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The non-linear second-order ordinary differential equation
 
The 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/v/v096/v096030/v0960301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\dot{x} dot - \mu ( 1 - x  ^ {2} )
 +
\dot{x} + x  = 0,\ \
 +
\mu = \textrm{ const } > 0,\ \
 +
\dot{x} ( t) \equiv  {
 +
\frac{dx}{dt}
 +
} ,
 +
$$
  
 
which is an important special case of the [[Liénard equation|Liénard equation]]. Van der Pol's equation describes the auto-oscillations (cf. [[Auto-oscillation|Auto-oscillation]]) of one of the simplest oscillating systems (the van der Pol oscillator). In particular, equation (1) serves — after making several simplifying assumptions — as a mathematical model of a generator on a triode for a tube with a cubic characteristic. The character of the solutions of equation (1) was first studied in detail by B. van der Pol .
 
which is an important special case of the [[Liénard equation|Liénard equation]]. Van der Pol's equation describes the auto-oscillations (cf. [[Auto-oscillation|Auto-oscillation]]) of one of the simplest oscillating systems (the van der Pol oscillator). In particular, equation (1) serves — after making several simplifying assumptions — as a mathematical model of a generator on a triode for a tube with a cubic characteristic. The character of the solutions of equation (1) was first studied in detail by B. van der Pol .
  
Equation (1) is equivalent to the following system of two equations in two phase variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v0960302.png" />:
+
Equation (1) is equivalent to the following system of two equations in two phase variables $  x, v $:
  
<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/v/v096/v096030/v0960303.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\dot{x}  = v,\ \
 +
\dot{v= - x + \mu ( 1 - x  ^ {2} ) v.
 +
$$
  
It is sometimes convenient to replace the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v0960304.png" /> by the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v0960305.png" />; equation (1) then becomes
+
It is sometimes convenient to replace the variable $  x $
 +
by the variable $  z( t) = {\int _ {0}  ^ {t} } x ( \tau )  d \tau $;  
 +
equation (1) then becomes
  
<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/v/v096/v096030/v0960306.png" /></td> </tr></table>
+
$$
 +
\dot{z} dot - \mu \left (
 +
\dot{z} -
 +
\frac{\dot{z}  ^ {3} }{3}
  
which is a special case of the [[Rayleigh equation|Rayleigh equation]]. If, together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v0960307.png" />, one also considers the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v0960308.png" />, introduces a new time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v0960309.png" /> and puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v09603010.png" />, one obtains the system
+
\right ) + z  = 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/v/v096/v096030/v09603011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
which is a special case of the [[Rayleigh equation|Rayleigh equation]]. If, together with  $  x $,
 +
one also considers the variable  $  y = - x + ( x  ^ {3} /3) + ( \dot{x} / \mu ) $,
 +
introduces a new time  $  \tau = t / \mu $
 +
and puts  $  \epsilon = \mu  ^ {-} 2 $,
 +
one obtains the system
  
instead of equation (1). For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v09603012.png" /> there exists a unique stable limit cycle in the phase plane of the system (2) to which all other trajectories (except for the equilibrium position at the coordinate origin) converge as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v09603013.png" />; this limit cycle describes the oscillations of the van der Pol oscillator [[#References|[2]]], [[#References|[3]]], [[#References|[4]]].
+
$$ \tag{3 }
 +
\epsilon x  ^  \prime  = \
 +
y - x +
 +
\frac{x  ^ {3} }{3}
 +
,\ \
 +
y  ^  \prime  = - x,\ \
 +
{}  ^  \prime  =  
 +
\frac{d}{d \tau }
 +
,
 +
$$
  
For small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v09603014.png" /> the auto-oscillations of the oscillator (1) are close to simple harmonic oscillations (cf. [[Non-linear oscillations|Non-linear oscillations]]) with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v09603015.png" /> and specified amplitude. In order to calculate the oscillation process more accurately, asymptotic methods are employed. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v09603016.png" /> increases, the auto-oscillations of the oscillator (1) deviate more and more from harmonic oscillations. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v09603017.png" /> is large, equation (1) describes [[Relaxation oscillation|relaxation oscillation]] with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v09603018.png" /> (to a first approximation). More accurate asymptotic expansions of magnitudes characterizing relaxation oscillations [[#References|[5]]] are known: The study of these oscillations is equivalent to the study of the solutions of the system (3) with a small coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v09603019.png" /> in front of the derivative [[#References|[6]]].
+
instead of equation (1). For any  $  \mu > 0 $
 +
there exists a unique stable limit cycle in the phase plane of the system (2) to which all other trajectories (except for the equilibrium position at the coordinate origin) converge as  $  t \rightarrow \infty $;
 +
this limit cycle describes the oscillations of the van der Pol oscillator [[#References|[2]]], [[#References|[3]]], [[#References|[4]]].
 +
 
 +
For small  $  \mu $
 +
the auto-oscillations of the oscillator (1) are close to simple harmonic oscillations (cf. [[Non-linear oscillations|Non-linear oscillations]]) with period $  2 \pi $
 +
and specified amplitude. In order to calculate the oscillation process more accurately, asymptotic methods are employed. As $  \mu $
 +
increases, the auto-oscillations of the oscillator (1) deviate more and more from harmonic oscillations. If $  \mu $
 +
is large, equation (1) describes [[Relaxation oscillation|relaxation oscillation]] with period $  1.614  \mu $(
 +
to a first approximation). More accurate asymptotic expansions of magnitudes characterizing relaxation oscillations [[#References|[5]]] are known: The study of these oscillations is equivalent to the study of the solutions of the system (3) with a small coefficient $  \epsilon $
 +
in front of the derivative [[#References|[6]]].
  
