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?