We consider limit cycles of a class of polynomial differential systems of the form $$\left\{\begin{array}{c}\dot{x}=y,\hfill \\ \dot{y}=-x-\epsilon (g_{21}\left(x\right)y^{2\alpha +1}+f_{21}\left(x\right)y^{2\beta })-{\epsilon }^2(g_{22}\left(x\right)y^{2\alpha +1}+f_{22}\left(x\right)y^{2\beta }),\hfill \end{array}\right.$$ where $\beta$ and $\alpha$ are positive integers, $g_{2j}$ and $f_{2j}$ have degree $m$ and $n$, respectively, for each $j=1,2$, and $\epsilon$ is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center $\dot{x}=y$, $\dot{y}=-x$ using the averaging theory of first and second order.

We first examine conditions implying monotonicity of the period function for potential systems with a center at 0 (in the whole period annulus). We also present a short comparative survey of the different criteria. We apply these results to quadratic Loud systems $\left(L_{D,F}\right)$ for various values of the parameters D and F. In the case of noncritical periods we propose an algorithm to test the monotonicity of the period function for $\left(L_{D,F}\right)$. Our results may be viewed as a contribution to proving (or disproving) a conjecture...

We are interested in conditions under which the two-dimensional autonomous system ẋ = y, ẏ = -g(x) - f(x)y, has a local center with monotonic period function. When f and g are (non-odd) analytic functions, Christopher and Devlin [C-D] gave a simple necessary and sufficient condition for the period to be constant. We propose a simple proof of their result. Moreover, in the case when f and g are of class C³, the Liénard systems can have a monotonic period function...