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A non-linear ordinary differential equation of order two:
 
A non-linear ordinary differential equation of order two:
  
<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/r/r077/r077740/r0777401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\dot{x} dot + F ( \dot{x} ) + x  = 0 ,\  \dot{x} =  
 +
\frac{dx}{dt}
 +
,
 +
$$
  
where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077740/r0777402.png" /> satisfies the assumption:
+
where the function $  F ( u) $
 +
satisfies the assumption:
  
<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/r/r077/r077740/r0777403.png" /></td> </tr></table>
+
$$
 +
u F ( u)  < 0 \  \textrm{ for  small  }  | u | ,
 +
$$
  
<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/r/r077/r077740/r0777404.png" /></td> </tr></table>
+
$$
 +
u F ( u)  > 0 \  \textrm{ for  large  }  | u | .
 +
$$
  
 
The Rayleigh equation describes a typical non-linear system with one degree of freedom which admits auto-oscillations (cf. [[Auto-oscillation|Auto-oscillation]]). This equation was named after Lord Rayleigh, who studied equations of this type related to problems in acoustics [[#References|[1]]].
 
The Rayleigh equation describes a typical non-linear system with one degree of freedom which admits auto-oscillations (cf. [[Auto-oscillation|Auto-oscillation]]). This equation was named after Lord Rayleigh, who studied equations of this type related to problems in acoustics [[#References|[1]]].
  
If one differentiates equation (*) and then puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077740/r0777405.png" />, one obtains the [[Liénard equation|Liénard equation]]
+
If one differentiates equation (*) and then puts $  y = \dot{x} $,  
 +
one obtains the [[Liénard equation|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/r/r077/r077740/r0777406.png" /></td> </tr></table>
+
$$
 +
\dot{y} dot + f ( y) \dot{y} + y  = 0 ,\ \
 +
f ( u)  = F ^ { \prime } ( u) .
 +
$$
  
 
The special case of the Rayleigh equation for
 
The special case of the Rayleigh equation for
  
<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/r/r077/r077740/r0777407.png" /></td> </tr></table>
+
$$
 +
F ( u)  = - \lambda
 +
\left ( u -  
 +
\frac{u  ^ {2} }{3}
 +
\right ) ,\ \
 +
\lambda = \textrm{ const } ,
 +
$$
  
 
is the [[Van der Pol equation|van der Pol equation]]. Sometimes the following special case of equation (*) is called the Rayleigh equation:
 
is the [[Van der Pol equation|van der Pol equation]]. Sometimes the following special case of equation (*) is called the Rayleigh 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/r/r077/r077740/r0777408.png" /></td> </tr></table>
+
$$
 +
\dot{x} dot - ( a - b \dot{x}  ^ {2} ) \dot{c} + x  = \
 +
0 ,\  a , b > 0 .
 +
$$
  
 
There is a large number of studies concerned with the existence and uniqueness conditions for a [[Limit cycle|limit cycle]] of the Rayleigh equation, that is, conditions under which auto-oscillations occur. The question of periodic solutions was studied also for different generalizations of the Rayleigh equation, e.g. for
 
There is a large number of studies concerned with the existence and uniqueness conditions for a [[Limit cycle|limit cycle]] of the Rayleigh equation, that is, conditions under which auto-oscillations occur. The question of periodic solutions was studied also for different generalizations of the Rayleigh equation, e.g. for
  
<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/r/r077/r077740/r0777409.png" /></td> </tr></table>
+
$$
 +
\dot{x} dot + F ( x , \dot{x} ) \dot{x} + g ( x)  = e ( t) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077740/r07774010.png" /> is a periodic function.
+
where $  e ( t) $
 +
is a periodic function.
  
 
The following equation is often called a Rayleigh-type system:
 
The following equation is often called a Rayleigh-type 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/r/r077/r077740/r07774011.png" /></td> </tr></table>
+
$$
 +
\dot{x} dot + F ( \dot{x} ) + G ( x)  = H ( t , x , \dot{x} ) ,
 +
$$
  
<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/r/r077/r077740/r07774012.png" /></td> </tr></table>
+
$$
 +
x  \in  \mathbf R  ^ {n} ,\  F : \mathbf R  ^ {n}  \rightarrow  \mathbf R  ^ {n} ,\  G : \mathbf R  ^ {n}  \rightarrow  \mathbf R  ^ {n} ;
 +
$$
  
 
moreover, it is usually assumed that
 
moreover, it is usually assumed that
  
<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/r/r077/r077740/r07774013.png" /></td> </tr></table>
+
$$
 +
=   \mathop{\rm grad}  f ,\  f: \mathbf R  ^ {n}  \rightarrow  \mathbf R ,\ \
 +
f  \in  C  ^ {1} ,
 +
$$
  
