A continuation principle for periodic solutions of forced motion equations on manifolds and applications to bifurcation theory

Abstract

We give a continuation principle for forced oscillations of second order differential equations on not necessarily compact diίferentiable manifolds. A topological sufficient condition for an equilibrium point to be a bifurcation point for periodic orbits is a straightforward consequence of such a continuation principle. Known results on open sets of euclidean spaces as well as a recent continuation principle for forced oscillations on compact manifolds with nonzero Euler-Poincare characteristic are also included as particular cases. 0. Introduction. Let M be a smooth (boundaryless) ra-dimensional manifold in W1 and consider on M a time dependent Γ-periodic tangent vector field, i.e. a continuous map /: R x M —> W1 with the property that, for all (ί,ί) € RxM,/(/,ί) is tangent to M at q and f(t + T , q) = f(t, q). The map / may be interpreted as a (periodic) force acting on a mass point q (of mass 1) constrained on M. A forced (or harmonic) oscillation on M is a Γ-periodic solution of the motion problem associated to the force /. In [FP4], in the attempt to solve the conjecture about the existence of forced oscillations for the spherical pendulum (i.e. for the case M = S2, the two dimensional sphere), we have studied the one-parameter motion problem associated to the force λf, λ > 0. In this context, we say that (λ, x) is a solution (pair) of the problem, if λ > 0 and x: R -> M is a forced oscillation corresponding to λf. Let us denote by X the set of all solution pairs. Since any point q e M is a rest point of the inertial problem (i.e. the motion problem with λ = 0), the constraint M may be regarded as a subset of X by means of the embedding q >-+ (0, q). With this in mind, we say that M is the manifold of trivial solutions of X and, consequently, any element of X\M will be a nontrivial solution (pair). We observe that in the nonflat case one may have nontrivial solutions even when λ = 0. Closed geodesies may be, in fact, Γ-periodic orbits if they have appropriate speed.