Speaker
Description
Stationary solutions of dynamical systems play an important role in engineering and science. These solutions are characterised by the fact that their underlying solution type does not change for infinite time. Very well-known stationary solutions are equilibria and periodic solutions. Beyond that, quasi-periodic solutions may occur. This solution type can be understood as an extension of periodic solutions, showing multiple rationally independent base frequencies. While equilibria and periodic solutions, as well as their entire trajectories, can be calculated directly using numerical methods such as shooting, this is not possible for quasi-periodic solutions as they do not exhibit a finite period. However, it can be shown that a quasi-periodic trajectory fills the surface of a two-dimensional torus densely as time approaches infinity. This enables the calculation of the underlying torus as invariant manifold in state space instead of parts of the solution trajectory. Since equilibria and periodic solutions can also be regarded as solutions on a lower-dimensional torus, all three solution types fit into a common framework.
Using a parametrization in torus coordinates (so-called “hyper-times”), the MATLAB toolbox CoSTAR (Continuation of Solution Torus AppRoximations), developed by the authors, aims at calculating parameter-dependent solution branches of the underlying tori. Currently, the toolbox features Fourier-Galerkin (i.e. Harmonic Balance) methods, Finite Difference methods as well as shooting methods for periodic and quasi-periodic solutions. As these methods exhibit individual advantages and disadvantages, a modular design allows to easily compare the results between the implemented approximation methods. Apart from that, stability analysis and bifurcation detection are of great interest in engineering and science. While the stability of equilibria and periodic solutions can be calculated using well-established methods, stability computation is still challenging for quasi-periodic solutions. A method for stability computation of quasi-periodic solutions, based on Lyapunov-exponents and developed by the research group, has been implemented in CoSTAR for the quasi-periodic shooting method [1,2]. This enables the continuation and stability calculation of each solution type within a unified framework.
To demonstrate the applicability of the toolbox, a continuation example is presented.
Bibliography
[1] Fiedler, R., Hetzler, H. & Bäuerle, S. Efficient numerical calculation of Lyapunov-exponents and stability assessment for quasi-periodic motions in nonlinear systems. Nonlinear Dyn 112, 8299–8327 (2024). https://doi.org/10.1007/s11071-024-09497-9
[2] Hetzler, H. & Bäuerle, S. 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. GAMM-Mitteilungen 46 (2023), e202300006.
https://doi.org/10.1002/gamm.202300006
