Liapunov direct method in approaching bifurcation problems

Abstract

A concept of total stability for continuous or discrete dynamical systems and a generalized definition of bifurcation are given: it is possible to show the link between an abrupt change of the asymptotic behaviour of a family of flows and the arising of new invariant sets, with determined asymptotic properties. The theoretical results are a contribution to the study of the behaviour of flows near an invariant compact set. They are obtained by means of an extension of Liapunov's direct method. © 1975 Fondazione Annali di Matematica Pura ed Applicata

Dynamics of commuting systems on two-dimensional manifolds

“…Every surface is homeomorphic to a sphere with handles, and two such spheres are homeomorphic if and only ff they have the same number of handles [GP,p. 124,W,Thin. 7.1].…”

On Bifurcations Arising from Unstable Equilibria and Invariant Sets

“…This line of research was initiated, in its general form, by Marchetti, Negrini, Salvadori and Scalia with the paper [5]. There it was proved that if for a family of dynamical systems, depending on a parameter *, an invariant set M is asymptotically stable for a certain value * 0 of *, and completely unstable for nearby values, then a bifurcation of M takes place at * 0 .…”

“…In the theory developed in [5] and [7], the fundamental underlying principle was that of persistence of asymptotic stability under perturbations, a consequence of which is the relation between extracritical loss of stability (defined above) and bifurcation.…”

“…DE983495 to the two-dimensional centre manifold on which the change of stability takes place.) In what follows, we will refer to the class of bifurcations considered in [5] as AS CI (asymptotic stability complete instability) bifurcations. In the paper [7] by the present authors, the requirement of [5] that M be completely unstable was weakened, and the theory was extended to the case of semidynamical systems, dropping the requirement of local compactness of the state space and assuming instead that the systems be asymptotically compact.…”

On certain generalizations of the Hopf bifurcation

“…With the paper [6], a new point of view was introduced into the theory of bifurcation (understood in the sense of the Hopf bifurcation and its generalizations). While in the traditional theory the existence of bifurcations is based on the behaviour of the eigenvalues of the linear part of a family of systems under a change of a parameter, in the paper mentioned, as well as in succeeding ones following the same general line [1-3, 5, 13, 15] (summarized in [11]) the same property is proved under the assumption of certain changes in the dynamical behaviour when a parameter reaches or passes some critical value.…”