Homoclinic Solutions for Autonomous Ordinary Differential Equations with Nonautonomous Perturbations

Nonautomonous ordinary differential equations, depending on two parameters μ1 and μ2, are considered in n. It is assumed that when both parameters are zero the differential equation is autonomous with a hyperbolic equilibrium and a homoclinic solution. No restriction is placed on the dimension of the phase space, n, or on the dimension of intersection of the stable and unstable manifolds. By means of the method of Lyapunov-Schmidt a bifurcation function, H, is constructed between two finite dimensional spaces where the zeros of H correspond to homoclinic solutions at nonzero parameter values. The independent variables of H consist of scalars μ1, μ2, ξ and a vector β where ξ is a phase angle and β corresponds to directions, other than along the original homoclinic solution, tangent to both the stable and unstable manifolds. When ξ is fixed the equation H = 0 yields, in general, several bifurcation curves through the origin in the μ1-μ2 plane along which there exists a homoclinic solution. When ξ is varied these become a number of wedge-shaped regions. The theory is applied to two examples, one in 6 where the invariant manifolds meet in dimension three and a second in 4 where these manifolds agree.