Complex oscillations in the delayed Fitzhugh-Nagumo equation

Abstract

Motivated by the dynamics of neuronal responses, we analyze the dynamics of the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system provides a canonical example of a canard explosion for sufficiently small delays. Beyond this regime, delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a delay-induced subcritical Bogdanov-Takens instability arising at the fold points of the S-shaped critical manifold. Underlying the transition from canard-induced to delay-induced dynamics is an abrupt switch in the nature of the Hopf bifurcation.

References in corpus (4)

• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• Inverse Scattering Transform for the Camassa-Holm equation
• On a kinetic FitzHugh-Nagumo model of neuronal network
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• The modified Camassa-Holm equation on a nonzero background: large-time asymptotics for the Cauchy problem
• On the well-posedness of a nonlocal (two-place) FORQ equation via a two-component peakon system
• A Riemann-Hilbert approach to the modified Camassa-Holm equation with step-like boundary conditions
• Liouville correspondence between the Short-Pulse Hierarchy and the Sine-Gordon Hierarchy