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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). |
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