This problem presents the famous H opj bifurcation for the almost linear system dx - =

Chapter 7, Problem 7.3.36

(choose chapter or problem)

This problem presents the famous H opj bifurcation for the almost linear system dx - = EX + y -x(x 2 + l), dt dy 2 2 dt = -x + Ey - y(x + y ), which has imaginary characteristic roots A = i if E = O. (a) Change to polar coordinates as in Example 6 of Section 7.2 to obtain the system rl = r(E - r2), (}I = -1. (b) Separate variables and integrate directly to show that if E ; 0, then ret) as t +00, so in this case the origin is a stable spiral point. (e) Show similarly that if E > 0, then ret) .jE as t +00, so in this case the origin is an unstable spiral point. The circle r(t) == .jE itself is a closed periodic solution or limit cycle. Thus a limit cycle of increasing size is spawned as the parameter E increases through the critical value 0.