Analytic Solution for a Complex Network of Chaotic Oscillators

Chaotic evolution is generally too irregular to be captured in an analytic solution. Nonetheless, some dynamical systems do have such solutions enabling more rigorous analysis than can be achieved with numerical solutions. Here, we introduce a method of coupling solvable chaotic oscillators that maintains solvability. In fact, an analytic solution is given for an entire network of coupled oscillators. Importantly, a valid chaotic solution is shown even when the coupling topology is complex and the population of oscillators is heterogeneous. We provide a specific example of a solvable chaotic network with star topology and a hub that oscillates much faster than its leaves. We present analytic solutions as the coupling strength is varied showing states of varying degrees of global organization. The covariance of the network is derived explicity from the analytic solution characterizing the degree of synchronization across the network as the coupling strength varies. This example suggests that analytic solutions may constitute a new tool in the study of chaotic network dynamics generally.