Melnikov distance

One of the main tools to determine the existence of non-existence of chaos in a perturbed hamiltonian system is the Melnikov theory. In this theory, the distance between the stable and ustable manifolds of the perturbed system is calculated up to the first order term.

Example

Consider a smooth dynamical system ${\ddot {x}}=f(x)+\epsilon g(t)$, with $\epsilon \geq 0$ 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 $\epsilon \neq 0$ 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 J and Holmes P 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Berlin: Springer)