Stability Analysis of Singularly Perturbed Systems using Slow and Fast Manifolds

Abstract

This paper uses slow and fast manifolds to find stability regions for nonlinear singularly perturbed systems. The analysis starts with a two-stage transformation to decouple the slow and fast dynamics. The first transformation based on the deviation from the slow manifold transforms a nonlinear singularly perturbed system to a partially decoupled system where the slow subsystem is driven by the fast variables and the fast subsystem has slowly time-varying coefficients. The second transformation using the fast manifold eliminates the effect of the fast variables on the slow subsystem. For the decoupled system, the region of stability is found using the stable slow and fast manifolds of the unstable equilibrium points of the subsystem. This stability region can then be transformed to the original state space to form the stability region of the original system. Two examples are given to illustrate the results.