A Numerical Bifurcation Function for Homoclinic Orbits

Abstract

In this chapter we adapt the discussion in Ashwin/Mei [26] and consider a numerical method for studying (quasi-) periodic solutions near a homoclinic orbit of the parameter-dependent autonomous differential equation

$$ \mathop x\limits^. \left( t \right) = f\left( {x\left( t \right),\lambda } \right) $$

((8.1))

where f : R ⁿ x R ^p → R ⁿ is a C ^k-continuous (k ≥ 2) mapping and λ∈R ^p represents the control parameters. This ODE may be thought of as a large system derived from a spatial discretization or from ansatz for traveling waves of reaction-diffusion equations. In one-parameter problems homoclinic orbits arise typically as limits of a branch of periodic solutions when the periods tend to infinity and the solutions stay bounded in phase space (cf. Guckenheimer/Holmes [144], Glendinning [126], Doedel/Kernevez [94] and Rinzel [251]). Dynamics near a homoclinic orbit reveals long time behavior of a system. It gives also hints on global bifurcations, namely bifurcation of homoclinic orbits. This is a complementary to the local bifurcations which we have studied with the Liapunov-Schmidt method and the center manifold theory.