Limit cycles of non-autonomous scalar ODEs with two summands

Abstract

We establish upper bounds for the number of limit cycles (isolated periodic solutions in the set of periodic solutions) of the two families of scalar ordinary differential equations $x'=(a(t) x +b(t)) f(x)$ and $x'=a(t) g(x) +b(t)f(x)$, where $f(x)$ and $g(x)$ are analytic funtions and $a(t)$, $b(t)$ are $T$--periodic continuous functions for which there exist $\alpha, \beta \in R$ such that $\alpha a(t)+\beta b(t)$ is not identically zero and does not change sign in $[0,T]$. As a consequence we obtain that generalized Abel equations $x'=a(t)x^n + b(t)x^m$, where $n> m \geq 1$ are natural numbers, have at most three limit cycles.