PlanetMath: van der Pol equation

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van der Pol equation

(Definition)

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

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

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]

Properties:

If $\mu=0$ then the origin is a center. In fact, if $\mu=0$ then
$\displaystyle \frac{d^2x}{dt^2}+x=0$
and if we suppose that the initial condition are $(x_0,\dot{x}_0)$ then the solution to the system is
$\displaystyle x(t)=x_0\cos t + \dot{x}_0\sin t.$
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.

$\includegraphics[scale=0.5]{vanderpol_mu1.eps}$
Phase portrait when $\mu=1$ .

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

$\includegraphics[scale=0.5]{vanderpol_mu0.eps}$
Phase portrait when $\mu=0$ .

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

$\includegraphics[scale=0.5]{vanderpol_mum1.eps}$
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.

"van der Pol equation" is owned by Daume.

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Other names:

van der Pol oscillator

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Cross-references: negative, grid, coordinate, equivalent, limit cycle, circles, periodic, solution, initial condition, center, origin, positive, parameter, real, first order, ordinary differential equation, Lienard system, equation, differential equation
There are 2 references to this entry.

This is version 9 of van der Pol equation, born on 2006-07-27, modified 2006-08-02.
Object id is 8177, canonical name is VanDerPolEquation.
Accessed 3397 times total.

Classification:

AMS MSC:	34-00 (Ordinary differential equations :: General reference works )
	34C07 (Ordinary differential equations :: Qualitative theory :: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramif)
	34C25 (Ordinary differential equations :: Qualitative theory :: Periodic solutions)

Pending Errata and Addenda

None.[ View all 3 ]

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