 
The equation
 
The 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/v/v096/v096030/v09603020.png" /></td> </tr></table>
+
$$
 +
\dot{x} dot - \mu ( 1 - x  ^ {2} )
 +
\dot{x} + x  = E _ {0} + E  \sin  \omega t
 +
$$
  
 
describes the behaviour of the van der Pol oscillator when acted upon by a periodic external disturbance. The most important in this context is the study of frequency capture (the existence of periodic oscillations), beats (the possibility of almost-periodic oscillations) and chaotic behaviour [[#References|[2]]], [[#References|[4]]].
 
describes the behaviour of the van der Pol oscillator when acted upon by a periodic external disturbance. The most important in this context is the study of frequency capture (the existence of periodic oscillations), beats (the possibility of almost-periodic oscillations) and chaotic behaviour [[#References|[2]]], [[#References|[4]]].
Line 29: Line 83:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  B. van der Pol,  "On oscillation hysteresis in a triode generator with two degrees of freedom"  ''Philos. Mag. (6)'' , '''43'''  (1922)  pp. 700–719</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  B. van der Pol,  ''Philos. Mag. (7)'' , '''2'''  (1926)  pp. 978–992</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">  S. Lefschetz,  "Differential equations: geometric theory" , Interscience  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.J. Stoker,  "Nonlinear vibrations in mechanical and electrical systems" , Interscience  (1950)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.A. Dorodnitsyn,  "Asymptotic solution of the van der Pol equation"  ''Priklad. Mat. Mekh.'' , '''11'''  (1947)  pp. 313–328  (In Russian)  (English abstract)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.F. Mishchenko,  N.Kh. Rozov,  "Differential equations with small parameters and relaxation oscillations" , Plenum  (1980)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  B. van der Pol,  "On oscillation hysteresis in a triode generator with two degrees of freedom"  ''Philos. Mag. (6)'' , '''43'''  (1922)  pp. 700–719</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  B. van der Pol,  ''Philos. Mag. (7)'' , '''2'''  (1926)  pp. 978–992</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">  S. Lefschetz,  "Differential equations: geometric theory" , Interscience  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.J. Stoker,  "Nonlinear vibrations in mechanical and electrical systems" , Interscience  (1950)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.A. Dorodnitsyn,  "Asymptotic solution of the van der Pol equation"  ''Priklad. Mat. Mekh.'' , '''11'''  (1947)  pp. 313–328  (In Russian)  (English abstract)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.F. Mishchenko,  N.Kh. Rozov,  "Differential equations with small parameters and relaxation oscillations" , Plenum  (1980)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v09603021.png" /> the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096030/v09603022.png" /> terms of the series for amplitude and period have been computed by symbolic calculation, see [[#References|[a1]]]. The computation of [[#References|[5]]] has been refined in [[#References|[a2]]]. For a recent survey of the free and forced van der Pol oscillator, see [[#References|[a3]]].
+
For small $  \mu $
 +
the first $  164 $
 +
terms of the series for amplitude and period have been computed by symbolic calculation, see [[#References|[a1]]]. The computation of [[#References|[5]]] has been refined in [[#References|[a2]]]. For a recent survey of the free and forced van der Pol oscillator, see [[#References|[a3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.B. Dadfar,  J. Geer,  C.M. Andersen,  "Perturbation analysis of the limit cycle of the free Van der Pol equation"  ''SIAM J. Appl. Math.'' , '''44'''  (1984)  pp. 881–895</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Bavinck,  J. Grasman,  "The method of matched asymptotic expansions for the periodic solution of the Van der Pol equation"  ''Int. J. Nonlin. Mech.'' , '''9'''  (1974)  pp. 421–434</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Grasman,  "Asymptotic methods for relaxation oscillations and applications" , Springer  (1987)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Guckenheimer,  P. Holmes,  "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.B. Dadfar,  J. Geer,  C.M. Andersen,  "Perturbation analysis of the limit cycle of the free Van der Pol equation"  ''SIAM J. Appl. Math.'' , '''44'''  (1984)  pp. 881–895</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Bavinck,  J. Grasman,  "The method of matched asymptotic expansions for the periodic solution of the Van der Pol equation"  ''Int. J. Nonlin. Mech.'' , '''9'''  (1974)  pp. 421–434</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Grasman,  "Asymptotic methods for relaxation oscillations and applications" , Springer  (1987)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Guckenheimer,  P. Holmes,  "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer  (1983)</TD></TR></table>

Revision as of 08:27, 6 June 2020


The non-linear second-order ordinary differential equation

$$ \tag{1 } \dot{x} dot - \mu ( 1 - x ^ {2} ) \dot{x} + x = 0,\ \ \mu = \textrm{ const } > 0,\ \ \dot{x} ( t) \equiv { \frac{dx}{dt} } , $$

which is an important special case of the Liénard equation. Van der Pol's equation describes the auto-oscillations (cf. Auto-oscillation) of one of the simplest oscillating systems (the van der Pol oscillator). In particular, equation (1) serves — after making several simplifying assumptions — as a mathematical model of a generator on a triode for a tube with a cubic characteristic. The character of the solutions of equation (1) was first studied in detail by B. van der Pol .

