# Limit cycle continuation folding over itself

I was doing some computations in MATCONT. First I continued equlibrium point. Hopf bifurcation occured for $$a = 6$$ and branching point/pitchfork for $$a = -2$$. Continuing forward and backward this branching point BP discover another Hops at $$a = -6$$. See diagram bellow

Then I started continuation of limit cycle in terms of $$a$$ again from these Hops.

Or more conventional view

1st question: It is impossible to continue these cycles further even with their number exceeds 10k :) Does it mean that I have to stop and just make only continuation of the period doublings (red ones)?

Then I did continuation in $$a$$ of Hopf for $$a>0$$.

I was not expecting this behaviour at all and I have hard time to interpret it or how to check obtained result by integration in time domain. 2nd question How do you interpret LC continuation which is going backward, then suddenly fold over itself and is going back. How can I verify these by time domain integration? Is it that there exist two limit cycles so I should be able to reach on of them using different initital value?

Should I check both backward and forward continuation as well? Does it make any sense to you?

• Did you choose to detect all bifurcations that happen along continuation path? It's hard to judge by your diagrams at this moment, but could it be that you turned off LPC bifurcation? It's a saddle-node bifurcation of limit cycles and it might produce folds on such diagrams too. Sep 25, 2021 at 14:16
• Yes, there is LPC Sep 25, 2021 at 15:04