Scaling invariance of the homoclinic tangle

The structure of the homoclinic tangle of 11 / 2 degrees of freedom Hamiltonian systems in the neighborhood of the saddle point is invariant under discrete rescaling of the system's parameters. The rescaling constant is derived from the separatrix map and the Melnikov formula. Invariant manifolds for the periodically modulated Duffing oscillator are computed numerically to confirm this property. The scaling is related to the recently found invariance of the separatrix map under a discrete renormalization group. A possibility to extend the scaling invariance to different systems is demonstrated. The equivalency conditions under which two systems have the similarity of their chaotic layer structure near the saddle are derived. A numerical example shows a Duffing oscillator and a pendulum (acted on by different periodic perturbations) with the same structure of the tangle.