Bifurcation of periodic solutions

Abstract

Mechanical and electrical phenomena which can be described mathematically by the bifurcation or appearance of periodic solutions of a nonlinear ordinary different equation when some parameter is varied have been well-known for many years. See Minorsky [7]. In recent years, it has been observed that a number of biological and chemical phenomena can be described by bifurcation of periodic solutions and the Hopf Bifurcation Theorem (see Hopf [4]) has been widely applied. See, for examples, Hsi.i and Kazarinoff [S], Othmer [S], Othmer and Tyson [9], Poore [I I], Troy [12]. For an extensive discussion of the Hopf Theorem and applications in fluid mechanics, see Marsden and McCracken [6]. The primary purpose of this paper is to describe extensions of the Hopf Bifurcation Theorem that are obtained by applying a general bifurcation theorem proved in an earlier paper [3]. The approach used is finite-dimensional: it is based on the classical approach of PoincarC [IO], techniques introduced by Coddington and Levinson [2], and the use of topological degree. Roughly speaking, the result says that if certain smoothness conditions are satisfied and if the higher order term satisfies a simply stated nonzero condition then bifurcation occurs. The usual hypotheses about the behavior of the eigenvalues are largely avoided, but the condition on the higher order term is essential. As might be expected since degree theory is used, only existence results are obtained, i.e., we do not obtain continuous families of solutions. In Section 2, the general bifurcation problem is described. In Section 3, we outline a finite-dimensional proof of the Hopf Bifurcation Theorem and then combine the technique of this proof with the bifurcation theorem in [3] to obtain extensions of the Hopf Bifurcation Theorem. Actually we show how the extension can be made in a specific case: the case in which