View source for Liénard equation

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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|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

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

(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====
<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>