Generalized Transversality, Exchange of Stability and Hopf Bifurcation
In Hopf bifurcation theory often an exchange of stability of an equilibrium gives rise to the creation of periodic orbits for a one parameter family of differential equations. In particular, let us consider the system in Rn given by [see pdf for notation] where µ E [0,^) for µ sufficiently small, a(µ), ß(µ) and Aµ are C°° in µ with a(0) = 0 and ß(0) = 1. Assume [see pdf for notation] where [see pdf for notation] and for each µ, X, Y, Z are of order greater than one at the origin. Finally, the eigenvalues [see pdf for notation] of the (n-2) x (n-2) matrix A0 satisfy the non-resonance condition [see pdf for notation]. We shall refer to the right hand sides of (10) and (1µ) as f0(x,y,z) and fµ (x,y,z) respectively.