Toward the calculation of higher-dimensional stable manifolds and stable sets for noninvertible and piecewise-smooth maps.(English)Zbl 1167.37039

Stable manifolds can be considered as fundamental building blocks of dynamical systems. Together with its sibling, the unstable manifold, each stable manifold might provide important information about a system’s global dynamics. The stable and unstable manifold may indicate, for example, boundaries of basins of attraction or, if a stable and unstable manifold intersect, the occurrence of chaotic dynamics. In this work the author presents a method to compute the stable manifolds and sets of maps in general. Neither the system’s inverse nor its Jacobian is required. It is also not limited to one-dimensional stable manifolds and sets. Theoretically, the method might be capable of computing manifolds independent of their dimension, though in practice it applied it only to get one- and two-dimensional stable manifolds and sets.
For a given map $$f:\mathbb{R}^d \to \mathbb{R}^d$$, the proposed method is capable of yielding large parts of stable manifolds and sets within a certain compact region $$M \subset \mathbb{R}^d$$. The algorithm divides the region $$M$$ in sets and uses an adaptive subdivision technique to approximate an outer covering of the manifolds. The use of suggested method is illustrated by computation of one- and two-dimensional stable manifolds and global stable sets.

MSC:

 37M20 Computational methods for bifurcation problems in dynamical systems 37M99 Approximation methods and numerical treatment of dynamical systems 37-04 Software, source code, etc. for problems pertaining to dynamical systems and ergodic theory 37C05 Dynamical systems involving smooth mappings and diffeomorphisms 37E99 Low-dimensional dynamical systems

AnT 4.669
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