Numerical detection of synchronization phenomena in quasi-periodic solutions

In science and technology, dynamical systems can show so-called quasi-periodic solutions. These solutions are composed of two or more base frequencies. The solution in the time domain can be represented by an invariant manifold. To parametrize the invariant manifold, we choose the hyper-time parametrization. If quasi-periodic solutions branches are continued by means of a path continuation, the phenomenon of synchronization may occur. This is important, because the hyper-time parametrization is only valid, as long as the number of base frequencies remains unchanged. Therefore, it is essential to detect a parametrization to a synchronization point. Synchronization can happen in different types. We address the mechanism of suppression, where one base frequency becomes suppressed until its amplitude vanishes. This corresponds to the quasi-periodic solution ending in a Neimark–Sacker bifurcation. We present a method to derive a scalar measure from the quasi-periodic solution in the hyper-time parametrization, to detect an approach to a Neimark–Sacker bifurcation while continuing the solution branch.