Existence of at most two limit cycles for some non-autonomous differential equations

Abstract

It is know that the non-autonomous differential equations $ {\rm d}x/{\rm d} t = a(t)+b(t)|x| $, where $ a(t) $ and $ b(t) $ are 1-periodic maps of class $ \mathcal{C}^1 $, have no upper bound for their number of limit cycles (isolated solutions satisfying $ x(0) = x(1) $). We prove that if either $ a(t) $ or $ b(t) $ does not change sign, then their maximum number of limit cycles is two, taking into account their multiplicities, and that this upper bound is sharp. We also study all possible configurations of limit cycles. Our result is similar to other ones known for Abel type periodic differential equations although the proofs are quite different.