# Van der Pol equation

The non-linear second-order ordinary differential equation (1)

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 : (2)

It is sometimes convenient to replace the variable by the variable ; equation (1) then becomes which is a special case of the Rayleigh equation. If, together with , one also considers the variable , introduces a new time and puts , one obtains the system (3)

instead of equation (1). For any 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 ; this limit cycle describes the oscillations of the van der Pol oscillator , , .

For small the auto-oscillations of the oscillator (1) are close to simple harmonic oscillations (cf. Non-linear oscillations) with period and specified amplitude. In order to calculate the oscillation process more accurately, asymptotic methods are employed. As increases, the auto-oscillations of the oscillator (1) deviate more and more from harmonic oscillations. If is large, equation (1) describes relaxation oscillation with period (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 in front of the derivative .

The equation 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 , .