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This paper focuses on investigating the maximum number of limit cycles bifurcating from the periodic orbits adapted to the cubic system given by ẋ=y−yx+a2,ẏ=−x+xx+a2,where a is a positive number with a≠1. The study specifically examines the perturbation of this system within the class of all septic polynomial differential systems. Our main result demonstrates that the first-order averaging theory associated with the perturbed system yields a maximum of twenty-two limit cycles.

By means of the averaging method of the first order, we introduce the maximum number of limit cycles which can be bifurcated from the periodic orbits of a Hamiltonian system. Besides, the perturbation has been used for a particular class of the polynomial differential systems.

In this work, we study the bifurcation of limit cycles from the period annulus surrounding the origin of a class of cubic polynomial differential systems; when they are perturbed inside the class of all polynomial differential systems of degree six, we obtain at most fifteenth limit cycles by using the averaging theory of first order.

In this paper, we study the number of limit cycles of a new class of polynomial differential systems, which is an extended work of two families of differential systems in systems considered earlier. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a center using the averaging theory of first and second order.

We apply the averaging theory of first and second order to a class of generalized Kukles polynomial differential systems to study the maximum number of limit cycles of these systems.