van der Pol equation

In 1920 the Dutch physicist Balthasar van der Pol studied a differential equationMathworldPlanetmath 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 equationMathworldPlanetmath equation is a case of a Lienard system and is expressed as a second order ordinary differential equation


or a first order planar ordinary differential equation

x˙ = y+μ(x-x3)
y˙ = -x

where μ is a real parameter. The parameter μ is usually considered to be positive since the the term -μ(1-x2) adds to the model a nonlinear damping. [C]


  • If μ=0 then the origin is a center. In fact, if μ=0 then


    and if we suppose that the initial conditionMathworldPlanetmath are (x0,x˙0) then the solution to the system is


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

  • If μ>0 the system has a unique limit cycleMathworldPlanetmath, and the limit cycle is attractive. This follows directly from Lienard’s theorem. [P]

  • The system is sometimes given under the form

    X˙ = -Y
    Y˙ = X+μ(1-X2)Y

    which equivalent to the previous planar system under the change of coordinate (X,Y)=(3x,-3(y+μ(x-x3))).[C]

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 μ=1, the system has an attractive limit cycle and the origin is a repulsive focus.

Phase portrait when μ=1.

When the parameter μ=0 the origin is a center.

Phase portrait when μ=0.

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

Phase portrait when μ=-1.


  • C Chicone, Carmen, Ordinary Differential Equations with Applications, Springer, New York, 1999.
  • P Perko, Lawrence, Differential Equations and Dynamical SystemsMathworldPlanetmathPlanetmath, Springer, New York, 2001.
Title van der Pol equation
Canonical name VanDerPolEquation
Date of creation 2013-03-22 16:06:42
Last modified on 2013-03-22 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 34-00
Synonym van der Pol oscillator