Liénard equation

A non-linear second-order ordinary differential equation

$$\tag{* } x ^ {\prime\prime} + f ( x) x ^ \prime + x = 0 .$$

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 $f$ has the property

$$f ( x) < 0 \ \ \textrm{ for small } | x | ,$$

$$f ( x) > 0 \ \textrm{ for large } | x | ,$$

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

$$x ^ \prime = v ,\ \ v ^ \prime = - x - f ( x) v$$

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

$$\frac{dv}{dx} = \frac{- x - f ( x) v }{v} .$$

If one introduces a new variable $y = x ^ \prime + F ( x)$, where $F ( x) = \int _ {0} ^ {x} f ( \xi ) d \xi$, then (*) goes into the system

$$x ^ \prime = y - F ( x) ,\ \ y ^ \prime = - x .$$

More general than the Liénard equation are the equations

$$x ^ {\prime\prime} + f ( x) x ^ \prime + g ( x) = 0 ,$$

$$x ^ {\prime\prime} + \phi ( x , x ^ \prime ) x ^ \prime + g ( x) = 0 .$$

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

$$x ^ {\prime\prime} + f ( x) x ^ \prime + x = e ( t)$$

and generalizations of it have also been studied in detail.

References

 [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: http://encyclopediaofmath.org/index.php?title=Li%C3%A9nard_equation&oldid=47673
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article