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Homoclinic bifurcation (b), FP1, FP1, FP1, FP1, FP1, FP2, FP2, FP2, FP2,…
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When \(\mu\) increases from negative to positive passing through 0, the system goes through a supercritical Hopf bifurcation -- the stable spiral becomes slower and slower close to the origin (as clearly seen by the extremely dense plot near the origin in the \(\mu=0\) plot) as we go from negative towards 0, and then as we go further the spiral becomes unstable with a small limit cycle appearing around the origin.
It continues growing as we continue increasing \(\mu\) and eventually intersects the saddle point when \(\mu=\mu_c\) -- turning into a homoclinic orbit (we have a Homoclinic bifurcation). As we increase further it overshoots the saddle point and turns into an ordinary trajectory.
