This repo presents a shooting toolbox for autonomous and non-autonomous dynamical systems.
It supports dynamical systems in both the first-order and the second-order forms. In particular,
- For first-order systems, numerical integration solvers of MATLAB are used for performing forward simulation.
- For second-order systems, Generalized alpha and Newmark integration schemes combined with Newton's iteration are used for performing forward simulation.
Automatic construction of adjoints for first-order systems is supported as well. Please refer to [1-2] for more details about the automatic generation of adjoints.
This toolbox can be used to solve two-point boundary-value problems with arbitrary boundary conditions. For example, it can be used to perform the continuation of periodic orbits. The variational equations are solved as a by product to obtain the Jacobian for the iterations of fixed points of Poincare maps. Applications of this toolbox for computing the forced response curves of high-dimensional mechanical systems with distributed nonlinearities can be found in [5-6] (1320 DOFs).
This toolbox is developed in the platform COCO (continuation core). For more details about COCO, please refer to [3-4].
When you use this toolbox and want to cite it, you may use the following
<@misc{coco-shoot, author = {Li, Mingwu and Dankowicz, Harry}, title = {A {COCO}-based shooting toolbox for dynamical systems}, howpublished = {\url{https://github.com/mingwu-li/forward}}, note = {Accessed: xxxx-xx-xx} }>
[1] Li, M., & Dankowicz, H. (2018). Staged construction of adjoints for constrained optimization of integro-differential boundary-value problems. SIAM Journal on Applied Dynamical Systems, 17(2), 1117-1151.
[2] Li, M., & Dankowicz, H. (2020). Optimization with equality and inequality constraints using parameter continuation. Applied Mathematics and Computation, 375, 125058.
[3] https://sourceforge.net/projects/cocotools/
[4] Dankowicz, H., & Schilder, F. (2013). Recipes for continuation. Society for Industrial and Applied Mathematics.
[5] Jain, S., & Haller, G. (2022). How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models. Nonlinear Dynamics, 107(2), 1417-1450.
[6] Li, M., Jain, S., & Haller, G. (2022). Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part I: Periodic response and forced response curve. Nonlinear Dynamics. https://doi.org/10.1007/s11071-022-07714-x