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 show that in normalized families of polynomial or rational maps, Misiurewicz maps (critically finite or infinite) unfold generically. For example, if $f_{{\lambda }_0}$ is critically finite with non-degenerate critical point $c_1\left({\lambda }_0\right),...,c_n\left({\lambda }_0\right)$ such that $f_{{\lambda }_0}^{k_i}\left(c_i\left({\lambda }_0\right)\right)=p_i\left({\lambda }_0\right)$ are hyperbolic periodic points for i = 1,...,n, then  IV-1. Age impartible......................................................................................................................................................................... 31  $\lambda \mapsto (f_{\lambda }^{k_1}\left(c_1\left(\lambda \right)\right)-p_1\left(\lambda \right),...,f_{\lambda }^{k_{d-2}}\left(c_{d-2}\left(\lambda \right)\right)-p_{d-2}\left(\lambda \right))$ is a local diffeomorphism...

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...