The numerical computation of unstable manifolds for infinite dimensional dynamical systems by embedding techniques
This provides a computational tool for analyzing chaotic dynamics in infinite-dimensional systems, but the extension is incremental.
The authors extend a framework for computing finite-dimensional unstable manifolds to infinite-dimensional dynamical systems, demonstrating feasibility on the Kuramoto-Sivashinsky equation and Mackey-Glass delay differential equation.
In this work we extend the novel framework developed by Dellnitz, Hessel-von Molo and Ziessler to the computation of finite dimensional unstable manifolds of infinite dimensional dynamical systems. To this end, we adapt a set-oriented continuation technique developed by Dellnitz and Hohmann for the computation of such objects of finite dimensional systems with the results obtained in the work of Dellnitz, Hessel-von Molo and Ziessler. We show how to implement this approach for the analysis of partial differential equations and illustrate its feasibility by computing unstable manifolds of the one-dimensional Kuramoto-Sivashinsky equation as well as for the Mackey-Glass delay differential equation.