Numerical detection of suppression of quasi‐periodic solutions

Abstract

AbstractDynamical systems can show so‐called quasi‐periodic solutions, which are composed of two or more so‐called incommensurable frequencies. Solving these systems in the time‐domain is not favorable, due to the fact that quasi‐periodic solutions have no finite periodicity, thus it is unclear how long simulations must be carried out in order to capture all relevant information. The time‐solution however fills a toroidal manifold densely. To calculate the invariant manifold directly one can use the hyper‐time parametrization. One can easily extract maximum values from the hyper‐time solution. The trade‐off however is, that the parametrization loses its validity if the quasi‐periodic solution synchronizes. When continuing quasi‐periodic solution branches it is essential to detect an approach to such synchronization points. We present an algorithm, with which one can detect synchronization by suppression of quasi‐periodic solutions, on the basis of a solution in the hyper‐time parametrization.