Cubic Lienard equation with quadratic damping. I
The authors study the general cubic Lienard system \[ \dot x=y-a_1x-a_2x^2-a_3x^3,\quad \dot y=-b_1x-b_2x^2-b_3x^3.\tag{1} \] It is assumed that \(a_2^2-3a_1a_3>0.\) By using the Zhang Zhifen theorem and the Poincaré-Bendixson theorem, conditions of existence and uniqueness of one limit cycle when \(a_1a_3<0\) for system (1) with one equilibrium are given. Applying Hopf bifurcation theory and qualitative analysis, sufficient conditions are obtained that system (1) has at least two limit cycles when \(a_1a_3>0\). Using a simulation program, three examples with one limit cycle and two limit cycles are presented to demonstrate the theoretical analysis.
scientific article; zbMATH DE number 1462493