On a stable torus in a 3D system with a saddle-focus

TL;DR

This paper investigates how a stable two-dimensional torus can arise in a three-dimensional slow-fast system near a saddle-focus and in the presence of a homoclinic loop, a scenario central to elliptic bursting in mathematical neuroscience. The approach builds a return map inside the Shilnikov funnel by coupling a local linear flow near the saddle-focus with a global return map, yielding a circle map whose diffeomorphic behavior under the condition $\nu=\lambda/\rho>1$ supports a smooth invariant torus $z=h(\varphi; \mu)$ for small $\mu>0$. The main contributions are three propositions: (I) existence of a stable invariant torus, (II) a hyperbolic set with Bernoulli-shift dynamics, and (III) a stable periodic orbit, together with the possibility of torus breakdown and complex resonant phenomena. The framework integrates Afraimovich-Shilnikov theory of torus breakdown and the Shilnikov funnel, offering a unified view of transitions among bursting, tonic spiking, and quiescence in neuronal models and other slow-fast systems. The results illuminate how geometric structures near saddle-focus homoclinic configurations organize both regular and chaotic dynamics and provide a template for analyzing similar bifurcations in applied settings.

Abstract

This paper proposes a conceptual model for the onset of a stable torus near a saddle-focus equilibrium. This bifurcation scenario is typical of slow-fast systems that generate elliptic bursting in a variety of neuronal models in mathematical neuroscience. Variants of the model also capture other dynamical regimes recurring in a neighborhood of the saddle-focus. We also discuss homoclinic bifurcations for which the model assumptions are feasible.

Figures (11)

• Figure 1: Leonid Shilnikov drafting this paper at Andrey's home shortly before Christmas in 2010.
• Figure 2: (A) Geometry of an elliptic burster in phase space. The surface M$_{\rm LC}$ is the slow-motion manifold composed of stable (outer part) and unstable (inner cone) periodic orbits of the fast subsystem of the model (\ref{['fhn']}). Its fold corresponds to a saddle-node bifurcation of periodic orbits. The black dot marks a subcritical AH bifurcation, to the left and right of which the quiescent equilibrium is a stable and unstable focus, respectively. Also shown is a trajectory converging to a stable two-dimensional torus in the three-dimensional phase space. (B) Voltage waveform of the elliptic burster. (C,D) Trajectories spiraling toward a stable equilibrium and a smooth invariant circle on a cross-section transverse to M$_{\rm LC}$, corresponding to a stable periodic orbit and a two-dimensional torus, respectively.
• Figure 3: (A) A homoclinic loop to a saddle-focus with a two-dimensional unstable manifold $W^{u}(O)$ and a one-dimensional stable manifold $W^{s}(O)$. in three-dimensional phase space. (B) A sketch of a Shilnikov funnel generated by $W^{u}(O)$ of a saddle–focus: it widens by spiraling outward, forming a funnel-shaped surface, wrapping around onto an attractor: simple with regular or non-trivial with chaotic dynamics; here it is a stable periodic orbit (a green circle) -- just like as a period-doubling cascade had started.
• Figure 4: (A) A half-neighborhood of the saddle-focus $O$ is bounded by its local unstable manifold $z=0$ and the cross-sections $S_0: \{z=0, r\leq 1\}$ and $S_1: \{z=1, r\leq 1\}$. (B) Orbits of the linear system (\ref{['eq2']}) define the return map $T_0: S_0 \to S_1$.
• Figure 5: At $n=1$, the global map sends (A) $W^u_{O,loc}\cap S_1$ to a circle surrounding the origin in the cross-section $S_0$, and (B) the entire cross-section $S_1$ to an annulus surrounding the origin.