On areas of attraction and repulsion in finite time dynamical systems and their numerical approximation

• Published in: Journal of difference equations and applications 2018-11, Vol.24 (11), p.1734-1755
• Main Authors: Girod, Alina, Hüls, Thorsten
• Format: Article
• Language: English
• Subjects: Approximation; areas of attraction and repulsion; Attraction; contour algorithm; Dynamical systems; finite time dynamical systems; finite time hyperbolicity; homoclinic orbits; Intersections; Invariant fibre bundles; Mathematical analysis; Mathematical models; Trajectories
• ISSN: 1023-6198; 1563-5120
• DOI: 10.1080/10236198.2018.1518438

Stable and unstable fibre bundles with respect to a fixed point or a bounded trajectory are of great dynamical relevance in (non)autonomous dynamical systems. These sets are defined via an infinite limit process. However, the dynamics of several real world models are of interest on a short time interval only. This task requires finite time concepts of attraction and repulsion that have been recently developed in the literature. The main idea consists in replacing the infinite limit process by a monotonicity criterion and in demanding the end points to lie in a small neighbourhood of the reference trajectory. Finite time areas of attraction and repulsion defined in this way are fat sets and their dimension equals the dimension of the state space. We propose an algorithm for the numerical approximation of these sets and illustrate its application to several two- and three-dimensional dynamical systems in discrete and continuous time. Intersections of areas of attraction and repulsion are also calculated, resulting in finite time homoclinic orbits.