An explicit expression of the first Lyapunov and period constants with applications (Q1363555)

By using and extending the method and results of a former paper of \textit{A. Gasull} and \textit{R. Prohens} [Rocky Mt. J. Math. 26, No. 1, 135-164 (1996; Zbl 0855.34032)], the authors got explicit expressions of the first three Lyapunov constants \(V_3\), \(V_5\), \(V_7\) and of the first two period constants \(p_2\), \(p_4\) for a general system: \[ \dot z= iz+\sum_{k\geq 2} F_k(z,\overline z),\quad z\in\mathbb{C} \] having the origin \(z= x+iy=0\) as a critical point with pure imaginary eigenvalues. Applications are made to three special polynomial systems and a Lotka-Volterra system. The authors obtain necessary and sufficient conditions for a center or an isochronous center, as well as the maximum order of a weak focus.