TITLE: How to locate non-circular limit cycles?
QUESTION [6 upvotes]: 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).

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

REPLY [3 votes]: 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.