Liénard equation

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A non-linear second-order ordinary differential equation


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 has the property

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). Sufficient conditions for the appearance of auto-oscillations in the system (*) were first proved by A. Liénard [1].

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

(a stable limit cycle on the phase plane is adequate for an auto-oscillating process in the system (*)), or the equivalent equation

If one introduces a new variable , where , then (*) goes into the system

More general than the Liénard equation are the equations

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

and generalizations of it have also been studied in detail.


[1] A. Liénard, Rev. Gen. Electr. , 23 (1928) pp. 901–912; 946–954
[2] A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Dover, reprint (1987) (Translated from Russian)
[3] G. Sansone, "Ordinary differential equations" , 2 , Zanichelli (1948) (In Italian)
[4] S. Lefschetz, "Differential equations: geometric theory" , Interscience (1957)
[5] R. Reissig, G. Sansone, R. Conti, "Nichtlineare Differentialgleichungen höherer Ordnung" , Cremonese (1969)
How to Cite This Entry:
Liénard equation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article