oscillator type and a given choice of oscillator variables to use in the coupling arrangement, the stability of each linear coupling scheme can be calculated from the stability of any other for symmetric coupling schemes. In particular, when there are desynchronization bifurcations our approach reveals interesting patterns and relations between desynchronous modes, including the situation in which for some systems there is a limit on the number of oscillators that can be coupled and still retain synchronous chaotic behavior.