Equation (1) is equivalent to the following system of two equations in two phase variables $ x, v $:

$$ \tag{2 } \dot{x} = v,\ \ \dot{v} = - x + \mu ( 1 - x ^ {2} ) v. $$

It is sometimes convenient to replace the variable $ x $ by the variable $ z( t) = {\int _ {0} ^ {t} } x ( \tau ) d \tau $; equation (1) then becomes

$$ \dot{z} dot - \mu \left ( \dot{z} - \frac{\dot{z} ^ {3} }{3} \right ) + z = 0, $$

which is a special case of the Rayleigh equation. If, together with $ x $, one also considers the variable $ y = - x + ( x ^ {3} /3) + ( \dot{x} / \mu ) $, introduces a new time $ \tau = t / \mu $ and puts $ \epsilon = \mu ^ {-} 2 $, one obtains the system

$$ \tag{3 } \epsilon x ^ \prime = \ y - x + \frac{x ^ {3} }{3} ,\ \ y ^ \prime = - x,\ \ {} ^ \prime = \frac{d}{d \tau } , $$

instead of equation (1). For any $ \mu > 0 $ there exists a unique stable limit cycle in the phase plane of the system (2) to which all other trajectories (except for the equilibrium position at the coordinate origin) converge as $ t \rightarrow \infty $; this limit cycle describes the oscillations of the van der Pol oscillator [2], [3], [4].

For small $ \mu $ the auto-oscillations of the oscillator (1) are close to simple harmonic oscillations (cf. Non-linear oscillations) with period $ 2 \pi $ and specified amplitude. In order to calculate the oscillation process more accurately, asymptotic methods are employed. As $ \mu $ increases, the auto-oscillations of the oscillator (1) deviate more and more from harmonic oscillations. If $ \mu $ is large, equation (1) describes relaxation oscillation with period $ 1.614 \mu $( to a first approximation). More accurate asymptotic expansions of magnitudes characterizing relaxation oscillations [5] are known: The study of these oscillations is equivalent to the study of the solutions of the system (3) with a small coefficient $ \epsilon $ in front of the derivative [6].

The equation

$$ \dot{x} dot - \mu ( 1 - x ^ {2} ) \dot{x} + x = E _ {0} + E \sin \omega t $$

describes the behaviour of the van der Pol oscillator when acted upon by a periodic external disturbance. The most important in this context is the study of frequency capture (the existence of periodic oscillations), beats (the possibility of almost-periodic oscillations) and chaotic behaviour [2], [4].

References

[1a] B. van der Pol, "On oscillation hysteresis in a triode generator with two degrees of freedom" Philos. Mag. (6) , 43 (1922) pp. 700–719
[1b] B. van der Pol, Philos. Mag. (7) , 2 (1926) pp. 978–992
[2] A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Dover, reprint (1987) (Translated from Russian)
[3] S. Lefschetz, "Differential equations: geometric theory" , Interscience (1957)
[4] J.J. Stoker, "Nonlinear vibrations in mechanical and electrical systems" , Interscience (1950)
[5] A.A. Dorodnitsyn, "Asymptotic solution of the van der Pol equation" Priklad. Mat. Mekh. , 11 (1947) pp. 313–328 (In Russian) (English abstract)
[6] E.F. Mishchenko, N.Kh. Rozov, "Differential equations with small parameters and relaxation oscillations" , Plenum (1980) (Translated from Russian)

Comments

For small $ \mu $ the first $ 164 $ terms of the series for amplitude and period have been computed by symbolic calculation, see [a1]. The computation of [5] has been refined in [a2]. For a recent survey of the free and forced van der Pol oscillator, see [a3].

References

[a1] M.B. Dadfar, J. Geer, C.M. Andersen, "Perturbation analysis of the limit cycle of the free Van der Pol equation" SIAM J. Appl. Math. , 44 (1984) pp. 881–895
[a2] H. Bavinck, J. Grasman, "The method of matched asymptotic expansions for the periodic solution of the Van der Pol equation" Int. J. Nonlin. Mech. , 9 (1974) pp. 421–434
[a3] J. Grasman, "Asymptotic methods for relaxation oscillations and applications" , Springer (1987)
[a4] J. Guckenheimer, P. Holmes, "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer (1983)
How to Cite This Entry:
Van der Pol equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Van_der_Pol_equation&oldid=49106
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article