# 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 [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

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

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 [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)

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].