VAN-DER-POL'S EQUATION

As an application of the Modern Taylor Series Method the solution of the well known Van-der-Pol's equation

$\frac{d^2y}{dt^2} + \mu (y^2-1)\frac{dy}{dt} + y = 0$

is described.

Let

$\frac{dy}{dt} = y_1.$

Then

$\frac{dy_1}{dt}= - y -\mu(y^2-1)y_1.$

The corresponding source text in TKSL/386 is

var
  y,y1;
const
  mi = 3, p1 = 0, p2 = 1,
  tmax = 10, eps = 1E-20;
system
  y' = y1                          & p1;
  y1'= mi*y1*(1-y*y)-y             & p2;
sysend.

Time functions ORD, y and y1 are in Fig.5.1. The aim of Fig.5.1 is to point out two things. First, the values of ORD are high and second, these values vary considerably during the computation.

 

Figure 5.1:

Fig.5.2 shows the solution in a better scale. The solution of the Van-der-Pol equation is shown using phase-plain in Fig.5.3.

 

Figure 5.2:

 

Figure 5.3: