The same distribution of limit cycles in a Hamiltonian system with nine seven-order perturbed terms
The authors investigate the number, distribution and accurate positions of limit cycles of nine perturbed planar Hamiltonian systems \[ \dot x = y(1+x^2 -ay^2) + \varepsilon p_i(x,y), \quad \dot y = -x(1-cx^2 +y^2) + \varepsilon q_i(x,y), \] where \(a>c>0\), \(ac>1\), \(0 < \varepsilon \ll 1\), the terms \(p_i(x,y)\) and \(q_i(x,y)\) are some polynomials in the ring \(\mathbb{R}[x,y]\) of degree \(7\) with coefficients depending on real parameters \(l,m,\lambda\), and \(i=1, \dots, 9\). Using qualitative analysis (e.g., methods of detection functions) and numerical simulation, the authors show that the above nine perturbed systems have the same bifurcation diagrams of limit cycles and have \(13\) limit cycles for some parameters. They illustrate the positions of \(13\) limit cycles and detection curves for some concrete parameters.
scientific article; zbMATH DE number 5278782