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About new properties of recurrent motions and minimal sets of dynamical systems

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The article presents a new property of recurrent motions of dynamical systems. Within a compact metric space, this property establishes the relation between motions of general type and recurrent motions. Besides, this property establishes rather simple behaviour of recurrent motions, thus naturally corroborating the classical definition given in the monograph [V.V. Nemytskii, V.V. Stepanov. Qualitative Theory of Differential Equations. URSS Publ., Moscow, 2004 (In Russian)]. Actually, the above-stated new property of recurrent motions was announced, for the first time, in the earlier article by the same authors [A.P. Afanas’ev, S. M. Dzyuba. On recurrent trajectories, minimal sets, and quasiperoidic motions of dynamical systems // Differential Equations. 2005, v. 41, № 11, p. 1544–1549]. The very same article provides a short proof for the corresponding theorem. The proof in question turned out to be too schematic. Moreover, it (the proof) includes a range of obvious gaps. Some time ago it was found that, on the basis of this new property, it is possible to show that within a compact metric space α- and ω-limit sets of each and every motion are minimal. Therefore, within a compact metric space each and every motion, which is positively (negatively) stable in the sense of Poisson, is recurrent. Those results are of obvious significance. They clearly show the reason why, at present, there are no criteria for existence of non-recurrent motions stable in the sense of Poisson. Moreover, those results show the reason why the existing attempts of creating non-recurrent motions, stable in the sense of Poisson, on compact closed manifolds turned out to be futile. At least, there are no examples of such motions. The key point of the new property of minimal sets is the stated new property of recurrent motions. That is why here, in our present article, we provide a full and detailed proof for that latter property. For the first time, the results of the present study were reported on the 28th of January, 2020 at a seminar of Dobrushin Mathematic Laboratory at the Institute for Information Transmission Problems named after A. A. Kharkevich of the Russian Academy of Sciences.

Keywords

dynamical systems; minimal sets; recurrent motions and motions stable in the sense of Poisson

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DOI

10.20310/2686-9667-2021-26-133-5-14

UDC

517.938

Pages

5-14

References

[1] V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, URSS Publ., Moscow, 2004 (In Russian). [2] J. K. Hale, Theory of Functional Differential Equations, Mir Publ., Moscow, 1984 (In Russian). [3] R. K. Miller, “Almost periodic differential equations as dynamical systems with application to existence of a.p. solutions”, Journal of Differential Equations, 1:3 (1965), 337–345. [4] L. G. Deysach, G. R. Sell, “On the existence of almost periodic motions”, The Michigan Math. J., 12:1 (1965), 87–95. [5] V. M. Millionshchikov, “Recurrent and almost periodic limit solutions of non-autonomous systems”, Differential Equations, 4:9 (1968), 1555–1559. [6] N.P. Bhatia, S.-N. Chow, “Weak attraction, minimality, recurrence, and almost periodicity in semisystems”, Funkcialaj Ekvacioj, 15 (1972), 35–59. [7] D. N. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, HPC, New York, 2009. [8] A.P. Afanas'ev, S. M. Dzyuba, On recurrent trajectories, minimal sets, and quasiperoidic motions of dynamical systems", Differential Equations, 41:11 (2005), 1544-1549. [9] A.P. Afanas'ev, S. M. Dzyuba, Weak periodic shift operator and generalized-periodic motions", Differential Equations, 49:1 (2013), 126-131. [10] A.P. Afanas'ev, S. M. Dzyuba, Poisson Stability in Dynamical and Continuous Periodic Systems, LKI Publ., Moscow, 2007 (In Russian). [11] L. Schwartz, Analisys. V. 2, Moscow, 1972 (In Russian). [12] E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, LKI Publ., Moscow, 2007 (In Russian).

Received

2020-09-08

Section of issue

Scientific articles

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