Phase Models Beyond Weak Coupling

Abstract

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.

Phase-amplitude reduction far beyond the weakly perturbed paradigm

“…Additionally systems with slowly varying parameters that describe adaptation [63] or memory [64] typically result in slowly decaying amplitude components and would be good candidates for this method of analysis. The implementation and analysis of high-accuracy phase reduction methods have led to the discovery and characterization of surprising behaviors observed in perturbed limit cycle oscillators that go beyond the weakly perturbed limit [65], [31], [30], [42], [13]. The reduction frameworks presented here represent a powerful strategy that can be used to understand nontrivial synchronization and entrainment behaviors that emerge in response to strong perturbations.…”

“…where T denotes the matrix transpose and Id is an appropriately sized identity matrix. The functions B k (θ) and C k j (θ) provide second order corrections to the dynamics as the state is perturbed from the periodic orbit, and have been used to identify bifurcations resulting from nonlinear interactions due to coupling [42] and to investigate how phase response depends on prior perturbations [29]. Strategies for computation of B k (θ) and C k j (θ) are discussed in detail in [28].…”

“…, ψ M ). The terms of the expansion I n (θ), I j n (θ), and I jj n (θ) can be obtained by computing periodic solutions to (41), (42), (43), respectively. All other terms can be found by computing the periodic solutions to equations of the form (44).…”

“…The relationships (41) and (42) were previously derived in [25] and [28], respectively. The above strategy allows for straightforward derivation for the relationships governing even higher order corrections to the isostable reduced equations using (44) and can be obtained with a symbolic computational package.…”

Quasi phase reduction of all-to-all strongly coupled λ−ω oscillators near incoherent states

“…Apart from incoherent states, there are more complex phenomena, such as QPS, modulated QPS, or pure collective chaos in which identical oscillators behave as "quasiphase oscillators" on top of an unsteady closed curve [19,28,29]. Recent advances extending standard phase reduction beyond the first order do not appear to be practical enough even to cover the moderate coupling regime [17,30]. Alternative methods based on phase-amplitude reduction or isostables fall short in the dimensionality reduction actually achieved [6,[31][32][33].…”