Geometric Desingularization of a Cusp Singularity in Slow–Fast Systems with Applications to Zeeman’s Examples

Abstract

The cusp singularity -a point at which two curves of fold points meet-is a prototypical example in Takens' classification of singularities in constrained equations, which also includes folds, folded saddles, folded nodes, among others. In this article, we study cusp singularities in singularly perturbed systems for sufficiently small values of the perturbation parameter, in the regime in which these systems exhibit fast and slow dynamics. Our main result is an analysis of the cusp point using the method of geometric desingularization, also known as the blow-up method, from the field of geometric singular perturbation theory. Our analysis of the cusp singularity was inspired by the nerve impulse example of Zeeman, and we also apply our main theorem to it. Finally, a brief review of geometric singular perturbation theory for the two elementary singularities from the Takens' classification occurring for the nerve impulse example -folds and folded saddles -is included to make this article self-contained.

Analysis of a slow–fast system near a cusp singularity

“…which serves as the phase space of the CDE (3) and as the set of equilibrium points of the layer equation (4). In the latter context, it is useful to recall the concept of Normally Hyperbolic Invariant Manifold (NHIM).…”

Multiple Geometric Viewpoints of Mixed Mode Dynamics Associated with Pseudo-plateau Bursting

“…The scenario depicted in Figure 5(d) shows that the singular orbits are close to the curve of cusps (since Γ F (0,δ) is so short), and consequently, the full system trajectories exhibit slow passage effects associated with a cusp. Currently, cusp singularities in slow/fast systems have been treated only for the case of a single cusp point [7], whereas our system presents a whole curve of them. Numerically, we observe that small perturbations to C change the geometry of the trajectory significantly (as highlighted by Figures 5(d) and (f)), but the precise details are unknown and left to future work.…”

Formal normal form of Ak slow–fast systems

“…For ε = 0, we can define a new time parameter τ by t = ετ . With this new time τ , we can write (1) as x = ε f (x, z, ε) z = g(x, z, ε), (2) where the prime denotes the derivative with respect to τ . An important geometric object in the study of SFSs is the slow manifold, which is defined by …”

“…In the rest of the document, we prefer to work with an SFS written as (2). Furthermore, to avoid working with an ε-parameter family of vector fields as in (2), we extend (2) by adding the trivial equation ε = 0.…”

“…In this case, the blow-up technique [4,9] can be applied to desingularize the SFS. This methodology has been successfully used in many cases, e.g., [2,7,10,11,15,16], where many of these deal with an A k -SFS with fixed k = 2 or k = 3. Briefly speaking, the blow-up technique consists in an appropriate change of coordinates under which the induced vector field is regular or has simpler singularities (hyperbolic or partially hyperbolic).…”