# How to plot Van der Pol equation's limit cycle? [duplicate]

The Van der Pol equation is

$$y''-\mu (1-y^2)y'+y=0, \,\, \mbox{where}\,\, \mu ≥ 0.$$

Use the Runge-Kutta Method to plot the limit cycle and some solutions in the phase plane (the $yy'$-plane) both inside and outside of the limit cycle for a few different values of µ.

• I'm voting to close this question as off-topic because it shows no effort whatsoever and sounds like it is an assignment from a lecture. – halirutan May 7 '17 at 1:30

The best option for you is to use NDSolve. Since you have not provided specific initial conditions, so I choose randomly.

sol[mu_?NumericQ] := First@NDSolve[{y''[t] - mu*(1 - y[t]^2) y'[t] + y[t] == 0, y[0] == 0,
y'[0] == 1}, y, {t, 0, 10}]

ParametricPlot[Evaluate[{y[t], y'[t]} /. sol[#] & /@ Range[0, 1, 0.2]], {t, 0, 10}]


You can also use ParametricNDSolve,

sol = ParametricNDSolveValue[{y''[t] - mu*(1 - y[t]^2) y'[t] + y[t] ==  0, y[0] == 0,
y'[0] == 1}, y, {t, 0, 10}, {mu}];

ParametricPlot[Evaluate[{sol[#][t], D[sol[#][t], t]} & /@ Range[0, 2, 0.1]], {t, 0, 10}]