Computing two-dimensional saddle slow manifolds and associated canard orbits
Mixed-mode oscillations (MMOs) of realistic models often appear in slow-fast systems of dimension higher than three [20, 38, 49, 64, 67, 72]. Transitions between small- and large-amplitude oscillations of MMOs in such higher-dimensional systems may also be organized by slow manifolds and associated canard orbits. In this chapter, we introduce a general numerical framework for computing two-dimensional saddle slow manifolds and their stable and unstable manifolds in the context of a higher-dimensional setting, namely, systems with two fast and two slow variables. We also introduce a general method for computing and continuing canard orbits in this setting. We then illustrate our methods throughout with an extended version of the normal form of a folded node.
General approach for computing two-dimensional slow manifolds in R4
This section presents a general approach for computing suitable families of orbit segments to represent the relevant parts of two-dimensional attracting and saddle slow manifolds in R4. We consider systems of the form (1.13) with n = 2 fast variables and m = 2 slow variables. We further assume that the critical manifold S of (1.13) has a two-dimensional attracting sheet Sa and a two-dimensional saddle sheet Ss that meet at a generic fold curve F . As was described in Section 1.4, we first rescale the vector field (1.13) to obtain (1.14), where the parameter T is the integration time associated with (1.13). We then implement a two-point boundary-value problem (2PBVP) with the boundary conditions given in (1.15), where⌅ 0 and⌅ 1 are codimension-i and codimension-j hypersurfaces with i + j = 4. The choice of ⌅0 and⌅ 1 determines a one-parameter family of solutions that represent a given two-dimensional invariant manifold.
Computing attracting and repelling slow manifolds in R4
where La is a one-dimensional curve on Sa suﬃciently away from F , and ⌃a is a codimension-one submanifold transverse to S »a. The computation of a repelling slow manifold can be performed in the same way by reversing time. This approach of computing attracting and repelling slow manifolds is a direct generalization to higher dimensions of the method used in [13, 14, 45] and in Chapter 2; it can also be used for computing attracting and repelling slow manifolds in any system with two slow and an arbitrary number n 1 of fast variables.
Computing saddle slow manifolds in R4
Computations of one-dimensional saddle slow manifolds and associated stable and un-stable manifolds were performed with shooting methods , iterative methods  and a 2PBVP setup . The flow on one-dimensional saddle slow manifolds is very simple because it has only one degree of freedom. In contrast, the flow on a two-dimensional saddle slow manifold may be quite intricate with subsets of trajectories that have di↵erent properties.
Here, we introduce a method for computing two-dimensional saddle slow manifolds in R4. This is a challenge because one needs to take into account the nature of the flow on the manifold. In addition, the existence of expanding and contracting fast normal directions means that a saddle slow manifold is not uniformly attracting in either forward or backward time. To deal with these issues, we define suitable 2PBVPs to compute di↵erent parts of a given saddle slow manifold. To set up such a method, one needs some information about the behavior of the slow flow of the reduced system, as well as about the stable and unstable eigendirections associated with the saddle sheet of the critical manifold.
We aim to represent the saddle slow manifold S »s in systems of the form (1.13) as a collection of one-parameter families of orbit segments that approach S »s along its stable manifold W s(S »s) and leave via its unstable manifold W u(S »s). This can be achieved by implementing suitable 2PBVPs. Specifically, we represent each part of S »s as orbit segments subject to following boundary conditions
Here, ⌃0 and ⌃1 are three-dimensional submanifolds that are chosen to be transverse to the stable manifold W s(S »s) and unstable manifold W u(S »s) of the saddle slow manifold S »s, respectively; their definitions are informed by the (local) eigendirections of the equilibria of the fast subsystem that lie on Ss. In contrast, the three-dimensional submanifolds ⌃0.
Extended normal form of a folded node
Our approach of computing two-dimensional saddle slow manifolds is general, but requires some knowledge of the slow flow on the critical manifold. To illustrate it, we now introduce a concrete example of a four-dimensional vector field to provide a case study. The normal form of a folded node is a system with one fast and two slow variables [28, 79]. Here, we introduce and consider the extended vector field given by which includes a second fast variable w representing a stable fast direction. Since w does not play a role in the dynamics of the other three variables, the dynamics of x, y and z of the original model, including the slow flow on the critical manifold, remains unchanged. Hence, this system is a good test-case model for computing two-dimensional saddle slow manifolds in R4. The critical manifold of (3.4) is given by the two-dimensional parabolic surface
S := (x, y, z, w) | x = z2, and z = w ,
which has an attracting sheet Sa and a saddle sheet Ss that meet at a nondegenerate fold curve F .
where 0 denotes the derivative with respect to the slow time scale ⌧ = « t. The origin is a folded singularity, which is an equilibrium of the desingularized reduced system on F but not of the full system. Its eigenvalues as an equilibrium of the desingularized reduced system are µ and 1; thus, the folded singularity is a node for µ > 0. We first consider µ= 9.2. The slow flow on the saddle sheet Ss is the image of that on Sa under the symmetry sheet Sa and the saddle sheet Ss of the critical manifold. The magenta trajectory is the strong singular canard ⇠s, which divides Sa into funnel region and jump region. There are three types of trajectories on Sa and Ss that behave di↵erently. Orange and green trajectories on Sa, which lie on the funnel region between ⇠s and F , move to Ss through the folded node in finite time. The black trajectories on Sa move towards F and then jump in the fast direction. The black and green trajectories on Ss move away from F all the way to infinity. On the other hand, orange trajectories on Ss leave from the vicinity of the folded node and make a return to the fold curve F , where they jump in the fast direction. Note that the slow flow in Fig. 3.1 around the folded node is indeed topologically the same as the sketch shown in Fig. 1.2.
1 Introduction and background
1.1 Geometric singular perturbation theory
1.2 Systems with one fast and two slow variables
.3 Systems with two slow variables and two fast variables
1.4 Two-point boundary-value problem setup for slow-fast systems
1.5 Aims and outline of the thesis
2 Slow manifolds and twin canard orbits in an autocatalytic model
2.1 Background on the autocatalator
2.2 Twin canard orbits and ribbons of the attracting slow manifold
2.3 Termination/creation of twin canard orbits at fold bifurcations
2.4 Conclusions for the autocatalator model
3 Computing saddle slow manifolds and associated canard orbits
3.1 General approach for computing two-dimensional slow manifolds in R4
3.2 Extended normal form of a folded node
3.3 Slow manifolds of the extended normal form
3.4 Computing stable and unstable manifolds of saddle slow manifolds
3.5 Stable and unstable manifolds of the saddle slow manifold of the extended normal form
3.6 Interaction between attracting and saddle slow manifolds in R4
3.7 Canard orbits in R4 and how to compute them
3.8 Continuation of canard orbit
4 Slow manifolds and canard orbits in the full Hodgkin–Huxley model
4.1 Bifurcation diagram of the Hodgkin–Huxley model
4.2 Slow-fast analysis of the Hodgkin–Huxley model
4.3 Computing the attracting slow manifold
4.4 Computing the saddle slow manifold
4.5 Interaction between attracting and saddle slow manifolds
4.6 Computing canard orbits of the Hodgkin–Huxley model
4.7 Ribbons of the attracting slow manifolds
4.8 Continuation of canard orbits
5 Discussion and final remarks
5.1 Summary of results
5.2 Directions for future research
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