Recent Advances in Coupled Oscillator Theory

Abstract

We review the theory of weakly coupled oscillators for smooth systems. We then examine situations where application of the standard theory falls short and illustrate how it can be extended. Specific examples are given to non-smooth systems with applications to the Izhikevich neuron. We then introduce the idea of isostable reduction to explore behaviors that the weak coupling paradigm cannot explain. In an additional example, we show how bifurcations that change the stability of phase locked solutions in a pair of identical coupled neurons can be understood using the notion of isostable reduction.

23 pages, 5 figures, review

References in corpus (8)

• Phase reduction approach to synchronization of nonlinear oscillators
• Chaos in generically coupled phase oscillator networks with nonpairwise interactions
• Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems
• Hopf normal form with $S_N$ symmetry and reduction to systems of nonlinearly coupled phase oscillators
• Phase-amplitude descriptions of neural oscillator models
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• Clusters in nonsmooth oscillator networks

Cited by in corpus (12)

• Generalized splay states in phase oscillator networks
• Nonlinear phase-amplitude reduction of delay-induced oscillations
• Estimating asymptotic phase and amplitude functions of limit-cycle oscillators from time series data
• Synchronization transitions in Kuramoto networks with higher-mode interaction
• Designing two-dimensional limit-cycle oscillators with prescribed trajectories and phase-response characteristics
• Optimal coupling functions for fast and global synchronization of weakly coupled limit-cycle oscillators
• Phase and amplitude description of complex oscillatory patterns in reaction-diffusion systems
• Phase autoencoder for rapid data-driven synchronization of rhythmic spatiotemporal patterns
• Sparse optimization of mutual synchronization in collectively oscillating networks
• Gaussian Process Phase Interpolation for estimating the asymptotic phase of a limit cycle oscillator from time series data
• Collective in-plane magnetization in a 2D XY macrospin system within the framework of generalized Ott-Antonsen theory
• High-Order Accuracy Computation of Coupling Functions for Strongly Coupled Oscillators