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Transversal Curves for Finding the Exact Number of Limit Cycles

  • ORDINARY DIFFERENTIAL EQUATIONS
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Abstract

For autonomous systems of differential equations with smooth right-hand sides, in a simply connected domain of the real phase plane we consider the problem of finding the exact number of limit cycles surrounding one or several stationary points with total Poincaré index \(+1 \). A two-step method is proposed for solving this problem. At the first step, using the Dulac–Cherkas test, we find nested closed curves that have no common points and split the simply connected domain into doubly connected subdomains, with each curve being transversal to the vector field of the system and surrounding all stationary points. This allows finding an upper bound for the number of limit cycles, because the system has exactly one limit cycle in each of the interior doubly connected subdomains, while the exterior doubly connected subdomain contains at most one limit cycle. The second step is performed to check the existence of a limit cycle in the exterior subdomain. At this step, using once more the Dulac–Cherkas test, or the Dulac test, or their generalizations, we construct an auxiliary closed curve transversal to the vector field and surrounding the previously found closed curves. The efficiency of the method is demonstrated by examples of polynomial Liénard systems, including a generalized van der Pol system, for which we establish the uniqueness of the limit cycle in the entire phase plane.

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Authors and Affiliations

  1. Yanka Kupala State University of Grodno, Grodno, 230023, Belarus

    A. A. Grin’

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  1. A. A. Grin’
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Correspondence to A. A. Grin’.

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Translated by V. Potapchouck

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Grin’, A.A. Transversal Curves for Finding the Exact Number of Limit Cycles. Diff Equat 56, 415–425 (2020). https://doi.org/10.1134/S0012266120040023

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  • DOI: https://doi.org/10.1134/S0012266120040023

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