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.

Figure 1

Above, the intersection of ${\epsilon }_a$ and ${\epsilon }_s$ as given by (9) define regions for which locking (either synchronous or antiphase) is stable for the CGL model (7) when $q=1$ and $(d,\epsilon )$ vary. Synchrony is only stable in regions I,II; antiphase is only stable in I,III. Thick solid and dashed lines show ${\epsilon }_{s,\text{approx}}$ and ${\epsilon }_{a,\text{approx}}$, respectively, determined from the higher order phase approximation.

Figure 2

Solid black lines in panels $A$ and $C$ show the phase differences from full simulations of (11) for different initial conditions and coupling strengths. Here, $\varphi$ is inferred directly from numerical simulations of the full model (11). Panels $B$ and $D$ illustrate corresponding coupling functions calculated according to (5). Dashed (respectively, solid) red lines denote unstable (respectively, stable) fixed points predicted by the second order accurate coupling functions. The loss of stability due to changes in coupling strength can never be predicted by first order accurate methods alone because the shape of the resulting coupling function cannot change with $\epsilon$. On the other hand, the second order accurate strategy used here reflects the observed behavior perfectly.