Weak Focus, Limit Cycles, and Bifurcations for Bounded Quadratic Systems

We study the bounded quadratic systems with either two weak foci or a weak focus of order 2. From the first case we obtain (1, 1)-configuration of limit cycles, and in the second case we prove that there is no limit cycle surrounding the weak focus of order 2. Also, we unfold the bounded quadratic system with a center, and the maximum number of limit cycles in this case is two.