Van der Pol system

The van der Pol oscillator is an oscillatory system with non-linear damping and self-excitation . For small amplitudes the damping is negative (the amplitude is increased); From a certain threshold value of the amplitude the damping becomes positive, the system stabilizes and goes into a limit cycle . The model was named after the Dutch physicist Balthasar van der Pol , who presented it in 1927 as the result of his research on oscillators with vacuum tubes .

application

The homogeneous (i.e. undisturbed) Van der Pol system fulfills the conditions of the Poincaré-Bendixson theorem , which is why chaos can not occur with it. In contrast, the conditions for the Poincaré-Bendixson theorem are no longer fulfilled in the inhomogeneous (i.e. disturbed) Van der Pol system , here deterministic chaos can occur.

Mathematical description

Homogeneous Van der Pol equation

Behavior of the homogeneous van der Pol equation

The dimensionless homogeneous differential equation of the second order

{\ displaystyle {\ ddot {x}} - \ varepsilon (1-x ^ {2}) {\ dot {x}} + x = 0}

with as a parameter and as a time-dependent variable describes the behavior of a free van der Pol oscillator over time. A closed solution does not exist. Stationary points are helpful to investigate the principle behavior . The following applies to: ${\ displaystyle \ varepsilon \ geq 0}$ ${\ displaystyle x}$ ${\ displaystyle x = \ mathrm {const}}$

{\ displaystyle {\ dot {x}} _ {s} = 0}

The linearization of the differential equation with

{\ displaystyle x (t) = x_ {s} + \ Delta x (t)}

results

{\ displaystyle \ Delta {\ ddot {x}} - \ varepsilon \ Delta {\ dot {x}} + \ Delta x = 0}

The characteristic equation is

{\ displaystyle \ lambda ^ {2} - \ varepsilon \ cdot \ lambda + 1 = 0}

with the solutions

{\ displaystyle \ lambda _ {1,2} = {\ frac {\ varepsilon} {2}} \ pm {\ frac {\ sqrt {\ varepsilon ^ {2} -4}} {2}}}

According to the size of, there are the following cases: ${\ displaystyle \ varepsilon}$

${\ displaystyle \ varepsilon> 2}$ ; exponential growth of the linearized system, d. H. the system is unstable around the stationary point
${\ displaystyle 0 <\ varepsilon <2}$ ; increasing vibrations
${\ displaystyle \ varepsilon = 0}$ ; harmonic oscillation .

The negative damping ( ) for small elongations of the oscillator becomes positive for larger elongations ( ). The oscillation is dampened in order to be stimulated again in the case of small elongations. ${\ displaystyle \ varepsilon> 0}$ ${\ displaystyle | x |> 1}$

Properties of the solution behavior are:

The period of the oscillation increases with the parameter .

{\ displaystyle \ varepsilon}

As it increases , the oscillation becomes more anharmonic and changes to tilting oscillations . ${\ displaystyle \ varepsilon \;}$

Regardless of the selected initial conditions , the system strives for a certain limit cycle.

The proof of the existence of an unambiguous, asymptotically stable limit cycle is made with the help of the Poincaré map .

Inhomogeneous Van der Pol equation

Behavior of the inhomogeneous Van der Pol equation

The dimensionless inhomogeneous differential equation of the second order

{\ displaystyle {\ ddot {x}} - \ varepsilon (1-x ^ {2}) {\ dot {x}} + x = F \ cdot \ sin (\ omega \ cdot t)}

describes the driven van der Pol oscillator with the amplitude and the angular frequency . ${\ displaystyle F}$ ${\ displaystyle \ omega}$

Some features of the solution:

For small amplitudes of the excitation, the system oscillates with the natural frequency .
For larger amplitudes, other frequencies occur in addition to the natural frequency and the excitation frequency. It shows quasiperiodic behavior: If one defines the following Poincaré cut with the time t

{\ displaystyle t = {\ frac {n \ cdot 2 \ pi} {\ omega}}, n \ in \ mathbb {N}}

the 2-dimensional (stroboscopic) image is obtained. One Lyapunov exponent is zero and the other is negative, which means quasi-periodic motion.

A further increase in the amplitude leads to locking: the system oscillates at the excitation frequency.

Web links

Commons : Van der Pol system - collection of pictures, videos and audio files

Individual evidence

↑ Gerald Teschl : Ordinary Differential Equations and Dynamical Systems (= Graduate Studies in Mathematics . Volume 140 ). American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( free online version ).

[1] Gerald Teschl : Ordinary Differential Equations and Dynamical Systems (= Graduate Studies in Mathematics . Volume 140 ). American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( free online version ).