Phys601/F11/Problem Set 6Due 10/17/11 (upgraded) GPS:Do Ch8 #1, #27 6.1HVan der Pol Oscillator Consider the Van der Pol Oscillator for x(t) d2x/dt2+ x = - 2ν(dx/dt) (1-x2). Let x ~ 1 (“of order unity”), i.e., make no assumptions on whether x is small or large. This means that the sign of the anti-friction term on the RHS could change. a. Suppose ν<< 1. Solve for x perturbatively by expanding in a series x = x0 + x1 + x2….. assuming that each term is smaller than the previous term in the series. Show that the regular perturbation series exhibits secular behaviour. b. Solve by assuming a multiple time scale solution, i.e., assuming that x = x(t, τ), where τis over a longer time scale, in particular d/dt = ∂/∂t + ∂/∂τ, where |∂/∂τ| << |∂/∂t|. Obtain solutions to lowest significant order, i.e., make sure you get to the interesting stuff. Make a sketch of x(t) for the two cases 0 < x(0) < 1 and 1 < x(0). You will have to solve an equation using the “energy method”. Look up the integrals needed.
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Celestial mechanics, van der Pol, der Pol oscillator, Multiscale Bertrand