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Recently I've encountered limit cycles in dynamical systems which are basically a cyclic trajectory.
From what I understand, there are different types of limit cycles for example stable and unstable limit cycle (for an $\epsilon$ environment we’ll be pulled into the cycle or pushed out of the cycle environment). One of the ways to locate a limit cycle is to present the dynamic system with polar coordinates and try to find the right values such that $r'$ will be $0$, which will point out a limit cycle with unchanging radius (a circle).

But I’ve seen that there are limit cycles which are not circular, for instance in the attached picture, we can see there is a limit cycle with changing radius ($r'$ is probably a function of $\theta$ as well).

Non circular limit cycle

My question is: How can we locate the limit cycle in these situations?

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Itay.V
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  • What software did you use for the phase portrait? Would you mind to share the code? Always looking for good plots :D. As Soon As I have access to a PC I will write you an answer. – MrYouMath Sep 29 '17 at 22:07
  • well i got that picture from wikipedia but you can do that with couple of lines in Maple its great software for mathamatics and for differential equations and dynamic systems in particular. i think i can find some code for plots like these (its really friendly-usable software for ploting things out you should try it out).

    oh and thanks in advance (:

     – Itay.V Sep 29 '17 at 22:09
  • Maple and Matlab plots are not very good looking. I always need to take the data and use tikz to make them pretty :D. – MrYouMath Sep 29 '17 at 22:39
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    Also don't forget that if your system comes as a member of some parameterized family, you should look for a Hopf bifurcation — this is one of the simplest bifurcations that give rise to limit cycles. Hopf bifurcation points is also a good starting point for collocation methods: you know where approximately solution is, that's the benefit. – Evgeny Sep 30 '17 at 15:37

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With the possible exception of some carefully crafted examples, you have to resort to numerical methods:

  • The trajectory of stable limit cycles can be found by integrating the differential equations. If the solution exhibits a periodic behaviour (after a while), you have a limit cycle. If an optical assessment of periodicity is not sufficient or possible, you can compute the Lyapunov exponents (if the largest one is zero and the next is negative, you have a limit cycle) or apply this periodicity test of mine.

  • To locate an unstable limit cycle, you can try integrating backwards in time. If this doesn’t work, e.g., because the backwards integration is not a well-behaving problem, you have to resort to collocation methods. Briefly, you set up a generic limit cycle composed of piecewise polynomials and try to choose the parameters such that the trajectory solves your differential equation at certain, fine-sampled points. For further reading, consider starting here.

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