The Existence of Transverse Homoclinic Solutions for Higher Order Equations

Differential equations are considered which contain a small parameter. When the parameter is zero the equation is autonomous with a hyperbolic equilibrium and a homoclinic solution. No restriction is placed on the dimension of the phase space or the dimension of intersection of the stable and unstable manifolds. Sufficient conditions on the perturbation terms for producing a transverse homoclinic solution and hence chaos are obtained. The technique follows the work of Palmer using exponential dichotomies and the method of Lyapunov–Schmidt.