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)

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)