Consider the following weakly nonlinear oscilator, x + x + (x 1) = 0, - where...

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Consider the following weakly nonlinear oscilator, x + x + (x 1) = 0, - where 0 < < 1. In this problem you will use two timing to find the location of closed trajectories for this equation. (a) Define the two time scales 7 = t and T = et, and write the solution x(T,T) as a perturbation series x = x0 + x1 + x + ... Using the order e terms, find an expression for xo expressed in terms of an amplitude R(T) and a phase (T). You may use the expressions for x and derived in class by the chain rule, = x1 +E +E +0(), Oxo +2e- axo Jx1 + +0(). 72 Consider the following weakly nonlinear oscilator, x + x + (x 1) = 0, - where 0 < < 1. In this problem you will use two timing to find the location of closed trajectories for this equation. (a) Define the two time scales 7 = t and T = et, and write the solution x(T,T) as a perturbation series x = x0 + x1 + x + ... Using the order e terms, find an expression for xo expressed in terms of an amplitude R(T) and a phase (T). You may use the expressions for x and derived in class by the chain rule, = x1 +E +E +0(), Oxo +2e- axo Jx1 + +0(). 72