Nonautonomous dynamics: classification, invariants, and implementation

V. Z. Grines ; L. M. Lerman

Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 68 (2022) no. 4, pp. 596-620

Résumé

The work is a brief review of the results obtained in nonautonomous dynamics based on the concept of uniform equivalence of nonautonomous systems. This approach to the study of nonautonomous systems was proposed in [10] and further developed in the works of the second author, and recently  — jointly by both authors. Such an approach seems to be fruitful and promising, since it allows one to develop a nonautonomous analogue of the theory of dynamical systems for the indicated classes of systems and give a classification of some natural classes of nonautonomous systems using combinatorial type invariants. We show this for classes of nonautonomous gradient-like vector fields on closed manifolds of dimensions one, two, and three. In the latter case, a new equivalence invariant appears, the wild embedding type for stable and unstable manifolds [14, 17], as shown in a recent paper by the authors [5].

Détail

Keywords: nonautonomous dynamics, nonautonomous vector field, gradient-like vector field, uniform equivalence, wild embedding.