Homoclinic Solutions for Autonomous Ordinary Differential Equations with Nonautonomous Perturbations

Abstract

Nonautomonous ordinary differential equations, depending on two parameters μ₁ and μ₂, are considered in $\text{R}$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, $\text{R}$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 $\text{R}$⁶ where the invariant manifolds meet in dimension three and a second in $\text{R}$⁴ where these manifolds agree.