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Nonautonomous Dynamics: Classification, Invariants, and Implementation

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Abstract

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 [24] 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 [5, 8], as shown in a recent paper by the authors [19].

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Authors and Affiliations

  1. National Research University “Higher School of Economics”, Nizhniy Novgorod, Russia

    V. Z. Grines & L. M. Lerman

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  1. V. Z. Grines
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  2. L. M. Lerman
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Correspondence to L. M. Lerman.

Additional information

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 68, No. 4, Differential and Functional Differential Equations, 2022.

V. Z. Grines is deceased.

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Grines, V.Z., Lerman, L.M. Nonautonomous Dynamics: Classification, Invariants, and Implementation. J Math Sci (2024). https://doi.org/10.1007/s10958-024-07238-2

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