Do you know any such system (preferably a simple one, ideally quadratic) which has more than one limit cycle? I would prefer it if there was at most one unstable equilibrium.
The general problem of finding the number of limit cycles for a specific dynamical system is related to the Hilbert’s 16th problem. Among the dynamical systems with several limit cycles you can find Lienard systems, being the van der Pol system a particular case. Some references and examples can be found in:
M. A. F. Sanjuan, Lienard Systems, Limit Cycles, and Melnikov Theory”, Physical Review E57(1), 340-344 (1998)
There is a non-countable number of systems with that property of having more than one limit cycle. Van Der Pol is not quadratic.
Edited: I mentioned Van Der Pol since the author referred Van Der Pol as an instance of a quadratic map. After my answer that part of the question was removed from the question by the author.
I didn't mention Van Der Pol as a quadratic map. I mentioned it as a system which has only one limit cycle. I removed it because it caused misunderstanding.
Can you please mention one of those non-countable number of systems?
Can have as many as you please limit cycles depending on the zeros of f. It can have even an infinite number of limit cycles. The number of such systems in non-countable. A trivial change of variables gets what you want.
The general problem of finding the number of limit cycles for a specific dynamical system is related to the Hilbert’s 16th problem. Among the dynamical systems with several limit cycles you can find Lienard systems, being the van der Pol system a particular case. Some references and examples can be found in:
M. A. F. Sanjuan, Lienard Systems, Limit Cycles, and Melnikov Theory”, Physical Review E57(1), 340-344 (1998)