Synchronization

Synchronization is a universal phenomenon that occurs in many scientific and technical disciplines. Dynamic systems can produce different types of solutions. Periodic solutions are of great importance in the technical sciences. These generally occur when a system is subject to one or more excitation mechanisms. External excitation, self-excitation and parameter excitation are of central importance here. If at least two mechanisms are present, so-called quasi-periodic solutions can occur, which have a discrete frequency spectrum but do not have a finite period. The calculation of quasi-periodic solutions as a whole requires special methods. It should be mentioned here that there are various approaches which require different levels of detailed system knowledge. In general, it is desirable for the calculation methods themselves to require as little system knowledge as possible. One such approach is the so-called “hyper-time parameterization”, which depends only on the solution type itself and the number of so-called base frequencies.

When investigating parameter-dependent systems that show quasi-periodic solutions, synchronization may occur. This is characterized by the fact that the number of base frequencies is reduced. This is of crucial importance, as the calculation approach depends on the number of base frequencies and loses its validity at the moment of synchronization. It is therefore crucial to be able to detect occurring synchronizations. Unfortunately, there is no closed bifurcation theory for quasi-periodic solutions (e.g. based on Lyapunov exponents), so detection with test functions is not possible.

The so-called hyper-time parameterization is used as parameterization. This allows the quasi-periodic solution to be calculated as a whole and independently of the solution stability. The resulting partial differential equation is called invariance equation. This equation is solved using standard discretization methods, such as the finite difference method, the multifrequency harmonic balance or the quasi-periodic shooting method. From the solution of the invariance equation, maximum amplitudes can be identified very easily, which are of particular interest in the engineering sciences. However, the hyper-time parameterization is only valid as long as the number of so-called base frequencies remains unchanged. This is particularly important if the solution is synchronized.

There are different types of synchronization, depending on the system and the selected parameter combination. Two important cases are the so-called frequency locking and the suppression of natural dynamics. The first step is to investigate the properties of the solution and identify measures that can be used to detect the imminent synchronization of the quasi-periodic solutions. In a second step, the possibility of defining test functions on the basis of these invariant measures, with which the synchronization can be predicted numerically, will be investigated.

• A. Seifert & H. Hetzler. Numerical detection of frequency locking of quasi-periodic solutions. Proceedings of ISMA Conference 2024, Leuven, Belgien
• A. Seifert & H. Hetzler. Numerical detection of suppression of quasi-periodic solutions (2024). Proceedings in Applied Mathematics and Mechanics - PAMM 2024. doi.org/10.1002/pamm.202400111
• A. Seifert, S. Bäuerle, & H. Hetzler. Numerical detection of synchronisation phenomena in quasi-periodic solutions (2023). Proceedings in Applied Mathematics and Mechanics - PAMM 2023. https://doi.org/10.1002/pamm.202300235