Limit Cycles of a Class of Perturbed Differential Systems Bifurcating From an Unperturbed Hamiltonian Center

We study the number of limit cycles of the following planar differential system: [Formula: see text] where [Formula: see text] are positive integers, for every [Formula: see text] the polynomial [Formula: see text] has degree [Formula: see text] with [Formula: see text] [Formula: see text] and [Formula: see text] are generalized trigonometric functions and [Formula: see text] is a small parameter. We provide an accurate upper bound of the maximum number of limit cycles such that the above system can have bifurcation from an unperturbed Hamiltonian center, using the averaging theory of the first order. These findings contribute to the broader understanding of nonlinear dynamics and have potential applications in various fields such as control systems, signal processing, and artificial intelligence.