Van der Pol's equation

Question

Consider the van der Pol equation below: (x′′)+c(x^2−1)(x′)+(x)=0 , which governs the flow of current in a vacuum tube with three internal elements. It can be written as the system x′= y , y′= -x-c(x^2 - 1).y By setting c = 2/3, x(0) = 1 and y(0) = 0,

i need to :

1. find x(1) and y(1).

2)graph the solution that satisfies these initial conditions, i) x(t) and y(t), ii) x(t) versus y(t)

3)present a version of the previous graph with control added to allow interactive manipulation of the value of c. Observe and explain the results.

Answer 1

A phase space plot of the homogeneous Van Der Pole equation is easily obtained. E.g.:

tmax = 15;
Manipulate[
 eq = {x''[t] + c (x[t]^2 - 1) x'[t] + x[t] == 0, x[0] == 1, 
   x'[0] == 0};
 f[t_] = x[t] /. NDSolve[eq, x[t], {t, 0, tmax}][[1]];
 ParametricPlot[{f[t], f'[t]}, {t, 0, tmax}]
 , {c, 0., 1}, TrackedSymbols :> {c}]

The rest you should be able to figure out yourself.