Effects of symmetry on instability-driven chaotic energy transport in near-integrable, Hamiltonian systems

Various remarkable nonlinear phenomena have been confirmed, discovered and explored through computer simulations of nearly integrable, Hamiltonian, large dimensional lattices with periodic boundary conditions. Here we pick the periodic sine-Gordon example of an “a priori unstable” integrable system which is then discretized spatially by an explicit non-integrable scheme. We then study the dynamics of this fixed, near-integrable Hamiltonian lattice and explore the transitions in short-time and long-time behavior as we initialize either far from or nearby integrable homoclinic structures. For this volume we specifically highlight the effects of symmetry. We illustrate a subtle phenomenon whereby small errors, on the order of machine arithmetic and roundoff error, may break a symmetry of the data and the equations, unleashing a symmetry-breaking instability absent in the symmetric subspace, and thereby exciting significant stochastic dynamics and energy transport nonexistent in the corresponding symmetric simulation.