Document Type
Report
Source Publication Title
Technical Report 153
Abstract
We are interested in obtaining an analysis of the bifurcating periodic orbits arising in the generalized Hopf bifurcation problems in Rn. The existence of these periodic orbits has often been obtained by using such techniques as the Lyapunov-Schmidt method or topological degree arguments (see Marsden and McCracken [8] and Hale [6] and their references). Our approach, on the other bend, is based upon stability properties of the equilibrium point of the unperturbed system. Andronov et. al. [1] showed the fruitfulness of this approach in studying bifurcation problems in R2 (for more recent papers see Negrini and Salvadori [9] and Bernfeld and Salvadori [2]). In the case of R2, in contrast to that of Rn, n > 2, the stability arguments can be effectively applied because of the Poincaré-Bendixson theory. Bifurcation problems in Rn can be reduced to that of R2 when two dimensional invariant manifolds are known to exist. The existence of such manifolds occurs, or example when the unperturbed system contains only two purely imaginary eigenvalues.
Disciplines
Mathematics | Physical Sciences and Mathematics
Publication Date
3-1-1981
Language
English
License
This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.
Recommended Citation
Salvadori, L. and Bernfeld, Stephen R., "Quasi-Invariant Manifolds, Stability, and Generalized Hopf Bifurcation" (1981). Mathematics Technical Papers. 41.
https://mavmatrix.uta.edu/math_technicalpapers/41