Tangencies Between Global Invariant Manifolds and Slow Manifolds Near a Singular Hopf Bifurcation

Abstract

Invariant manifolds of equilibria and periodic orbits are key objects that organise the behaviour of a dynamical system both locally and globally. If multiple time scales are present in the dynamical system, there also exist so-called slow manifolds, that is, manifolds along which the flow is very slow compared with the rest of the dynamics. In particular, slow manifolds are known to organise the number of small oscillations of what are known as mixed-mode oscillations (MMOs). Slow manifolds are locally invariant objects that may interact with invariant manifolds, which are globally invariant objects; such interactions produce complicated dynamics about which only little is known from a few examples in the literature. We study the transition through a quadratic tangency between the unstable manifold of a saddle-focus equilibrium and a repelling slow manifold in a system where the corresponding equilibrium point undergoes a supercritical singular Hopf bifurcation. We compute the manifolds as families of orbits segments with a twopoint boundary value problem setup and track their intersections, referred to as connecting canard orbits, as a parameter is varied. We describe the local and global properties of the manifolds, as well as the role of the interaction as an organiser of large-amplitude oscillations in the dynamics. We find and describe recurrent dynamics in the form of MMOs, which can be continued in parameters to Shilnikov homoclinic bifurcations. We detect and identify two such homoclinic orbits and describe their interactions with the MMOs.

Bifurcations and Slow-Fast Analysis in a Cardiac Cell Model for Investigation of Early Afterdepolarizations

“…The equilibrium point, which is a saddle-node of type (1,2) with eigenvalues λ 1 = −3.91295, λ 2 = 0.0928806, and λ 3 = 4.27207, is attracting the orbit along the stable manifold W s (eq) of the equilibrium (in green) at the same time that the canard orbits generate new EADs until the orbit progresses to the other attracting sheet (see inset in Figure 3e). This phenomenon has been previously observed in other 1-fast, 2-slow variables systems [27][28][29]. Figure 4 shows how further variations in the bifurcation parameter G k lead to higher numbers of EADs as the orbit approaches the equilibrium point shown in black.…”

Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem

“…Similar analysis can be possibly carried out in this model in the regime far from the fold-Hopf bifurcation, where the SAOs in a MMO trajectory are distinctly detectable. Further analysis has been done on the Koper model to study the existence of a homoclinic orbit [19,37]. The detection of trajectories with an extraordinarily large number of small oscillations between large amplitude oscillations and a chaotic return map suggest that the trajectories possibly lie in a small neighborhood of a Shil'nikov homoclinic orbit in this model.…”

Geometric Analysis of Mixed-mode Oscillations in a Model of Electrical Activity in Human Beta-cells

“…We see that the system goes through a period-doubling cascade as g hERG increases, entering chaos, which is interrupted at g hERG ≈ 0.1537 nS/pF where a sharp transition of the lowest minimum occurs. This bifurcation, which leads to MMOs, is most likely due to the intersection of the unstable manifold of the saddle-focus and the repelling sheet of the slow manifold [11,18].…”