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Localizing sets and trajectory behavior

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Abstract

The problem is considered of finding domains in the phase space in which the trajectories of a system have a fairly simple behavior determined by a typical scenario. The problem is solved by applying the method of localization of compact invariant sets of the system. It is proved that localizing sets separate simple and complex dynamics of nonlinear systems, namely, in the complement of any localizing set, the trajectory behavior of a system admits a standard description in the form of several scenarios, while, in localizing sets and their intersections, the trajectory behavior of the system can be very complex. Specifically, it is shown that the α- and ω-limit sets of any trajectory are contained in localizing sets.

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Authors and Affiliations

  1. Bauman Moscow State Technical University, Vtoraya Baumanskaya ul. 5, Moscow, 105005, Russia

    A. N. Kanatnikov & A. P. Krishchenko

  2. Institute for Systems Analysis, Federal Research Center “Computer Science and Control,”, Russian Academy of Sciences, pr. 60-letiya Oktyabrya 9, Moscow, 117312, Russia

    A. N. Kanatnikov & A. P. Krishchenko

Authors
  1. A. N. Kanatnikov
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  2. A. P. Krishchenko
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Correspondence to A. N. Kanatnikov.

Additional information

Original Russian Text © A.N. Kanatnikov, A.P. Krishchenko, 2016, published in Doklady Akademii Nauk, 2016, Vol. 470, No. 2, pp. 133–136.

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Kanatnikov, A.N., Krishchenko, A.P. Localizing sets and trajectory behavior. Dokl. Math. 94, 506–509 (2016). https://doi.org/10.1134/S1064562416050070

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  • DOI: https://doi.org/10.1134/S1064562416050070

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