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

Fundinger, Danny

J. Nonlinear Sci. 18, No. 4, 391-413 (2008).

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.

Reviewer: Irina V. Konopleva (Ul’yanovsk)

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

Keywords:

stable manifold; global stable set; algorithms; nonlinear dynamical systems; noninvertible map; adaptive subdivision

Software:

AnT 4.669

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