# Van der Pol equation

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. 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

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

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 , , .

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  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 .

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 , .

How to Cite This Entry:
Van der Pol equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Van_der_Pol_equation&oldid=52372
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article