Math_345_April_2006

# Math_345_April_2006 - Be sure that this examination has...

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Unformatted text preview: Be sure that this examination has three pages. The University of British Columbia Final Examinations April 2006 Mathematics 345 Applied Nonlinear Dynamics and Chaos Instructor: W. Nagata Time: 2 1 2 hours A non-programmable calculator, and one 8 1 2 00 11 00 page of notes may be used. No other aids are permitted. [25] 1. Consider the one-dimensional equation for ( t ) in the phase circle S 1 = R / (2 Z ), given by = f ( , r ) = r sin - sin 2 , (1) where r R is a parameter. (a) Observing in (1) that f (0 , r ) = 0 for all r , use linear stability analysis at the fixed point * = 0 (mod 2 ) to verify that the stability changes as r is increased through r c = 0. (b) What type of bifurcation (saddle-node, transcritical, or pitchfork) occurs at ( * , r c ) = (0 (mod 2 ) , 0)? Find the normal form of this bifurcation. ( Hint: an alternate expression for the vector field, f ( , r ) = r sin - 1 2 + 1 2 cos(2 ), may simplify calculations.) (c) Plot the set Z = { ( , r ) S 1 R : f ( , r ) = 0 } (horizonal axis r , vertical axis ). Noting that Z divides the r-cylinder into a finite number of open regions where f ( , r ) &gt; 0 or f ( , r ) &lt; 0, sketch all the qualitatively different phase portraits in S 1 that are possible for (1), for r only . ( Hints: f ( , r ) = sin ( r- sin ); it is not necessary to sketch the graphs of versus ; it is not necessary to solve for...
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## Math_345_April_2006 - Be sure that this examination has...

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