Abstract
We study the dynamics arising when two identical oscillators are coupled near a Hopf bifurcation where we assume a parameter ϵ uncouples the system at epsilon =0. Using a normal form for N=2 identical systems undergoing Hopf bifurcation, we explore the dynamical properties. Matching the normal form coefficients to a coupled Wilson–Cowan oscillator network gives an understanding of different types of behaviour that arise in a model of perceptual bistability. Notably, we find bistability between in-phase and anti-phase solutions that demonstrates the feasibility for synchronisation to act as the mechanism by which periodic inputs can be segregated (rather than via strong inhibitory coupling, as in the existing models). Using numerical continuation we confirm our theoretical analysis for small coupling strength and explore the bifurcation diagrams for large coupling strength, where the normal form approximation breaks down.
Highlights
The Hopf bifurcation is a generic and well-characterised transition that a nonlinear system can undergo to create temporal patterns of behaviour on changing a parameter
For larger network systems composed of similar subsystems that undergo oscillatory instability, when coupled together, this can lead to the formation of non-trivial spatiotemporal patterns
We emphasise that we explore a special case of two identical Hopf bifurcations that has symmetries and is close to 1 : 1 resonance
Summary
The Hopf bifurcation is a generic and well-characterised transition that a nonlinear system can undergo to create temporal patterns of behaviour on changing a parameter. Pérez-Cervera et al Journal of Mathematical Neuroscience (2019) 9:7 thorough and generic analysis of the low-dimensional dynamics of coupled oscillator systems, by means of normal form theory We use this analysis to understand the behaviour of a pair of Wilson–Cowan oscillators that arise in a model of perceptual bistability, which complements the results in [11]. A similar analysis was performed in [6] for the case of a linear coupling term, considering a particular sub-case of the normal form studied here We apply this theory to understand the appearance of a variety of oscillatory patterns in a model of perceptual bistability. We analyse the oscillatory solutions So±sc that arise from the bifurcation curves CH±B of system (3)
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