Asymptotic analysis of subcritical Hopf–homoclinic bifurcation

Abstract

This paper discusses the mathematical analysis of a codimension two bifurcation determined by the coincidence of a subcritical Hopf bifurcation with a homoclinic orbit of the Hopf equilibrium. Our work is motivated by our previous analysis of a Hodgkin-Huxley neuron model which possesses a subcritical Hopf bifurcation [5]. In this model, the Hopf bifurcation has the additional feature that trajectories beginning near the unstable manifold of the equilibrium point return to pass through a small neighborhood of the equilibrium, that is, the Hopf bifurcation appears to be close to a homoclinic bifurcation as well. This model of the lateral pyloric (LP) cell of the lobster stomatogastric ganglion was analyzed for its ability to explain the phenomenon of spike-frequency adaptation [5], in which the time intervals between successive spikes grow longer until the cell eventually becomes quiescent. The presence of a subcritical Hopf bifurcation in this model was one identified mechanism for oscillatory trajectories to increase their period and finally collapse to a non-oscillatory solution. The analysis presented here explains the apparent proximity of homoclinic and Hopf bifurcations. We also develop an asymptotic theory for the scaling properties of the interspike intervals in a singularly perturbed system undergoing subcritical Hopf bifurcation that may be close to a codimension two subcritical Hopf-homclinic bifurcation.

Mixed-Mode Oscillations in Three Time-Scale Systems: A Prototypical Example

“…also section 1. Other mechanisms that do not explicitly involve canards have been proposed to explain MMOs; examples include break-up of an invariant torus [21], loss of stability of a Shilnikov orbit [16], slow passage through Hopf bifurcation [22], and subcritical Hopf-homoclinic bifurcation [12,13]. While these other mechanisms are consistent with some of the characteristic features of MMOs, they cannot typically explain all of them; see [2].…”

Dynamics of nearly inviscid Faraday waves in almost circular containers

“…Note that since the Hopf bifurcations are subcritical the unstable periodic orbits created at the bifurcation are not involved in this behavior, cf. [20].…”

Mixed-Mode Oscillations with Multiple Time Scales

“…Guckenheimer and Willms [95] observed that a subcritical (ordinary) Hopf bifurcation may result in large regions of the parameter space being funneled into a small neighborhood of a saddle equilibrium with unstable complex eigenvalues. After trajectories come close to the equilibrium, SAOs grow in magnitude as the trajectory spirals away from the equilibrium.…”

“…In this case, the returns are likely to come close enough to q that they will give rise to long epochs of small, slowly growing oscillations. See Guckenheimer and Willms [95] for a three-dimensional example and Guckenheimer et al [89] for a high-dimensional example occurring in a neural model. We remark that, although this mechanism for creating MMOs applies to a single-time system, the Hopf bifurcation naturally introduces a slow time scale in the system associated with the real parts of the unstable complex eigenvalues.…”