Let's define a parametric vector $(f(t),g(t))$.
If there a way to determinate how many times this function will come back to a certain point ?
For instance here is the plot of $(f(x),g(x))=\left(-2 x-\log (1-x)+0.1 \log (x)-0.9,\log (1 - x)+x^2+0.9 x\right)$ in the interval $x\in[0,0.7]$
And we count one loop, and one intersection in this loop.
Or if you look at $(f(x),g(x))=(x \sin(x),x \cos(x))$ in the interval $x\in[-10,10]$ you observe 3 points of intersection.
However for the function : $(f(x),g(x))=(\sin(x), x \cos(x))$, there are two points of intersection, but each point counts many passages :
So my question is wether there is a way to determine wether there exists a loop and one many time is the curve going through the intersection in an interval.
The last functions are easier to handle but I'm looking for a general approach and criteria, for more complex functions like the first one. The final result could be gotten numerically. But I don't know what is the best way to pose the problem, and I assume there exists already some litterature about it.
Any references are welcome.
EDIT :
One trivial necessary condition is the following : there must exist 1 inflexion point for each of the two functions.