# sol3 - MAE294B/SIO203B Methods in Applied Mechanics Winter...

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MAE294B/SIO203B: Methods in Applied Mechanics Winter Quarter 2010 http://maecourses.ucsd.edu/mae294b Solution III 1 Draw the loci where ˙ x vanishes and ˙ x is singular. The former are ﬁxed points and correspond to μ = x 2 and x + μ 2 = 2. The latter correspond to x + μ 2 and are not ﬁxed points. However they do latter separate regions where ˙ x has different signs. There are four intersection points of these loci. The points ( 1 , 1 ) and ( 1 . 8312 , - 1 . 3532 ) are standard transcritical bifurcations. The points ( 0 , 0 ) and ( 1 , - 1 ) are not normal bifurcations since they contain the singular curve. However they look like transcritical bifurcations since two “stable” and two “unstable” singularities/ﬁxed points meet. Figure 1 shows the diagram. 2 The simplest way to do this is to use the iteration u n + 1 = nt - ε sin u n , u 0 = 0 . Then u 1 = nt , u 2 = nt - ε sin nt and u 3 = nt - ε sin [ nt - ε sin nt ] = nt - ε [ sin nt - ε sin nt cos nt + O ( ε 2 ) = nt - ε sin nt

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## This note was uploaded on 03/16/2010 for the course MAE 294b taught by Professor Young,w during the Winter '08 term at UCSD.

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sol3 - MAE294B/SIO203B Methods in Applied Mechanics Winter...

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