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On the Conditions for Invariance of the Sets Belonging to the Phase Portraits of Nonlinear Dynamic Systems

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Abstract

Consideration was given to the conditions for invariance of the sets belonging to the phase portraits of the nonlinear dynamic systems, that is, to the conditions under which these sets consist of entire trajectories of the dynamic systems following nonlinear autonomous vector ordinary differential equations of an arbitrary order. The necessary and sufficient conditions for invariance of the phase sets as well as the sufficient conditions for nonexistence of the invariant sets were presented.

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REFERENCES

  1. Zhukov, V.P., On Periodic Modes in the Nonlinear Systems, Avtom. Telemekh., 1981, no. 7, pp. 45–50.

  2. Zhukov, V.P., On One Method of Studying Periodic Modes in Nonlinear Systems, Avtom. Telemekh., 1992, no. 7, pp. 3–9.

  3. Zhukov, V.P., Analogs of the Bendixson and Dulac Criteria for Arbitrary-order Dynamic Systems, Avtom. Telemekh., 1999, no. 10, pp. 46–64.

  4. Zhukov, V.P., Divergent Conditions for Nonexistence of Self-oscillations in the Nonlinear Dynamic Systems of the Second Order, Avtom. Telemekh., 2000, no. 1, pp. 21–28.

  5. Zhukov, V.P., Method of Radial Drift for Qualitative Study of the Properties of the Nonlinear Dynamic Systems. III. On Existence of Invariant Closed Loops, Avtom. Telemekh., 2001, no. 8, pp. 21–40.

  6. Blanchini, F., Set Invariance in Control, Automatica, 1999, vol. 35, pp. 1747–1767.

    Article  MathSciNet  Google Scholar 

  7. Arnol’d, V.I., Obyknovennye differentsial’nye uravneniya (Ordinary Differential Equations), Moscow: Nauka, 1971.

    Google Scholar 

  8. Matematicheskaya entsiklopediya (Mathematical Encyclopedia), Moscow: Sovetskaya Entsiklopediya, 1985, vol. 5.

  9. Andronov, A.A., Vitt, A.A., and Khai’kin, S.E., Teoriya kolebanii (Oscillation Theory), Moscow: Fizmatlit, 1959.

    Google Scholar 

  10. Mishchenko, A.S. and Fomenko, A.T., Kurs differentsial’noi geometrii i topologii (A Course of Differential Geometry and Topology), Moscow: Faktorial, 2000.

    Google Scholar 

  11. Matematicheskaya entsiklopediya (Mathematical Encyclopedia), Moscow: Sovetskays Entsiklopediya, 1984, vol. 4.

  12. Arnol’d, V.I., Matematicheskie metody klassicheskoi mekhaniki (Mathematical Methods of Classical Mechanics), Moscow: Nauka, 1974.

    Google Scholar 

  13. Dieudonne, J., Foundations of Modern Analysis, New York: Academic, 1960. Translated under the title Osnovy sovremennogo analiza, Moscow: Mir, 1964.

    Google Scholar 

  14. Erugin, N.P., Some General Issues of the Stable Motion Theory, Prikl. Mat. Mekh., 1951, vol. XI, no.2, pp. 227–236.

    Google Scholar 

  15. Kolmogorov, A.N. and Fomin, S.I., Elementy teorii funktsii i funktsional’nogo analiza (Elements of the Theory of Functions and Functional Analysis), Moscow: Nauka, 1972.

    Google Scholar 

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Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    V. P. Zhukov

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  1. V. P. Zhukov
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Translated from Avtomatika i Telemekhanika, No. 6, 2005, pp. 19–29.

Original Russian Text Copyright © 2005 by Zhukov.

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Zhukov, V.P. On the Conditions for Invariance of the Sets Belonging to the Phase Portraits of Nonlinear Dynamic Systems. Autom Remote Control 66, 866–875 (2005). https://doi.org/10.1007/s10513-005-0130-1

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  • DOI: https://doi.org/10.1007/s10513-005-0130-1

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