Numerical detection of suppression of quasi-periodic solutions

Dynamical 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.