Hopf Bifurcations Near 0:1 Resonance William F. Langford Kaijun Zhan Abstract The interactions of two Hopf bifurcations are explored, in the case for which the frequency associated with one of the Hopf bifucations is arbitrarily near 0, while the second remains bounded away from 0 (and without loss of generality is rescaled to 1). The unfolding of this codimension-three bifurcation reveals interesting codimension-one and -two bifurcations. KEYWORDS: Hopf bifurcation, resonance, codimension-three, torus, cou- pled oscillators. 1 Introduction Consider a smooth system of differential equations of the general form ˙ y = f (y,μ) (1) with state variables y ∈ R n , parameters μ ∈ R p , and an equilibrium point at the origin f (0,μ) = 0; such that equation (1) is a perturbation of one for which the linearization has eigenvalues {0, 0, i, -i} at μ = 0. In other words, we assume that the linearization D y f (0,μ) has two pairs of complex conjugate eigenvalues, near 0 and ±i respectively, and further assume that in the limit as μ → 0 these eigenvalues approach 0 and ±i in the complex plane. (We rescale time if necessary to make the second frequency 1 in the limit μ → 0). This is the meaning of the phrase “0:1 resonance” in the title. We assume that all other eigenvalues of the linearization of (1) have negative BTNA’98 Proceedings Chen, Chow, and Li (eds.), pp. 1–18. Copyright c 1999 by Springer All rights of reproduction in any form reserved. put here ISBN 1