van der Pol Oscillator on the phase plane

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The 2nd order ODE for van der Pol oscillator

$\ddot{x}+(x^2-1)\dot{x}+x=0$ is written as a system of first order ODEs

$\begin{equation} \dot{x} = y +x-\frac{x^3}{3},\qquad \dot{y} = -x. \end{equation}$ Notice the unusual choice of the variable

$y=\dot{x}-x+x^3/3$ , which is more convenient to show the existence of the periodic solution (call limit cycle) than the conventional one

$y=\dot{x}$ . You can click your mouse on the plane to see how the points converge to the periodic solution.