Document Type
Report
Source Publication Title
Technical Report 172
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).
Disciplines
Mathematics | Physical Sciences and Mathematics
Publication Date
11-1-1981
Language
English
License
This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.
Recommended Citation
Moauro, V. and Bertotti, M. L., "Bifurcation and Total Stability" (1981). Mathematics Technical Papers. 29.
https://mavmatrix.uta.edu/math_technicalpapers/29