Phase Models Beyond Weak Coupling.

We use the theory of isostable reduction to incorporate higher order effects that are lost in the first order phase reduction of coupled oscillators. We apply this theory to weakly coupled complex Ginzburg-Landau equations, a pair of conductance-based neural models, and finally to a short derivation of the Kuramoto-Sivashinsky equations. Numerical and analytical examples illustrate bifurcations occurring in coupled oscillator networks that can cause standard phase-reduction methods to fail.