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New properties of recurrent motions and limit sets of dynamical systems

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In the earlier article by the authors [A.P. Afanas’ev, S. M. Dzyuba “About new properties of recurrent motions and minimal sets of dynamical systems”, Russian Universities Reports. Mathematics, 26:133 (2021), 5–14] a connection between general motions and recurrent motions in a compact metric space is established, and a very simple behavior of recurrent motions is proved. Based on these results, we introduce here a new definition of recurrent motion which, in contrast to the one widely used in modern literature, provides fairly complete information about the structure of a recurrent motion as a function of time and, therefore, is more illustrative. At the same time, we show that in an abstract metric space, the proposed definition is equivalent to Birkhoff’s definition and is equivalent to the generally accepted modern definition in a complete metric space. Necessary and sufficient conditions for recurrence (in the sense of the definition proposed in the article) of a motion in a compact metric space are obtained. It is proved that α- and ω-limit sets of any motion are minimal in a compact metric space (this assertion was announced in an earlier paper by the authors). From the minimality of α- and ω-limit sets, it is deduced that in a compact metric space, each positively (negatively) Poisson-stable point lies on the trajectory of a recurrent motion, i.e. is a point of a minimal set, and thus, in a compact metric space with a finite positive invariant measure almost all points are points of minimal sets.

Keywords

dynamical systems, minimal sets, recurrent and stable in the sense of Poisson Motions

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DOI

10.20310/2686-9667-2022-27-137-5-15

UDC

517.938

Pages

5-15

References

[1] A.A. Markov, “Sur une proprietґe gґenґerale des ensembles minimaux de Birkhoff”, C.R. Acad. Sci., 193 (1931), 823–825. [2] V.V. Nemytskii, V.V. Stepanov, Qualitative Theory of Differential Equations, URSS Publ., Moscow, 2004 (In Russian). [3] G.D. Birkhoff, Dynamical Systems, Udm. University Publ., Izhevsk, 1999 (In Russian). [4] A.P. Afanas’ev, S.M. Dzyuba, “About new properties of recurrent motions and minimal sets of dynamical systems”, Russian Universities Reports. Mathematics, 26:133 (2021), 5–14 (In Russian). [5] A.P. Afanas’ev, S.M. Dzyuba, “Method for constructing minimal sets of dynamical systems”, Differential Equations, 51:7 (2015), 831–837. [6] A.P. Afanas’ev, S.M. Dzyuba, Poisson Stability in Dynamical and Continuous Periodic Systems, LKI Publ., Moscow, 2007 (In Russian). [7] J.K. Hale, Theory of Functional Differential Equations, Mir Publ., Moscow, 1984 (In Russian). [8] E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, LKI Publ., Moscow, 2007 (In Russian). [9] S.Kh. Aranson, “The absence of nonclosed Poisson-stable semitrajectories and trajectories doubly asymptotic to a double limit cycle for dynamical systems of the first degree of structural instability on orientable two-dimensional manifolds”, Math. USSR-Sb., 5:2 (1968), 205–219.

Received

2021-12-17

Section of issue

Scientific articles

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