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 .
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.
|||A. Liénard, Rev. Gen. Electr. , 23 (1928) pp. 901–912; 946–954|
|||A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Dover, reprint (1987) (Translated from Russian)|
|||G. Sansone, "Ordinary differential equations" , 2 , Zanichelli (1948) (In Italian)|
|||S. Lefschetz, "Differential equations: geometric theory" , Interscience (1957)|
|||R. Reissig, G. Sansone, R. Conti, "Nichtlineare Differentialgleichungen höherer Ordnung" , Cremonese (1969)|
Liénard equation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Li%C3%A9nard_equation&oldid=23384