Half-stable limit cycle

The most important dynamic bifurcation is Hopf bifurcation. This occurs when Ai and A2 cross the imaginary axis into the right half-plane of C as the bifurcation parameter g changes. At the crossing point both roots are purely imaginary with det(A) > 0 and tr(A) = 0, making Ay2 = i y/det(A). At this value of g, periodic solutions (stable limit cycles) start to exist as depicted in Figures 10 and 11 (A-2). 

If all neighboring trajectories approach the limit cycle, we say the limit cycle is stable or attracting. Otherwise the limit cycle is unstable, or in exceptional cases, half-stable. 

At p a half-stable cycle is born out of the clear blue sky. As p. increases it splits into a pair of limit cycles, one stable, one unstable. Viewed in the other direction, a stable and unstable cycle collide and disappear as p decreases through p, Notice that the origin remains stable throughout it does not participate in this bifurcation. 

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