# Quasi-periodic Oscillations

## Background

Quasi-periodic oscillations occur as a result of the superposition of oscillations with incommensurable frequencies. Typical examples from practical applications are often due to simultaneous self-excitation and external excitation, where the self-excitation frequency is determined by the nonlinear inherent dynamics, while the frequency of the external excitation is imposed on the system from outside. If both frequencies are approximately commensurable, synchronization can occur and the system response becomes periodic - in the general case, however, quasi-periodic signals are to be expected. Typical examples from application originate from rotor dynamics, laser dynamics, population dynamics, etc. In mathematics, suitable methods for the description of quasiperiodic oscillations have already been derived, but so far these have hardly been used for the investigation of engineering applications.

The current focus is on the investigation and validation of different methods for the determination of quasi-periodic solutions. Here, the approach via invariant manifolds is chosen.

A path continuation algorithm has been developed, which can track periodic as well as quasi-periodic solutions. Furthermore, a method has been developed at the department which can determine the stability of quasi-periodic solutions based on the method of characteristics. This allows a detection of bifurcation points, which serve as a basis for an investigation of synchronization effects.

With respect to operating points of practical applications, the determination and tracking of quasi-periodic attractors or repellors is of interest. Furthermore, the implemented methods should also be able to detect synchronization effects.

## Methodology

Currently, approaches for the description of stationary solutions are pursued by means of motion invariants. These can be represented as solutions of nonlinear partial differential equations, which can be solved numerically by standard methods (FEM, FD). The invariant manifolds determined in this way can then be visualized, for example, as a torus (see figure). Furthermore, current investigations aim at the properties of the motion invariants with respect to their stability. In addition, synchronization effects and the associated dimensional changes of the invariant torus play a role in current research.

The stability is determined by a calculation of the Lyapunov exponents. At the department a method has been developed that allows the efficient calculation of Lyapunov exponents for quasi-periodic solutions. Here, a mapping function is identified which describes the evolution of perturbations.

The investigation and validation of the method could already be carried out on simple models, including an academic one (externally excited Van der Pol equation).