Slow Unfoldings of Contact Singularities in Singularly Perturbed Systems Beyond the Standard Form

Abstract

We develop the contact singularity theory for singularly perturbed (or ‘slow–fast’) vector fields of the general form $z' = H(z,\varepsilon )$, $z\in {\mathbb {R}}^n$ and $0 < \varepsilon \ll 1$. Our main result is the derivation of computable, coordinate-independent defining equations for contact singularities under an assumption that the leading-order term of the vector field admits a suitable factorization. This factorization can in turn be computed explicitly in a wide variety of applications. We demonstrate these computable criteria by locating contact folds and, for the first time, contact cusps in general slow–fast models of biochemical oscillators and the Yamada model for self-pulsating lasers.

Appendix

We write down basic definitions for jet spaces and the contact group of diffeomorphisms (see, e.g., Golubitsky and Guillemin 1973; Izumiya et al. 2015 for a full treatment of the standard singularity theory).

Definition 16

The k-jet space $J^{k}(n,m)$ of smooth germs $f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}^m$ is defined by

$$\begin{aligned} J^{k}(n,m)= & {} {\mathscr {M}}_n \cdot {\mathscr {E}}(n,m)/ {\mathscr {M}}^{k+1}_n \cdot {\mathscr {E}}(n,m), \end{aligned}$$

where

$$\begin{aligned} {\mathscr {E}}(n,m)= & {} ({\mathscr {E}}_n)^m \end{aligned}$$

is the direct product of m copies of the set ${\mathscr {E}}_n$ of smooth germs from ${\mathbb {R}}^n$ to ${\mathbb {R}}$,

$$\begin{aligned} {\mathscr {M}}_n= & {} {\mathscr {E}}_n \cdot \{x_1, \ldots , x_n\} \end{aligned}$$

is the unique maximal ideal of germs vanishing at the origin, and

$$\begin{aligned} {\mathscr {M}}^k_n= & {} {\mathscr {E}}_n \cdot \{ x_1^{i_1}, \ldots , x_n^{i_n}, i_1 + \cdots + i_n = k\} \end{aligned}$$

is the set of germs with vanishing partial derivatives of order less than or equal to $k-1$ at the origin.

Remark 16

The set $J^k(n,m)$ may be identified with the set of polynomials of total degree less than or equal to k.

The definition of contact classes used in the paper is due to Mather:

Definition 17

The contact group $\mathscr {K}$ is the set of germs of diffeomorphisms of $({\mathbb {R}}^n \times {\mathbb {R}}^m, (0,0))$ which can be written in the form

$$\begin{aligned} H(x,y)= & {} (h(x),H_1(x,y)), \end{aligned}$$

where h acts on the right (i.e., $h \cdot f = f \circ h^{-1}$) and $H_1(x,0) = 0$ for x near 0. We say that f is $\mathscr {K}$-equivalent to g if g lies in the group orbit of f. We refer to this as the contact class of f.

Remark 17

Suppose $f,g \in {\mathscr {M}}_n \cdot {\mathscr {E}}(n,m)$ and $k = (h,H) \in \mathscr {K}$. Then $g = k \cdot f$ if and only if

$$\begin{aligned} (x,g(x))= & {} H(h^{-1}(x),f(h^{-1}(x))). \end{aligned}$$

Observe that H sends the graph of f to the graph of g near 0 (i.e., the zero sets of $\mathscr {K}$-equivalent germs are diffeomorphic).