Melnikov distance

One of the main tools for determining the existence of (or non-existence of) chaos in a perturbed Hamiltonian system is Melnikov theory. In this theory, the distance between the stable and unstable manifolds of the perturbed system is calculated up to the first-order term. Consider a smooth dynamical system $\ddot x = f(x) + \epsilon g(t)$, with $\epsilon \ge 0$ and $g(t)$ periodic with period $T$. Suppose for $\epsilon = 0$ the system has a hyperbolic fixed point x₀ and a homoclinic orbit $\phi(t)$ corresponding to this fixed point. Then for sufficiently small $\epsilon \ne 0$ there exists a T-periodic hyperbolic solution. The stable and unstable manifolds of this periodic solution intersect transversally. The distance between these manifolds measured along a direction that is perpendicular to the unperturbed homoclinc orbit $\phi(t)$ is called the Melnikov distance. If $d(t)$ denotes this distance, then $d(t) = \epsilon (M(t) + O(\epsilon))$. The function $M(t)$ is called the Melnikov function.

References

• Guckenheimer, John; Holmes, Philip (1983), Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Applied Mathematical Sciences 42, New York: Springer-Verlag, doi:10.1007/978-1-4612-1140-2, ISBN 0-387-90819-6, MR 709768.