Invariant manifolds of singularly perturbed ordinary differential equations (Q1087696)

Consider an autonomous system \[ (1)\quad dx/dt=f(x,y,\epsilon),\quad \epsilon dy/dt=g(x,y,\epsilon), \] where \(f: D\to {\mathbb{R}}^ m\), \(g: D\to {\mathbb{R}}^ n\), \(D=D_ 1\times D_ 2\times (-\epsilon_ 0,\epsilon_ 0)\) is a bounded domain in \({\mathbb{R}}^{m+n+1}\), and \(D_ 1\) is star shaped with \(C^{\nu +1}\) boundary, \(\nu\geq 1\). The author imposes certain stability assumption on (1) and proves the existence of an asymptotically stable (unstable) invariant manifold. This invariant manifold is \(\epsilon\)-close to the so-called reduced manifold of (1). The result is applied to a three-dimensional autonomous system describing a model for nerve impulse suggested by E. C. Zeeman: \[ (2)\quad \dot x_ 1=-1-x_ 2,\quad \dot x_ 2=-2x_ 2-2y,\quad \epsilon \dot y=- (x_ 1+x_ 2y+y^ 3). \]