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The non-linear second-order ordinary differential equation

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

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 .

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

$$ 
\ddot{z}  - \mu \left (
\dot{z} - 
\frac{\dot{z}   ^ {3} }{3}

\right ) + z  =  0,
$$

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

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

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 [[#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

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

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]]].

====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>

====Comments====
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====
<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>