Universality in the chaotic dynamics associated with saddle-centers critical points

We describe a new statistical pattern characteristic of saddle-center critical points in the dynamics of non-integrable Hamiltonian systems, associated with the partition of the Hamiltonian into rotational motion energy and hyperbolic motion energy pieces in a linear neighborhood of the saddle-center. The distribution of hyperbolic energy of orbits visiting a neighborhood of the saddle-center defines a statistical law that, for a large range of values of the hyperbolic energy, has the form p(z) z−γ, in a regime of high nonintegrability. We present numerical evidence to support the conjecture that this pattern is universal for any Hamiltonian system with one or several saddle-centers. The parameter γ appears as a universal characteristic associated with the chaotic dynamics of saddle-centers.