# 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].

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

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]

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

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