Interacting Global Invariant Manifolds and Slow Manifolds in Slow-Fast Systems

Abstract

Invariant manifolds of equilibria and periodic orbits are key objects that organize the behavior of a dynamical system both locally and globally. In slow-fast systems, that is, systems with sets of variables that evolve on two different time scales, there also exist so-called slow manifolds, along which the flow is very slow compared with the rest of the dynamics. Slow-fast systems are often found in various applications, including models for nerve conduction, physiological models, chemical models, ecological models and climate models, and slow manifolds play an important role for the overall dynamics. In particular, slow manifolds are known to organize the number of small oscillations of what are known as mixed-mode oscillations (MMOs). For slow-fast systems in R3 with two slow and one fast variables, slow manifolds are surfaces that can be either attracting or repelling, and intersections of two slow manifolds of different types are known as canard orbits. In addition, slow manifolds are locally invariant objects that may interact with invariant manifolds of equilibrium and periodic orbits, which are globally invariant objects. Such interactions may produce complicated dynamics about which only little is known from a few examples in the literature. This thesis focuses on such interactions in a slow-fast system with one fast and two slow variables. We present a numerical approach that allows for the systematic detection of canard orbits and intersections between other manifolds. The aim is to understand the consequences of the interaction between a global invariant manifold and a slow manifold for the overall dynamics in a slow-fast system. Specially, we study the generic situation of a quadratic tangency between the unstable manifold of a saddle-focus equilibrium and a repelling slow manifold. This scenario occurs in a system where the corresponding equilibrium undergoes a so-called supercritical singular Hopf bifurcation. We compute these manifolds as families of orbits segments with a two-point 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 their interaction as an organizer of large-amplitude oscillations in the dynamics. In particular, 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 Shilnikov orbits and describe their interactions with the MMOs. Furthermore, we study the overall dynamics organized by these global orbits. This involves the study of the invariant manifolds of a saddle periodic orbit to reveal different sources of recurrent dynamics, including the existence of a chaotic attractor.