Module

for

The van der Pol System

 

The van der Pol Equation

    The van der Pol equation is

        [Graphics:Images/VanDerPolMod_gr_1.gif],  
where  
        [Graphics:Images/VanDerPolMod_gr_2.gif]  is a constant.  
        
When  [Graphics:Images/VanDerPolMod_gr_3.gif]  the equation reduces to  [Graphics:Images/VanDerPolMod_gr_4.gif],   and has the familiar solution  [Graphics:Images/VanDerPolMod_gr_5.gif].  Usually the term  [Graphics:Images/VanDerPolMod_gr_6.gif]  in equation (1) should be regarded as friction or resistance, and this is the case when the coefficient [Graphics:Images/VanDerPolMod_gr_7.gif]  is positive.  However, if the coefficient  [Graphics:Images/VanDerPolMod_gr_8.gif]  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  

    [Graphics:Images/VanDerPolMod_gr_9.gif],   
and  
    [Graphics:Images/VanDerPolMod_gr_10.gif].    

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 [Graphics:Images/VanDerPolMod_gr_11.gif].

Extension to 2D. The Runge-Kutta method is easily extended to solve a system of D.E.'s over the interval  [Graphics:Images/VanDerPolMod_gr_12.gif].

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  

    
[Graphics:Images/VanDerPolMod_gr_13.gif]  with  [Graphics:Images/VanDerPolMod_gr_14.gif],  
    
[Graphics:Images/VanDerPolMod_gr_15.gif]  with  [Graphics:Images/VanDerPolMod_gr_16.gif],  

over the interval  
[Graphics:Images/VanDerPolMod_gr_17.gif]  at a discrete set of points.

[Graphics:Images/VanDerPolMod_gr_18.gif]

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  [Graphics:Images/VanDerPolMod_gr_19.gif].  
    [Graphics:Images/VanDerPolMod_gr_20.gif].  
Solution 1.

 

Example 2.  Solve the van der Pol equation with  [Graphics:Images/VanDerPolMod_gr_29.gif].  
    [Graphics:Images/VanDerPolMod_gr_30.gif].  
Solution 2.

 

Example 3.  Solve the van der Pol equation with  [Graphics:Images/VanDerPolMod_gr_39.gif].  
    [Graphics:Images/VanDerPolMod_gr_40.gif].  
Solution 3.

 

Example 4.  Solve the van der Pol equation with  [Graphics:Images/VanDerPolMod_gr_49.gif].  
    [Graphics:Images/VanDerPolMod_gr_50.gif].  
Solution 4.

 

Example 5.  Solve the van der Pol equation with  [Graphics:Images/VanDerPolMod_gr_60.gif].  
    [Graphics:Images/VanDerPolMod_gr_61.gif].  
Solution 5.

 

Example 6.  Solve the van der Pol equation with  [Graphics:Images/VanDerPolMod_gr_70.gif].  
    [Graphics:Images/VanDerPolMod_gr_71.gif].  
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