midsol[1]

midsol[1] - MAE294B/SIO203B: Methods in Applied Mechanics...

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MAE294B/SIO203B: Methods in Applied Mechanics Winter Quarter 2010 http://maecourses.ucsd.edu/mae294b Midterm solution 1 Sketch the bifurcation diagram for the equation ˙ x = 4 - 4 x 2 - μ 2 4 + 4 x 2 + μ 2 ( μ - x 2 ) 2 - 1 ( μ - x 2 ) 2 + 1 and state the nature of the bifurcations. A graphical answer will suffice. See Figure 1. There are saddle-node bifurcations at ( - 2 , 0 ) , ( 0 , 0 ) , ( 1 , 0 ) and ( 2 , 0 ) . There are transcritical bifurcations at ( 0 , - 1 ) , ( 0 , 1 ) , ( μ c , x c ) and ( μ c , - x c ) , where μ c = 2 ( 3 - 1 ) and x c = p 2 3 - 2 satisfy μ = 1 + x 2 and x 2 + μ 2 / 4 = 1. 2 Skip the naive expansion. The leading-order solution is x 0 = A ( T ) e i t + A * e - i t + c . c . At O ( ε ) , one finds x 1 tt + x 1 + 2 x 0 tT + x 2 0 x 0 t + x 3 0 = 0 . Secular terms look like e ± i t so look at x 2 0 ( x 0 t + x 0 ) = ( A e i t + A * e - i t ) 2 ( i A e i t - i A * e - i t + A e cit + A * e - i t ) = ( 3 + i ) A 2 A * e i t + ··· . The amplitude equation is
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midsol[1] - MAE294B/SIO203B: Methods in Applied Mechanics...

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