On a general Conley index continuation principle for singular perturbation problems

Abstract

In singular perturbation problems one frequently has a family (\pi_\eps)_{\eps\in]0,1]} of semiflows on a space X and a ‘singular limit flow’ \pi_0 of this family which is defined only on a subspace X_0 of X. Moreover, \pi_\epsilon converges to \pi_0 only in some ‘singular’ sense. Such a situation occurs, for example, in fast–slow systems of differential equations or in evolution equations on thin spatial domains.In this paper we prove a general singular Conley index continuation principle stating that every isolated invariant set K_0 of \pi_0 can be continued to a nearby family K_\epsilon of isolated invariant sets of \pi_\epsilon with the same Conley index. We illustrate this continuation result with damped wave equations on thin domains. This extends some results from our previous work.