Math_345_April_2006

Math_345_April_2006 - Be sure that this examination has...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ) > 0 or f ( , r ) < 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...
View Full Document

Page1 / 3

Math_345_April_2006 - Be sure that this examination has...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online