Liénard systems and potential-Hamiltonian decomposition. I: Methodology
A system of ordinary differential equations on the plane, \(\dot x = f(x,y)\), \(\dot y = g(x,y)\), where \(f\) and \(g\) are polynomials, is called potential-Hamiltonian decomposable (PH-decomposable) if there exist polynomial functions \(P, H : \mathbb{R}^2 \to \mathbb{R}\) such that \(f = -P_x + H_y\) and \(g = -P_y - H_x\), so that the correspodning vector field decomposes as the sum of a gradient and a Hamiltonian vector field, \(-\text{grad}\,P + JdH\). The authors obtain an explicit PH-decomposition for any polynomial system. They indicate its usefulness in studying limit cycles that arise in Liénard systems. [See also the following parts: part II: ibid., No. 3, 191--194 (2007 ; Zbl 1111.37010) and part III: ibid., No. 4, 253--258 (2007; Zbl 1111.37012)].
scientific article; zbMATH DE number 5130343