View source for Liénard equation

A non-linear second-order ordinary differential equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l0587901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

This equation describes the dynamics of a system with one degree of freedom in the presence of a linear restoring force and non-linear damping. If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l0587902.png" /> has the property

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that is, if for small amplitudes the system absorbs energy and for large amplitudes dissipation occurs, then in the system one can expect self-exciting oscillations (the appearance of auto-oscillations, cf. [[Auto-oscillation|Auto-oscillation]]). Sufficient conditions for the appearance of auto-oscillations in the system (*) were first proved by A. Liénard [[#References|[1]]].

The Liénard equation is closely connected with the [[Rayleigh equation|Rayleigh equation]]. An important special case of it is the [[Van der Pol equation|van der Pol equation]]. Instead of equation (*) it is often convenient to consider the system

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(a stable limit cycle on the phase plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l0587906.png" /> is adequate for an auto-oscillating process in the system (*)), or the equivalent equation

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If one introduces a new variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l0587908.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l0587909.png" />, then (*) goes into the system

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More general than the Liénard equation are the equations

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The main interest is in the determination of possibly more general sufficient conditions under which these equations have a unique stable periodic solution. The non-homogeneous Liénard equation

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and generalizations of it have also been studied in detail.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Liénard,   ''Rev. Gen. Electr.'' , '''23'''  (1928)  pp. 901–912; 946–954</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Andronov,   A.A. Vitt,   A.E. Khaikin,   "Theory of oscillators" , Dover, reprint  (1987)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Sansone,   "Ordinary differential equations" , '''2''' , Zanichelli  (1948)  (In Italian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Lefschetz,   "Differential equations: geometric theory" , Interscience  (1957)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R. Reissig,   G. Sansone,   R. Conti,   "Nichtlineare Differentialgleichungen höherer Ordnung" , Cremonese  (1969)</TD></TR></table>