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A non-linear second-order ordinary differential equation
 
A non-linear second-order ordinary differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l0587901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
x  ^ {\prime\prime} + f ( x) x  ^  \prime  + x  = 0 .
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l0587902.png" /> has the property
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l0587903.png" /></td> </tr></table>
+
$$
 +
f ( x)  < 0 \ \
 +
\textrm{ for  small  }  | x | ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l0587904.png" /></td> </tr></table>
+
$$
 +
f ( x)  > 0 \  \textrm{ for  large  }  | x | ,
 +
$$
  
 
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]]].
 
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]]].
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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
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l0587905.png" /></td> </tr></table>
+
$$
 +
x  ^  \prime  = v ,\ \
 +
v  ^  \prime  = - x - f ( x) v
 +
$$
 +
 
 +
(a stable limit cycle on the phase plane  $  x , v $
 +
is adequate for an auto-oscillating process in the system (*)), or the equivalent equation
  
(a stable limit cycle on the phase plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l0587906.png" /> is adequate for an auto-oscillating process in the system (*)), or the equivalent equation
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l0587907.png" /></td> </tr></table>
+
\frac{dv}{dx}
 +
  =
 +
\frac{- x - f ( x) v }{v}
 +
.
 +
$$
  
If one introduces a new variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l0587908.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l0587909.png" />, then (*) goes into the system
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l05879010.png" /></td> </tr></table>
+
$$
 +
x  ^  \prime  = y - F ( x) ,\ \
 +
y  ^  \prime  = - x .
 +
$$
  
 
More general than the Liénard equation are the equations
 
More general than the Liénard equation are the equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l05879011.png" /></td> </tr></table>
+
$$
 +
x  ^ {\prime\prime} + f ( x) x  ^  \prime  + g ( x)  = 0 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l05879012.png" /></td> </tr></table>
+
$$
 +
x  ^ {\prime\prime} + \phi ( x , x  ^  \prime  ) x  ^  \prime  + g ( x)  = 0 .
 +
$$
  
 
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
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058790/l05879013.png" /></td> </tr></table>
+
$$
 +
x  ^ {\prime\prime} + f ( x) x  ^  \prime  + x  = e ( t)
 +
$$
  
 
and generalizations of it have also been studied in detail.
 
and generalizations of it have also been studied in detail.

Latest revision as of 22:17, 5 June 2020


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