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On the Stability of Integral Manifolds of a System of Ordinary Differential Equations in the Critical Case

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Abstract

In this paper, we consider the stability problem for nonzero integral manifolds of a nonlinear, finite-dimensional system of ordinary differential equations whose right-hand side is a vector-valued function containing a parameter and periodic in an independent variable. We assume that the system possesses a trivial integral manifold for all values of the parameter and the corresponding linear subsystem does not possess the exponential dichotomy property.We find sufficient conditions for the existence of a nonzero integral manifold in a neighborhood of the equilibrium of the system and conditions for its stability or instability. For this purpose, based of the ideas of the Lyapunov method and the method of transform matrices, we construct operators that allow one to reduce the solution of this problem to the search for fixed points.

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References

  1. Yu. N. Bibikov, Multifrequency Nonlinear Oscillations and Their Bifurcations, Leningrad (1991).

  2. N. N. Bogolyubov, Statistical Methods in Mathematical Physics, L’vov (1945).

  3. M. I. Kuptsov, Existence of integral manifolds and periodic solutions of a system of ordinary differential equations [in Russian], thesis, Izhevsk (1997).

  4. M. I. Kuptsov, “Existence of an integral manifold of a system of differential equations,” Differ. Uravn., 34, No. 6, 855 (1998).

    Google Scholar 

  5. M. I. Kuptsov, “Local integral manifold of a system of differential equations,” Differ. Uravn., 34, No. 7, 1005–1007 (1998).

    MathSciNet  MATH  Google Scholar 

  6. M. I. Kuptsov, “Local integral manifold of a system of differential equations depending on a parameter,” Differ. Uravn., 35, No. 11, 1579–1580 (1999).

    Google Scholar 

  7. M. I. Kuptsov and V. A. Minaev, “Stability integral manifold of the differential equations system in critical case,” J. Phys. Conf. Ser., 973, 012055 (2018).

    Article  Google Scholar 

  8. M. I. Kuptsov, M. T. Terekhin, and V. V. Tenyaev, “On the problem of the existence of integral manifolds for systems of differential equations not solved with respect to derivatives,” Zh. Srednevolzhsk. Mat. Obshch., 19, No. 2, 76–84 (2017).

    MATH  Google Scholar 

  9. S. Z. Kurbanshoev and M. A. Nusayriev, “Construction of optimal integral manifolds for nonlinear differential equations,” Dokl. Akad. Nauk. Tajikistan, 57, No. 11–12, 807–812 (2014).

    Google Scholar 

  10. Yu. A. Mitropol’sky and O. B. Lykova, Integral Manifolds in Nonlinear Mechanics, Nauka, Moscow (1973).

  11. I. Prigogine, “The philosophy of instability,” Futures, No. 21, 396–400 (1989).

  12. V. V. Rumyantsev and A. S. Oziraner, Stability and Stabilization of Motion over a Part of Variables, Nauka, Moscow (1987).

    MATH  Google Scholar 

  13. A. M. Samoilenko, Foundations of the Mathematical Theory of Multifrequency Oscillations. Invariant Tori, Nauka, Moscow (1987).

    Google Scholar 

  14. V. Sobolev, “Slow integral manifolds and control problems in critical and twice critical cases,” J. Phys. Conf. Ser., 727, 012017 (2016).

    Article  MathSciNet  Google Scholar 

  15. D. Yu. Volkov, “Bifurcation of invariant tori from an equilibrium state in the presence of zero characteristic numbers,” Vestn. Leningrad. Univ., 1, No. 2, 102–103 (1988).

    Google Scholar 

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Authors and Affiliations

  1. Ryazan State Radio Engineering University, Ryazan, Russia

    M. I. Kuptsov

  2. Bauman Moscow State Technical University, Moscow, Russia

    V. A. Minaev

  3. Academy of Law and Management of the Federal Penal Service of Russia, Ryazan, Russia

    A. O. Faddeev & S. L. Yablochnikov

Authors
  1. M. I. Kuptsov
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  2. V. A. Minaev
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  3. A. O. Faddeev
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  4. S. L. Yablochnikov
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Correspondence to M. I. Kuptsov.

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 168, Proceedings of the International Conference “Geometric Methods in the Control Theory and Mathematical Physics” Dedicated to the 70th Anniversary of Prof. S. L. Atanasyan, 70th Anniversary of Prof. I. S. Krasil’shchik, 70th Anniversary of Prof. A. V. Samokhin, and 80th Anniversary of Prof. V. T. Fomenko. Ryazan State University named for S. Yesenin, Ryazan, September 25–28, 2018. Part I, 2019.

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Kuptsov, M.I., Minaev, V.A., Faddeev, A.O. et al. On the Stability of Integral Manifolds of a System of Ordinary Differential Equations in the Critical Case. J Math Sci 262, 825–834 (2022). https://doi.org/10.1007/s10958-022-05861-5

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  • DOI: https://doi.org/10.1007/s10958-022-05861-5

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