The van der Pol Equation

    The van der Pol equation is

        ,  
where  
          is a constant.  
        
When    the equation reduces to  ,   and has the familiar solution  .  Usually the term    in equation (1) should be regarded as friction or resistance, and this is the case when the coefficient   is positive.  However, if the coefficient    is negative then we have the case of "negative resistance."  In the age of "vacuum tube" radios, the "tetrode vacuum tube" (cathode, grid, plate),  was used for a power amplifier and was known to exhibit "negative resistance."  The mathematics is amazing too, and  van der Pol, Balthasar (1889-1959) is credited with developing equation (1).  The solution curves exhibits orbital stability.

    The van der Pol equation can be written as a second order system  

    ,   
and  
    .    

Any convenient numerical differential equation solver such as the Runge-Kutta method can be used to compute the solutions.  

Proof van der Pol System  van der Pol System  

Background. The Runge-Kutta method is used to numerically solve O.D.E.'s over .

Extension to 2D. The Runge-Kutta method is easily extended to solve a system of D.E.'s over the interval  .

Computer Programs  van der Pol System  van der Pol System  

Program (Runge-Kutta Method in 2D space)  To compute a numerical approximation for the solution of the initial value problem  

      with  ,  
      with  ,  

over the interval    at a discrete set of points.

Note.  The Runge-Kutta method in 2D is a "vector form" of the one-dimensional method, here the function f is replaced with F.

Example 1.  Solve the van der Pol equation with  .  
    .  
Solution 1.

Example 2.  Solve the van der Pol equation with  .  
    .  
Solution 2.

Example 3.  Solve the van der Pol equation with  .  
    .  
Solution 3.

Example 4.  Solve the van der Pol equation with  .  
    .  
Solution 4.

Example 5.  Solve the van der Pol equation with  .  
    .  
Solution 5.

Example 6.  Solve the van der Pol equation with  .  
    .  
Solution 6.

Research Experience for Undergraduates

The van der Pol Equation  The van der Pol Equation  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook The van der Pol System

(c) John H. Mathews 2004