Revealing the intrinsic geometry of finite dimensional invariant sets of infinite dimensional dynamical systems

2019/02/23 by Gerlach, Raphael, Koltai, Péter, Dellnitz, Michael
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.08824

Abstract

Embedding techniques allow the approximations of finite dimensional attractors and manifolds of infinite dimensional dynamical systems via subdivision and continuation methods. These approximations give a topological one-to-one image of the original set. In order to additionally reveal their geometry we use diffusion mapst o find intrinsic coordinates. We illustrate our results on the unstable manifold of the one-dimensional Kuramoto--Sivashinsky equation, as well as for the attractor of the Mackey-Glass delay differential equation.

Related

• Embedding of global attractors and their dynamics
• Explicit Construction of a Robust Family of Compact Inertial Manifolds
• Spatial discretizations of generic dynamical systems
• Manifold learning of Poincaré sections reveals the topology of high-dimensional chaotic flows
• The topology of a chaotic attractor in the Kuramoto-Sivashinsky equation