A Coordinate-Independent Version of Hoppensteadt’s Convergence Theorem

Abstract

The classical theorems about singular perturbation reduction (due to Tikhonov and Fenichel) are concerned with convergence on a compact time interval (in slow time) as a small parameter approaches zero. For unbounded time intervals Hoppensteadt gave a convergence theorem, but his criteria are generally not easy to apply to concrete given systems. We state and prove a convergence result for autonomous systems on unbounded time intervals which relies on criteria that are relatively easy to verify, in particular for the case of a one-dimensional slow manifold. As for applications, we discuss several reaction equations from biochemistry.

Appendix: Hoppensteadt’s Conditions

For the reader’s convenience we recall here the conditions and the main result from Hoppensteadt’s original paper [13]. Recall the notation $S_R$ from (2.3). Hoppensteadt considers a non-autonomous system that is given in Tikhonov standard form

$$\begin{aligned}&y_1'= f(\tau ,y_1,y_2,\varepsilon ) \end{aligned}$$

(5.1)

$$\begin{aligned}&y_2'=\varepsilon ^{-1}g(\tau ,y_1,y_2,\varepsilon ) \end{aligned}$$

(5.2)

with f and g defined on an open set

$$\begin{aligned}{}[0,\infty )\times D \times [0,\varepsilon _0)\subseteq [0,\infty )\times {\mathbb {R}}^s\times {\mathbb {R}}^r\times [0,\varepsilon _0)\rightarrow {\mathbb {R}}^r \end{aligned}$$

which satisfies $S_R\subseteq D$ for some $R>0$, and $f,\,g$ having values in ${\mathbb {R}}^s$ and ${\mathbb {R}}^r$, respectively. Assume that the following conditions hold:

• (I) The system

  $$\begin{aligned} y_1'&=f(\tau ,y_1,y_2,0) \end{aligned}$$

  (5.3)
  $$\begin{aligned} 0&=g(\tau ,y_1,y_2,0) \end{aligned}$$

  (5.4)

  admits a solution $Y:\,[0,\infty )\rightarrow {\mathbb {R}}^{s+r}$, $\tau \mapsto Y(\tau )$. With a suitable transformation of (5.3)–(5.4) one may assume that ${\widetilde{Y}}\equiv 0$ is a solution of the transformed system. From here on, it will be assumed that (5.3)–(5.4) admits the solution ${\widetilde{Y}}\equiv 0$.

• (II) The functions f, g and their partial derivatives with respect to $\tau$, $y_1$, $y_2$ respectively satisfy

  $$\begin{aligned} f,g,D_1f,D_2f,\partial _{\tau }g,D_1g,D_2g \in C([0,\infty )\times S_R\times [0,\varepsilon _0]). \end{aligned}$$

• (III) There exists an isolated and bounded $C^2$-solution $Y_2=Y_2(\tau ,y_1)$ of the implicit equation

  $$\begin{aligned} g(\tau ,y_1,Y_2(\tau ,y_1),0)=0 \end{aligned}$$

  for all $\tau \in [0,\infty )$ and $y_1\in S_1$. By a transformation ${\widetilde{y}}_1=y_1$ and ${\widetilde{y}}_2+Y(\tau ,y_1)=y_2$, one may obtain that $Y_2(\tau ,{\widetilde{y}}_1)=0$ for all $(\tau ,{\widetilde{y}}_1)\in [0,\infty )\times S_1$. This will be assumed for the original system (5.1)–(5.2) in the following.

• (IV) $f(\cdot ,\cdot ,0,0)$ is uniformly continuous in $[0,\infty )\times S_{1,R}$, and moreover $f(\cdot ,\cdot ,0,0)$ and $D_1f(\cdot ,\cdot ,0,0)$ are bounded in $[0,\infty )\times S_{1,R}$.

• (V) $g(\cdot ,\cdot ,\cdot ,0)$ is uniformly continuous in $[0,\infty )\times S_R$, and moreover $g(\cdot ,\cdot ,\cdot ,0)$, $\partial _{\tau }g(\cdot ,\cdot ,\cdot ,0)$, $D_1g(\cdot ,\cdot ,\cdot ,0)$ $D_2g(\cdot ,\cdot ,\cdot ,0)$ are bounded in $[0,\infty )\times S_R$.

• (VI) The solution ${\widetilde{Y}}\equiv 0$ of

  $$\begin{aligned} y'=f(\tau ,y,0,0) \end{aligned}$$

  (5.5)

  is uniformly asymptotically stable in the following sense: There exist a continuous, strictly increasing function

  $$\begin{aligned} d:[0,\infty )\rightarrow [0,\infty )\quad \text {with }d(0)=0 \end{aligned}$$

  and a continuous, strictly decreasing function $\sigma :[0,\infty )\rightarrow [0,\infty )$ with $\lim _{s\rightarrow \infty }\sigma (s)=0$, such that for every solution $\Phi (\tau ,z)$ of (5.5) with initial value $y_1(0)=z\in S_{1,R}$ and all $\tau \ge 0$ one has

  $$\begin{aligned} \left| \Phi (\tau ,z)\right| _1\le d(\left| z\right| _1)\cdot \sigma (\tau ). \end{aligned}$$

• (VII) For all $(\alpha ,\beta )\in [0,\infty )\times S_{1,R}$ the solution ${\widetilde{Y}}\equiv 0$ of

  $$\begin{aligned} {\dot{x}}=g(\alpha ,\beta ,x,0) \end{aligned}$$

  is uniformly asymptotically stable in the following sense: There exist a continuous, strictly increasing function

  $$\begin{aligned} e:[0,\infty )\rightarrow [0,\infty )\quad \text {with }e(0)=0 \end{aligned}$$

  and a continuous, strictly decreasing function $\rho :[0,\infty )\rightarrow [0,\infty )$ with $\lim _{s\rightarrow \infty }\rho (s)=0$, such that for every solution $\Psi (t,x_{0};\alpha ,\beta )$ of the equation with initial value $x(0)=x_{0}\in S_{2,R}$ and parameters $(\alpha ,\beta )\in [0,\infty )\times S_{1,R}$ and all $t\ge 0$ one has

  $$\begin{aligned} \left| \Psi (t,x_0;\alpha ,\beta )\right| _1\le e(\left| x_0\right| _1)\cdot \rho (t). \end{aligned}$$

Given these assumptions, Hoppensteadt’s main result [13] can be stated as follows:

Theorem 5.1

There exists a compact neighborhood $K\subset S_R$ of 0 and $\varepsilon _0^*\in (0,\varepsilon _0)$ such that the solution $\Phi (t,y_0,\varepsilon )$ of (5.1)–(5.2) with initial value $y(0)=y_0:=(y_{1,0},y_{2,0})\in K$ at $\tau =0$ exists for all positive times provided that $0<\varepsilon <\varepsilon _0^*$. Moreover $\Phi (t,y_0,\varepsilon )$ converges uniformly on all closed subsets of $(0,\infty )$ towards the solution of (5.3)–(5.4) with initial value $y_1(0)=y_{1,0}$, as $\varepsilon \rightarrow 0$.