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)