I am currently taking a course in dynamical systems and am planning to do a project on Hilbert's 16th problem/counting limit cycles. Unfortunately, every major related theorem I've found is too advanced for me (fourth year applied math major).
I am currently searching for related results that I would be able to prove (bounds on $H(n)$, existence theorems for limit cycles, etc.) I've found a few promising leads for lower bounds for $H(n)$, but I would love to hear any suggestions you might have. I am especially interested in proofs that use tools that appear frequently in the study of dynamical systems.