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 usedto model other phenomenon such as the human heartbeat byJohannes van der Mark[C].

The van der Pol equationequation is a case of a Lienard system and is expressed asa 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 sincethe 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

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

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

• If $\mu >0$ the system has a unique limit cycle, and the limitcycle 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 systemunder 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 doneby taking initial condition froman equally spaced grid and calculating the solution for positive andnegative time.

For the parameter $\mu =1$, the system hasan 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 isan attractive focus.

Phase portrait when $\mu =-1$.