Rescaling Invariance of Hamiltonian Systems Near Saddle Points

Abstract

In typical Hamiltonian systems chaotic motion appears because of the separatrix destruction by any small time-periodic perturbation (see Sect. 7.1.3). It forms a stochastic layer, a zone, of chaotically unstable motion near the unperturbed separatrix. In this section we study important properties of the stochastic layer, namely, a rescaling invariance of phase space of systems near the saddle points. This property of motion is generic for typical Hamiltonian systems subjected to time-periodic perturbations. The rescaling invariance consists of that the phase space (x,p) of system near hyperbolic saddle points is invariant with respect to the scaling transformation of the perturbation parameter ∈ → λ _∈ , the shift of perturbation phase λ → ϰ + π, and the phase space coordinates (x,p) → (λ^½_x, λ^1/2 _p ). The rescaling parameter λ depends only on the frequency of perturbation, Ω, and the divergence exponent γ of unperturbed orbits near the saddle point, λ = exp(2πγ/Ω). It means that the topology of phase space near the saddle point is a periodic function of log ∈ with the certain period, log λ.