Effective and Computational Chaos

Mark Ablowitz
Department of Applied Mathematics
University of Colorado at Boulder

Abstract: A class of integrable systems which arise in physical problems, such as the nonlinear Schrodinger and Sine-Gordon equations, exhibit "effective" chaos. The complicated dynamical motion evolves from initial data which is proximate to certain geometric strcutures, called homoclinic manifolds. Perturbations of any size, depending on inital data, can induce chaotic dynamics. The phenomena can be easily observed by considering discrete dynamical systems which serve to perturb the continuous models. The perturbation can be due to truncation or roundoff effects. The chaotic evolution can be intiated with or without the evolution passing through an unperturbed homoclinic manifold. We are working with Hammack and Henderson in order to "observe" this effect in water waves experiments.