Stationary solutions in applied dynamics: A unified framework for the numerical calculation and stability assessment of periodic and quasi-periodic solutions based on invariant manifolds

Abstract

The determination of stationary solutions of dynamical systems as well as analyzing their stability is of high relevance in science and engineering. For static and periodic solutions a lot of methods are available to find stationary motions and analyze their stability. In contrast, there are only few approaches to find stationary solutions to the important class of quasi-periodic motions–which represent solutions of generalized periodicity–available so far. Furthermore, no generally applicable approach to determine their stability is readily available. This contribution presents a unified framework for the analysis of equilibria, periodic as well as quasi-periodic motions alike. To this end, the dynamical problem is changed from a formulation in terms of the trajectory to an alternative formulation based on the invariant manifold as geometrical object in the state space. Using a so-called hypertime parametrization offers a direct relation between the frequency base of the solution and the parametrization of the invariant manifold. Over the domain of hypertimes, the invariant manifold is given as solution to a PDE, which can be solved using standard methods as Finite Differences (FD), Fourier-Galerkin-methods (FGM) or quasi-periodic shooting (QPS). As a particular advantage, the invariant manifold represents the entire stationary dynamics on a finite domain even for quasi-periodic motions – whereas obtaining the same information from trajectories would require knowing them over an infinite time interval. Based on the invariant manifold, a method for stability assessment of quasi-periodic solutions by means of efficient calculation of Lyapunov-exponents is devised. Here, the basic idea is to introduce a Generalized Monodromy Mapping, which may be determined in a pre-processing step: using this mapping, the Lyapunov-exponents may efficiently be calculated by iterating this mapping.