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

Timoteo Carletti, Gabriele Villari

Research output: Contribution to journalArticle

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
Pages (from-to)1
Number of pages15
JournalElectronic Journal of Qualitative Theory of Differential Equations
Volume2
DOIs
Publication statusPublished - 10 Jan 2020

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Phase Plane
Limit Cycle
Curvature
Damping
Periodic Solution
Arbitrary
Term
Generalization
Standards
Form

Keywords

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

Cite this

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title = "Existence of limit cycles for some generalisation of the Li{\'e}nard equations: the relativistic and the prescribed curvature cases",
abstract = "We study the problem of existence of periodic solutions for some generalisations of the relativistic Li{\'e}nard equation and the prescribed curvature Li{\'e}nard 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.",
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T1 - Existence of limit cycles for some generalisation of the Liénard equations

T2 - the relativistic and the prescribed curvature cases

AU - Carletti, Timoteo

AU - Villari, Gabriele

PY - 2020/1/10

Y1 - 2020/1/10

N2 - 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.

AB - 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.

KW - periodic orbits

KW - limit cycles

KW - Liénard relativistic equation

KW - Liénard curvature equation

U2 - https://doi.org/10.14232/ejqtde.2020.1.2

DO - https://doi.org/10.14232/ejqtde.2020.1.2

M3 - Article

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SP - 1

JO - Electronic Journal of Qualitative Theory of Differential Equations

JF - Electronic Journal of Qualitative Theory of Differential Equations

SN - 1417-3875

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