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Given the following van der Pol equation.

$$y''+u(y^{2}-1)y'+y=0$$

By numerically integrating the equation using any computer programme (such as Matlab) describe and deduce the nonlinear dynamics of the following system. (Observe the phase portrait at different values of $u$ and initial conditions such as in $0.1

What I tried/know:

  • By trying different values of $u$, I got a different phase portrait such as when $u=0.1$, I got a centre around the point $(0,0)$ and as $u$ increases the phase portrait slowly changes from a centre to a spiral.

  • I do know that at $|x|<1$ it acts like a ordinary damping and at $|x|>1$ the term acts like a negative damping, but I don’t quite get what the phase portrait means and its physical implication.

Could anyone show me the phase portrait and explain what this question means?

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    What do you mean by *centre?* A cycle or a single point? – 2017-02-17
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    A cycle along the point $(0,0)$ something like drawing a circle with the centre point $(0,0)$. – 2017-02-17
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    That would be a limit cycle. Please [edit] your question to add this information. – 2017-02-17
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    Please don't delete a question when someone has given a decent answer. – 2017-02-18

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In this context, phase portrait almost certainly refers to a bunch of trajectories of the dynamics in phase space. In your case, this would be for example a bunch of curves $(y(t), y'(t))$ for different initial conditions (usually equidistantly distributed over some area).

Looking at such a phase portrait can (empirically) answer you a lot of questions the dynamics:

  • What invariants sets (fixed points, limit cycles, strange invariant sets, …) are there, where are they located, and how many are they?

  • Are these sets attracting or repelling?

  • How do these sets depend on the parameters? In particular, are there any bifurcations (i.e., parameter values, at which these sets change their nature)?

From your question, it seems that you have already obtained single trajectories, from which there only is a small step to the phase portrait. Also, you already seem to have observed some invariant sets (or their absence).