Stability and Generalized Hopf Bifurcation Through a Reduction Principle
Bernfeld, Stephen R.
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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 Liapunov-Schmidt method or topological degree arguments (see  and its references). Our approach, on the other hand, is based upon stability properties of the equilibrium point of the unperturbed system. Andronov et. al.  showed the fruitfulness of this approach in studying bifurcation problems in R2 (for more recent papers see Negrini and Salvadori 161 and Bernfeld and Salvadori ). 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, for example when the unperturbed system contains only two purely imaginary eigenvalues.