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# PS6 - Phys601/F11/Problem Set 6 Due(upgraded GPS Do...

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Phys601/F11/Problem Set 6 Due 10/17/11 (upgraded) GPS: Do Ch8 #1, #27 6.1H Van der Pol Oscillator Consider the Van der Pol Oscillator for x(t) d 2 x/dt 2 + x = - 2 ν (dx/dt) (1-x 2 ). 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 = x 0 + x 1 + x 2 ….. 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 τ is 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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