On the Stability of Periodic Orbits for Nonlinear Oscillator Systems in Regions Exhibiting Stochastic Behavior

A computer has been used to determine the stability character of periodic orbits for the Hamiltonian oscillator system $\mathrm{H=}\frac{1}{2}{\mathrm{(p}}_1^2{\mathrm{+p}}_2^2{\mathrm{+q}}_1^2{\mathrm{+q}}_2^2{\mathrm{)+q}}_1^2{q}_2-\frac{1}{3}{q}_2^3$ . Using procedures developed by Greene [J. Math. Phys. 9, 760 (1968)], empirical evidence has been obtained indicating that this system has a dense or near dense set of unstable periodic orbits throughout its stochastic (unstable) regions of phase space. The extent to which such stochastic regions exhibit C‐system behavior, i.e., ergodicity and mixing, is discussed. Finally, the above Hamiltonian system is shown to be intimately related to the Fermi‐Pasta‐Ulam system as well as to the Toda lattice.