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]

•
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
$$x(t)={x}_{0}\mathrm{cos}t+{\dot{x}}_{0}\mathrm{sin}t.$$ 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 $\mu =1$.
References
 C Chicone, Carmen, Ordinary Differential Equations with Applications, Springer, New York, 1999.
 P Perko, Lawrence, Differential Equations and Dynamical Systems^{}, Springer, New York, 2001.
Title  van der Pol equation 

Canonical name  VanDerPolEquation 
Date of creation  20130322 16:06:42 
Last modified on  20130322 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 3400 
Synonym  van der Pol oscillator 