I'm studying for an upcoming for an exam and I found a question I'm having trouble with in a past paper.

a) Assume a non-linear flow $\dot x = F_\mu(x)$. At $\mu = 4$ the only stable dynamics is a periodic orbit which has a Poincare map of period $3$. How will the dynamics change for increasing and decreasing values of $\mu$?

b) What is the minimum dimension of the state space of the flow $\dot x = F_\mu(x)$?

c) Assume the solution of a non-linear, deterministic, dissipative flow is non-periodic. State the two main features that characterize the dynamics of the flow.

d) List other possibilities that might lead to non-periodic dynamics even in linear systems.

Now in part a) I understand that there's a period $3$ limit cycle. So does that mean that reducing $\mu$ make the limit cycle contract onto a fixed point for a supercritical Hopf bifurcation? And increasing $\mu$ the dynamics don't really change?

For part b) I believe the answer is $3$ dimensions, since the limit cycle has to cross back to its start again without intersecting itself.

Parts c) and d) I just don't really have any idea where to start or what information the questions are asking for.

Can anybody help me understand what is going on in these questions?

  • $\begingroup$ Is there a reason why you are discussing only Hopf bifurcations? Have you discussed other possibilities in your class? $\endgroup$ – OnceUponACrinoid Jun 7 '15 at 17:46
  • $\begingroup$ I didn't mention the other types because I didn't think they were relevant in a scenario where there is a limit cycle? $\endgroup$ – null0pointer Jun 7 '15 at 18:07
  • 1
    $\begingroup$ There are indeed many kinds of codimension 1 bifurcations (See scholarpedia.org/article/Periodic_orbit). To understand the language of parts (c) and (d) you can see google.com/… Please post your attempt if you make progress and are still stuck. $\endgroup$ – OnceUponACrinoid Jun 7 '15 at 19:10

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