(a) Van der Pol Equation. Determine the type of the critical point at (0, 0) when µ > 0, µ = 0, µ > 0. Show that if µ → 0, the isoclines approach straight lines through the origin. Why is this to be expected?
Rayleigh equation, show that the so-called Rayleigh equation5 y’' - µ(1 – 1/3 y'2)y' + y = 0 (µ > 0) also describes self-sustained oscillations and that by differentiating it and setting y = y' one obtains the van der Pol equation.
(c) Duffing equation the duffine equation is y’' + w02y + βy3 = 0 where usually |β| is small, thus characterizing a small deviation of the restoring force from linearity, β > 0 and β > 0 are called the cases of a hard spring and a soft spring, respectively. Find the equation of the trajectories in the phase plane. (Note that for β > 0 all these curves are closed).