This repo presents a torus toolbox for autonomous systems and non-autonomous systems subject to harmonic external forcing. It calculates a family of tori based on parameter continuation. This toolbox includes
- ode_TR2tor: continuation of tori arising from a Neimark-Sacker or torus (TR) bifurcation periodic orbit;
- ode_isol2tor: continuation of tori from an initial solution guess;
- ode_tor2tor: continuation of tori from a saved torus solution;
- ode_BP2tor: continuation of tori along a secondary branch passing through a branch point.
Finding torus is formulated as a boundary-value problem (BVP) governed by a partial-differential equation (PDE). With the method of characteristics, the PDE is reduced to an ordinary differential equation (ODE) and the BVP problem is reduced to a multi-segments BVP, which is governed by a set of ODEs with all-to-all boundary coupling condition.
This toolbox relies on continuation package COCO, a MATLAB-based toolbox for numerical continuation. Please refer https://sourceforge.net/projects/cocotools/ for the info and installation of COCO. The multi-segments BVP is encoded with the bvp-toolbox in COCO.
You can find more details about this toolbox in the doc and examples folder. If you use this torus toolbox, please refer to [1-2]. If you have any questions, you are welcome to put your questions at Discussions (https://github.com/mingwu-li/torus_collocation/discussions) or reach me at mingwli@ethz.ch.
[1] Li, M. Tor: a toolbox for the continuation of two-dimensional tori in autonomous systems and non-autonomous systems with periodic forcing. arXiv preprint arXiv:2012.13256
[2] Dankowicz, H., & Schilder, F. (Eds.). (2013). Recipes for continuation. Society for Industrial and Applied Mathematics.