Derivation of the Hopf Bifurcation Formula Using Lindstedt’s Perturbation Method and MACSYMA

Abstract

This paper involves the sudden appearance and growth of a periodic motion called a limit cycle in an autonomous system of two nonlinear first order ordinary differential equations. The bifurcation occurs as a parameter is tuned so that an equilibrium point goes from a stable focus to an unstable focus. The resulting limit cycle will generically occur either i) when the equilibrium is stable (in which case the limit cycle is unstable), or ii) when the equilibrium is unstable (in which case the limit cycle is stable). The Hopf bifurcation formula determines which of these two cases occurs in a given system, and depends in a complicated way on the second and third derivatives of the right-hand sides of the differential equations.

While the Hopf formula itself is well-known to many users, the usual derivations are complicated and less accessible. In this paper the Hopf formula is derived in a straightforward fashion using Lindstedt’s well-known perturbation method in conjunction with MACSYMA.