Bifurcation and Total Stability

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Bifurcation and Total Stability

Show simple item record Moauro, V. en Bertotti, M. L. en 2010-06-09T15:21:06Z en 2010-06-09T15:21:06Z en 1981-11 en
dc.identifier.uri en
dc.description.abstract In this paper we are concerned with the problem of bifurcation of invariant sets from an invariant set with respect to a family of flows. In particular, we will suppose that such flows are defined by a one-parameter family of ordinary differential equations: [see pdf for notation] where [see pdf for notation], f is locally Lipschitzian with respect to x, f(µ,0) = 0. As is well known, bifurcation phenomenon is often associated with a drastic change of suitable stability properties. For example, let suppose that the origin 0 of Rn be, with respect to (1), asymptotically stable for µ = 0 and completely unstable (that is asymptotically stable in the past) for µ > 0. Then, in a fixed neighborhood of 0, new compact invariant sets arise for µ > 0 and µ small enough. These sets are disjoint from the origin, asymptotically stable and tend to the origin as µ tends to 0. Also these sets can be taken as the largest compact invariant sets, disjoint from the origin, contained in a fixed neighborhood of the origin. The above result is a corollary of a theorem given in [1,2] where the general phenomenon of bifurcation of invariant sets from an invariant set is considered with respect to a one-parameter family of dynamical systems (not necessarily defined by differential equations). en
dc.language.iso en_US en
dc.publisher University of Texas at Arlington en
dc.relation.ispartofseries Technical Report;172 en
dc.subject Bifurcation en
dc.subject Invariant sets en
dc.subject Flows en
dc.subject Stability properties en
dc.subject.lcsh Mathematics Research en
dc.title Bifurcation and Total Stability en
dc.type Technical Report en
dc.publisher.department Department of Mathematics en

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