BifurcationsThe term bifurcation generally refers to something splitting apart. With

Chapter 6, Problem 36

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BifurcationsThe term bifurcation generally refers to something splitting apart. With regard to differential equations or systems involving a parameter, it refers to abrupt changes in the character of the solutions as the parameter is changed continuously. 33 through 36 illustrate sensitive cases in which small perturbations in the coefcients of a linear or almost linear system can change the type or stability (or both) of a critical point.This problem presents the famous Hopf bifurcation for the almost linear systemdx dt D *xCy !x.x2 Cy2/; dy dt D! x C*y !y.x2 Cy2/; which has imaginary characteristic roots $ D i if * D 0.(a) Change to polar coordinates as in Example 5 of Section 6.1 to obtain the system r0 Dr.*!r2/, &0 D! 1. (b) Separate variables and integrate directly to show that if * 5 0, thenr.t/!0 as t !C1, so in this case the origin is a stable spiral point. (c) Show similarly that if *>0 , then r.t/!p* as t !C1, so in this case the origin is an unstable spiral point. The circle r.t/$ p* itself is a closed periodic solution or limit cycle. Thus a limit cycle of increasing size is spawned as the parameter * increases through the critical value 0.

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