Attraction properties and non-asymptotic stability of simple heteroclinic cycles and networks in R4

Abstract

This thesis deals with stability properties of compact invariant sets, aiming at a better understanding of non-asymptotic stability for heteroclinic cycles and networks. In the first chapter we recall general aspects concerning the theory of heteroclinic cycles (such as the equivariant setting in which they are robust), and review well-known results by Krupa and Melbourne, e.g. from [28], on their asymptotic stability. Moreover, the stability index of Podvigina and Ashwin [33] is discussed together with various forms of non-asymptotic stability. The central result in this part is a connection between the signs of local stability indices and predominant asymptotic stability of a heteroclinic cycle. Building on results by all authors mentioned above, we use this relationship to characterize non-asymptotic stability of very simple heteroclinic cycles in R^4 , giving necessary and sufficient conditions for predominant asymptotic stability. The second chapter is concerned with heteroclinic networks. Some results from before are transferred without much difficulty to this context. We derive a list of all very simple networks in R^4 , and stability indices with respect to subcycles and the whole network (c- and n-indices) enable us to investigate competition within these. Results in this chapter have partially been obtained in collaboration with Sofia Castro and may be viewed as an extension and generalization of the Kirk and Silber study [23] of competition in a (B_3^−,B_3^-)-network.