On singularly perturbed problems and exchange of stabilities.

Abstract

Singular perturbation theory has been used for about a century to describe models displaying different timescales, that arise in applied sciences; particularly, models displaying two timescales, namely slow time and fast time. Different techniques have been developed over time in order to analyze the limit behaviour and the stabilities of their solutions when the small parameter tends to zero. The nature of the limit equation obtained when the small parameter tends to zero plays a major role in understanding the behaviour of the solution of singularly perturbed problems. In this thesis, we analyze the behaviour of the solution of singularly perturbed problems in the following cases. First, when the limit equation displays the Allee effect. Next, when the limit equation is structurally stable or non-structurally stable and the standard Tikhonov theorem is applicable and finally, when the quasi-steady states of the degenerate equation intersect causing an exchange of stabilities. Furthermore, we perform numerical simulations in each case to support the analytic results.