Global center manifolds in singular systems

Abstract

The problem of existence of aglobal center manifold for a system of O.D.E. like

$$\left\{ {\begin{array}{*{20}c} {\dot x = A(y)x + F(x,y)} \\ {\dot y = G(x,y), (x,y) \in \mathbb{R}^n \times \mathbb{R}^m ,} \\ \end{array} } \right.$$

((*))

is considered. We give conditions onA(y), F(x, y), G(x, y) in order that a functionH: ℝ^m→ℝⁿ, with the same smoothness asA(y), F(x, y), G(x, y), exists and is such that the manifoldC={(x,y)∈ℝⁿ×ℝ^m∣x=H(y),y∈ℝ^m} is an invariant manifold for (*), and there exists ρ>0 such that any solution of (*) satisfying sup _t∈ℝ∣x(t)∣ <ρ must belong toC. This is why we callC global center manifold. Applications are given to the problem of existence of heteroclinic orbits in singular systems.