van der Pol equation

In 1920 the Dutch physicist Balthasar van der Pol studied a differential equation that describes the circuit of a vacuum tube. It has been used to model other phenomenon such as the human heartbeat by Johannes van der Mark[C].

The van der Pol equation equation is a case of a Lienard system and is expressed as a second order ordinary differential equation

$$\frac{d^{2}x}{d{t}^2}-\mu (1-x^2)\frac{dx}{dt}+x=0$$

or a first order planar ordinary differential equation

$\dot{x}$ $=$ $y+\mu (x-x^3)$

$\dot{y}$ $=$ $-x$

where $\mu$ is a real parameter. The parameter $\mu$ is usually considered to be positive since the the term $-\mu (1-x^2)$ adds to the model a nonlinear damping. [C]

Properties:

• If $\mu =0$ then the origin is a center. In fact, if $\mu =0$ then

  $$\frac{d^{2}x}{d{t}^2}+x=0$$

  and if we suppose that the initial condition are $(x_0,{\dot{x}}_0)$ then the solution to the system is

  $$x(t)=x_0\cos t+{\dot{x}}_0\sin t.$$

  All solutions except the origin are periodic and circles. See phase portrait below.

• If $\mu >0$ the system has a unique limit cycle, and the limit cycle is attractive. This follows directly from Lienard’s theorem. [P]

• The system is sometimes given under the form

  $\dot{X}$ $=$ $-Y$

  $\dot{Y}$ $=$ $X+\mu (1-X^2)Y$

  which equivalent to the previous planar system under the change of coordinate $(X,Y)=(\sqrt{3}x,-\sqrt{3}(y+\mu (x-x^3)))$.[C]

Example:
The geometric representation of the phase portrait is done by taking initial condition from an equally spaced grid and calculating the solution for positive and negative time.

For the parameter $\mu =1$, the system has an attractive limit cycle and the origin is a repulsive focus.

Phase portrait when $\mu =1$.

When the parameter $\mu =0$ the origin is a center.

Phase portrait when $\mu =0$.

For the parameter $\mu =-1$, the system has a repulsive limit cycle and the origin is an attractive focus.

Phase portrait when $\mu =-1$.

Title	van der Pol equation
Canonical name	VanDerPolEquation
Date of creation	2013-03-22 16:06:42
Last modified on	2013-03-22 16:06:42
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	13
Author	Daume (40)
Entry type	Definition
Classification	msc 34C25
Classification	msc 34C07
Classification	msc 34-00
Synonym	van der Pol oscillator