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With the aid of computer algebra system Mathematica 8.0 and by the integral factor method, for a family of generalized nilpotent systems, we first compute the first several quasi-Lyapunov constants, by vanishing them and rigorous proof, and then we get sufficient and necessary conditions under which the systems admit analytic centers at the origin. In addition, we present that seven amplitude limit cycles can be created from the origin. As an example, we give a concrete system with seven limit cycles via parameter perturbations to illustrate our conclusion. An interesting phenomenon is that the exponent parameter n controls the singular point type of the studied system. The main results generalize and improve the previously known results in Pan.

Limit Cycles and Analytic Centers for a Family of 4n-1 Degree Systems with Generalized Nilpotent Singularities