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Lyapunov Quantities in the Problem about Local Bifurcations of Non-Autonomous Periodic Dynamical Systems

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Abstract

The article discusses the problems about the main scenarios of dynamical systems’ local bifurcations described by non-autonomous differential equations with periodic right-hand side. New formulae for calculating Lyapunov quantities are proposed. The proposed formulae are obtained on the basis of the general operator method for studying local bifurcations and do not require a transition to normal forms and the usage of the theorems about the central manifold. The indicated method allowed for obtaining new bifurcation formulae for studying the main scenarios of local bifurcations. The paper shows how these bifurcation formulae lead to new formulae for calculating Lyapunov quantities in the problems about bifurcations of forced oscillations and Andronov–Hopf bifurcations.

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Authors and Affiliations

  1. Department of Differential Equations, Faculty of Mathematics and Information Technology, Bashkir State University, 450076, Ufa, Russia

    M. G. Yumagulov

  2. Department of Applied Mathematics and Computer Science, Institute of Natural Science and Standardization, Nosov Magnitogorsk State Technical University, 455000, Magnitogorsk, Chelyabinsk oblast, Russia

    S. V. Akmanova

  3. Foreign Languages Department, Engineering and Economics Faculty, Ryazan State Radio Engineering University named after V. F. Utkin, 390005, Ryazan, Russia

    N. A. Kopylova

Authors
  1. M. G. Yumagulov
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  2. S. V. Akmanova
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  3. N. A. Kopylova
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Corresponding authors

Correspondence to M. G. Yumagulov, S. V. Akmanova or N. A. Kopylova.

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(Submitted by E. K. Lipachev)

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Yumagulov, M.G., Akmanova, S.V. & Kopylova, N.A. Lyapunov Quantities in the Problem about Local Bifurcations of Non-Autonomous Periodic Dynamical Systems. Lobachevskii J Math 41, 1918–1923 (2020). https://doi.org/10.1134/S1995080220090346

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  • DOI: https://doi.org/10.1134/S1995080220090346

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