In this paper, we consider the limit cycles of a class of polynomial differential systems of the form {ẋ = y − ε(g11 (x) y 2α+1 + f11 (x) y 2α) − ε 2 (g12 (x) y 2α+1 + f12 (x) y 2α), y = −x − ε(g21 (x) y 2α+1 + f21 (x) y 2α) − ε 2 (g22 (x) y 2α+1 + f22 (x) y 2α), where m, n, k, l and α are positive integers, g1κ, g2κ, f1κ and f2κ have degree n, m, l and k, respectively for each κ = 1, 2, and ε is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear centerẋ = y,ẏ = −x using the averaging theory of first and second order.
Up until now all the polynomial differential systems for which nonalgebraic limit cycles are known explicitly have degree odd. Here we show that that there are polynomial systems of even degree with explicit no-algebraic limit cycles. To our knowledge, there are no such type of examples in the literature.
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