# MATLAB Toolbox CoSTAR

## Background

Both in theoretical and applied sciences, stationary solutions of dynamic systems are investigated. Often one wants to represent properties of stationary solutions as a function of a parameter, the so-called bifurcation parameter. This is important for the investigation of technical, biological, physical, chemical and economic systems. At the Department of Engineering Dynamics, the focus is on equilibrium solutions, as well as periodic and quasi-periodic solutions.

## Methodology

For the efficient calculation of bifurcation diagrams, often so-called predictor-corrector methods are used. These methods can compute curves defined by a root-finding-problem. Established software solutions for curve tracing usually use one single specific approximation approach (e.g. shooting, harmonic balance, collocation) to compute stationary solutions. The Matlab toolbox CoSTAR (Continuation of Solution Torus AppRoximations) developed at the department follows a modular approach, where a predictor-corrector algorithm can be combined with different approximation approaches. The research side goal here is the direct comparison of different methods in a common framework. On the application side, the modular structure allows the choice of problem-specific approximation approaches.

Determination of stationary solutions

Different approaches for the computation of equilibrium solutions, as well as for the computation of periodic and quasi-periodic solutions are implemented. For each approach to the solution an individual root-finding-problem is provided. Currently, shooting methods and Fourier-Galerkin methods have been implemented for periodic and quasi-periodic solutions.

Predictor-corrector method

To use the predictor-corrector method for curve tracking, a known curve point is first assumed. From this point, a new point near the curve is determined by means of a predictor (e.g., tangential to the curve). Using a method for solving nonlinear root-finding-problems, the next curve point is then determined in a corrector step.

The algorithm of the predictor-corrector method was programmed in such a way that it can be used independently of the solution method used.

Determination of the stability

For the determination of the stability of equilibrium solutions, as well as periodic and quasi-periodic solutions, different methods are currently implemented.

The stability investigation of equilibrium solutions can be carried out directly by means of the eigenvalue theory.

If periodic solutions are determined by means of single or multiple shooting methods, the monodromy matrix is obtained as a by-product. Thus, the stability determination is directly possible in this case. The situation is different for periodic solutions determined by a Fourier-Galerkin method. Currently, at the department an algorithm is being developed, which determines the Floquet multipliers from the solution based on the Hill method.

For the stability investigation of quasi-periodic solutions, a method is currently being implemented which can efficiently determine the Lyapunov exponents of a quasi-periodic solution using the method of characteristics. This algorithm was developed at the department.

## Outlook

In the future, further modules will be implemented, which will allow e.g., to track bifurcation points and to create stability maps.

## Source Code

The Matlab toolbox CoSTAR will be made available via GitHub for use in research and education.

The department is currently looking for student assistants to support the development of the CoSTAR Toolbox. For details, please see the job advertisement.
Are you interested? Then send us an email with your application including a transcript of grades to baeuerle[at]uni-kassel[dot]de

S. Bäuerle, A. Seifert, J. Kappauf & H. Hetzler. A continuation framework for quasi‐periodic solution branches based on different torus discretization strategies. Proceedings of ISMA Conference, Leuven, Belgium, 12.-14. September 2022.