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The center problem and bifurcations of limit cycles for the Lyapunov system are continuously studied. We shall prove that we can construct successively a formal series such that the Lyapunov system is reduced a half-normal form. From the coefficients of the half-normal form, we obtain directly the Lyapunov constants of the origin. As examples, for two classes of cubic systems, the center and focus problem, and multiple bifurcations of limit cycles are studied. [ABSTRACT FROM AUTHOR]

This paper proves that a Z6-equivariant planar polynomial vector field of degree 5 has at least six symmetric centers, if and only if it is a Hamiltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z6-equivariant perturbations, the conclusion that the perturbed system has at least 24 limit cycles with the scheme 〈 4 ∐ 4 ∐ 4 ∐ 4 ∐ 4 ∐ 4〉 is rigorously proved. Two schemes of distributions of limit cycles are given. [ABSTRACT FROM AUTHOR]

By introducing a concept of the time-angle difference for planar complex analytic systems, two recursive formulas of the function of the time-angle difference and period constants of complex isochronous centers can be given. Using these results to a class of real cubic system, the first five period constants and the isochronous center conditions of the origin are obtained. One open problem posed by [Cairo et al., 1999] is proved. [ABSTRACT FROM AUTHOR]