# (a) Determine the type of the critical point at (0, 0) when > 0, = 0, < 0. (b) Rayleigh equation. Show that the Rayleigh equation 0) class=fr-fic fr-dib>also describes self-sustained oscillations and that by differentiating it and setting y = Y' one obtains the van der Pol equation. (c) The Duffing equation is Y

Chapter 4, P R O B L E M S E T 4 . 5 #14

(a) Determine the type of the critical point at (0, 0) when Î¼ > 0, Î¼ = 0, Î¼ < 0.

(b) Rayleigh equation. Show that the Rayleigh equation

0) " class="fr-fic fr-dib">also describes self-sustained oscillations and that by differentiating it and setting y = Y' one obtains the van der Pol equation.

(c) The Duffing equation is

where usually |Î²| is small, thus characterizing a small deviation of the restoring force from linearity. Î² > 0 and Î² < 0 are called the cases of a hard spring and a soft spring, respectively. Find the equation of the trajectories in the phase plane.For Î² > 0 all these curves are closed.)

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