Existence of limit cycles for some generalisation of the Liénard equations : the relativistic and the prescribed curvature cases

Carletti, Timoteo and Villari, Gabriele: Existence of limit cycles for some generalisation of the Liénard equations : the relativistic and the prescribed curvature cases. Electronic journal of qualitative theory of differential equations 2. pp. 1-15. (2020)

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Abstract

We study the problem of existence of periodic solutions for some generalisations of the relativistic Liénard equation d dt x˙ 1 − x˙ 2 f(x, x˙)x˙ + g(x) = 0 , and the prescribed curvature Liénard equation d dt x˙ 1 + x˙ 2 f(x, x˙)x˙ + g(x) = 0 , where the damping function depends both on the position and the velocity. In the associated phase-plane this corresponds to a term of the form f(x, y) instead of the standard dependence on x alone. By controlling the continuability of the solutions, we are able to prove the existence of at least a limit cycle in the associated phase-plane for both cases, moreover we provide results with a prefixed arbitrary number of limit cycles. Some examples are given to show the applicability of these results.

Item Type: Article
Journal or Publication Title: Electronic journal of qualitative theory of differential equations
Date: 2020
Number: 2
Page Range: pp. 1-15
ISSN: 1417-3875
DOI: https://doi.org/10.14232/ejqtde.2020.1.2
Uncontrolled Keywords: Liénard egyenlet, Differenciaegyenlet
Additional Information: Bibliogr.: p. 14-15. ; összefoglalás angol nyelven
Date Deposited: 2020. Jan. 28. 09:44
Last Modified: 2020. Jan. 28. 09:44
URI: http://acta.bibl.u-szeged.hu/id/eprint/66368

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