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

Timoteo Carletti, Gabriele Villari

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Abstract

We study the problem of existence of periodic solutions for some generalisations of the relativistic Liénard equation and the prescribed curvature Liénard equation 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.
Original languageEnglish
Article number2
Pages (from-to)1
Number of pages15
JournalElectronic Journal of Qualitative Theory of Differential Equations
Volume2020
DOIs
Publication statusPublished - 10 Jan 2020

Keywords

  • periodic orbits
  • limit cycles
  • Liénard relativistic equation
  • Liénard curvature equation
  • Periodic orbits
  • Limit cycles

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