Stability and chaos in Hamiltonian dynamics

Numerical studies of integrable (Toda lattice) and nonintegrable (modified Toda lattice and Fermi-Pasta-Ulam model) Hamiltonian systems indicate the possibility of interpreting analogies and differences between the two in terms of dynamical stability properties. At large energy densities, according to Siegel's theorem, a small perturbation of an integrable system produces chaotic behavior and equipartition of energy. On the other hand, at low energy densities all the models we have considered show a similar ordered behavior and the absence of equipartition. In this region, soliton solutions of the corresponding continuum equations can provide a unified dynamical description of the properties of the discretized systems.