# 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

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

or a first order planar ordinary differential equation

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

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

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

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

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

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 VanDerPolEquation 2013-03-22 16:06:42 2013-03-22 16:06:42 Daume (40) Daume (40) 13 Daume (40) Definition msc 34C25 msc 34C07 msc 34-00 van der Pol oscillator