MAT 119A Study Guide - Midterm Guide: Hopf Bifurcation, Bifurcation Diagram, Limit Cycle
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Winter 2008: (41 points) consider the equation x = x x. What type of bifurcation is it? (c) what type of bifurcation would occur if x = x x. O( 2). (c) calculate the averaged equation and nd the amplitude and frequency of any limit cycles for the original system. Solve the averaged equations explicitly to nd x(t, ) and show that it is consistent with the exact solution that you obtained in part a: (45 points) consider the nonlinear system of odes: X = y x((x2 + y2)2 (x2 + y2) 1) 1) And use the poincar e-bendixson theorem to show that there exist a stable limit cycle. (c) show that a hopf bifurcation occurs at xed point (0, 0) and = 1.
