A Numerical Bifurcation Function for Homoclinic Orbits

We present a numerical method to locate periodic orbits near homoclinic orbits. Using a method of X.-B. Lin and solutions of the adjoint variational equation, we get a bifurcation function for periodic orbits whose period is asymptotic to innnity on approaching a homoclinic orbit. As a bonus, a linear predictor for continuation of the homoclinic orbit is easily available. Numerical approximation of the homoclinic orbit and the solution of the adjoint variational equation are discussed. We consider a class of methods for approximating the latter equation such that an conserved quantity is preserved. We also consider a context where the eeects of continuous symmetries of equations can be incorporated. Applying the method to an ordinary diierential equation on R 3 studied by Freire et al. we show the bifurcation function can give good agreement with path-followed solutions even down to low period. As an example of an application to a parabolic partial diierential equation, we examine the bifurcation function for a homoclinic orbit in the Kuramoto-Sivashinsky equation.