On the Stability of Isolating Integrals. II. Ergodic Components in the Stochastic Region

Abstract

Computer calculations on a classical dynamical system of two degrees of freedom with the hamiltonian, \(H_{\varepsilon}=\frac{1}{2}(p_{1}^{2}+p_{2}^{2})+\frac{1}{2}(q_{1}^{2}+q_{2}^{2})+q_{1}^{2}q_{2}+\frac{1}{3}(1-2\varepsilon)q_{2}^{3}\) are presented. The Poincare mapping on ( q 2 , p 2 ) plane of this system around e =0.5 revealed that there exist several ergodic components in the stochastic region, and furthermore they have a hierarchy structure. The Lyapunov characteristic numbers are also shown to have different values for different ergodic components.