PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 1, January 2008, Pages 111–118 S 0002-9939(07)09088-0 Article electronically published on September 27, 2007 BIFURCATION OF HOMOCLINICS JACOBO PEJSACHOWICZ (Communicated by Carmen C. Chicone) Abstract. We show that homoclinic trajectories of nonautonomous vector fields parametrized by a circle bifurcate from the stationary solution when the asymptotic stable bundles of the linearization at plus and minus infinity are “twisted” in different ways. 1. Introduction The purpose of this paper is to explain the appearance of homoclinic solutions of nonautonomous differential equations in terms of the asymptotic behavior of their linearization. The functional analytic methods used in proofs of existence of homo- clinic trajectories of differential equations are mainly of three types. The first uses Melnikov functions in order to prove the persistence of homoclinic orbits under a small change of parameter. The second, typical of Hamiltonian systems, reduces the problem of existence of a homoclinic orbit to the one of existence of a nontrivial critical point of the action functional and then applies various generalizations of the mountain pass theorem. Here we will use the third approach which parallels the analysis of Hopf-bifurcation of periodic orbits from an equilibrium. Instead of focusing on the existence of a single homoclinic, we will consider a family of dif- ferential equations parametrized by a circle. Further, using a general bifurcation principle for Fredholm maps, we will show that a branch of homoclinics bifurcat- ing from the stationary solution appears whenever the asymptotic stable bundles of the linearization at plus and minus infinity are twisted differently, i.e., noniso- morphic. The study of homoclinics based on bifurcation theory is far from being new [13]. What is new here is that the appearance of homoclinics is a consequence of the nontrivial topology of the circle. It has been observed elsewhere that the topology of the parameter space produces interesting and, sometimes, unexpected global effects on dynamics. For example, it is accountable for the appearance of Berry’s phase in the adiabatic approximation of linear Hamiltonian systems when the Hamiltonian moves around a closed loop in the parameter space [3]. In [9] the authors construct a refined version of the Conley index associated to a family of flows parametrized by a circle, which encodes information about invariant sets of the flows that cannot be obtained from a local analysis. Our arguments here will be of the same type. We first translate the problem into one of bifurcation of zeroes of a family of Fredholm maps. Then we will consider the index bundle of the family Received by the editors August 3, 2006. 2000 Mathematics Subject Classification. Primary 34C23, 58E07; Secondary 37G20, 47A53. Key words and phrases. Differential equations, homoclinics, bifurcation, index bundle. c 2007 American Mathematical Society 111