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We consider limit cycles of a class of polynomial differential systems of the form
where and are positive integers, and have degree and , respectively, for each , and is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center , 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 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 . 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...
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