Homoclinic Solutions for Autonomous Dynamical Systems in Arbitrary Dimension

An autonomous differential equation in Rⁿ with two parameters, μ ₁, μ ₂, is considered and it is assumed that when both parameter values are zero the equation has a hyperbolic equilibrium and a homoclinic solution. Curves are sought through the origin in the μ ₁-μ ₂ plane along which the homoclinic solution persists for nonzero parameter values. By using the method of Lyapunov–Schmidt, a function, H, is obtained between two finite-dimensional spaces where the zeros of H represent homoclinic solutions for nonzero parameter values. The implicit function theorem is applied to H in various cases. For $n = 2$ a single curve is obtained as in the work of Melnikov. When $n > 2$ and the stable and unstable manifolds of the hyperbolic equilibrium have an intersection of dimension one, a result of Palmer is achieved which, again, yields a single curve. When this dimension of intersection is greater than one, original results give multiple curves. The various cases of the theory are illustrated by seven examples: one in R², one in R³, four in R⁴, and one in R⁶.