<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/r/r077/r077740/r07774014.png" /></td> </tr></table>
+
$$
 +
=   \mathop{\rm grad}  g ,\  g : \mathbf R  ^ {n}  \rightarrow  \mathbf R ,\  g  \in  C  ^ {2} ,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077740/r07774015.png" /> is a bounded vector function that is periodic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077740/r07774016.png" />. The problem of obtaining sufficient conditions for the existence of periodic solutions of such systems is of considerable interest.
+
and $  H $
 +
is a bounded vector function that is periodic in $  t $.  
 +
The problem of obtaining sufficient conditions for the existence of periodic solutions of such systems is of considerable interest.
  
 
See also the references to [[Liénard equation|Liénard equation]].
 
See also the references to [[Liénard equation|Liénard equation]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.W. [Lord Rayleigh] Strutt,  "Theory of sound" , '''1''' , Dover, reprint  (1945)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Cesari,  "Asymptotic behavior and stability problems in ordinary differential equations" , Springer  (1959)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.W. [Lord Rayleigh] Strutt,  "Theory of sound" , '''1''' , Dover, reprint  (1945)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Cesari,  "Asymptotic behavior and stability problems in ordinary differential equations" , Springer  (1959)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.J. Stoker,  "Nonlinear vibrations in mechanical and electrical systems" , Interscience  (1950)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.J. Stoker,  "Nonlinear vibrations in mechanical and electrical systems" , Interscience  (1950)</TD></TR></table>

Latest revision as of 08:10, 6 June 2020


A non-linear ordinary differential equation of order two:

$$ \tag{* } \dot{x} dot + F ( \dot{x} ) + x = 0 ,\ \dot{x} = \frac{dx}{dt} , $$

where the function $ F ( u) $ satisfies the assumption:

$$ u F ( u) < 0 \ \textrm{ for small } | u | , $$

$$ u F ( u) > 0 \ \textrm{ for large } | u | . $$

The Rayleigh equation describes a typical non-linear system with one degree of freedom which admits auto-oscillations (cf. Auto-oscillation). This equation was named after Lord Rayleigh, who studied equations of this type related to problems in acoustics [1].

If one differentiates equation (*) and then puts $ y = \dot{x} $, one obtains the Liénard equation

$$ \dot{y} dot + f ( y) \dot{y} + y = 0 ,\ \ f ( u) = F ^ { \prime } ( u) . $$

The special case of the Rayleigh equation for

$$ F ( u) = - \lambda \left ( u - \frac{u ^ {2} }{3} \right ) ,\ \ \lambda = \textrm{ const } , $$

is the van der Pol equation. Sometimes the following special case of equation (*) is called the Rayleigh equation:

$$ \dot{x} dot - ( a - b \dot{x} ^ {2} ) \dot{c} + x = \ 0 ,\ a , b > 0 . $$

There is a large number of studies concerned with the existence and uniqueness conditions for a limit cycle of the Rayleigh equation, that is, conditions under which auto-oscillations occur. The question of periodic solutions was studied also for different generalizations of the Rayleigh equation, e.g. for

$$ \dot{x} dot + F ( x , \dot{x} ) \dot{x} + g ( x) = e ( t) , $$

where $ e ( t) $ is a periodic function.

The following equation is often called a Rayleigh-type system:

$$ \dot{x} dot + F ( \dot{x} ) + G ( x) = H ( t , x , \dot{x} ) , $$

$$ x \in \mathbf R ^ {n} ,\ F : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} ,\ G : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} ; $$

moreover, it is usually assumed that

$$ F = \mathop{\rm grad} f ,\ f: \mathbf R ^ {n} \rightarrow \mathbf R ,\ \ f \in C ^ {1} , $$

$$ G = \mathop{\rm grad} g ,\ g : \mathbf R ^ {n} \rightarrow \mathbf R ,\ g \in C ^ {2} , $$

and $ H $ is a bounded vector function that is periodic in $ t $. The problem of obtaining sufficient conditions for the existence of periodic solutions of such systems is of considerable interest.

See also the references to Liénard equation.

References

[1] J.W. [Lord Rayleigh] Strutt, "Theory of sound" , 1 , Dover, reprint (1945)
[2] L. Cesari, "Asymptotic behavior and stability problems in ordinary differential equations" , Springer (1959)

Comments

References

[a1] J.J. Stoker, "Nonlinear vibrations in mechanical and electrical systems" , Interscience (1950)
How to Cite This Entry:
Rayleigh equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rayleigh_equation&oldid=48446
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article