Global dynamical behavior of a generalized Muthuswamy-Chua-Ginoux system

This paper is dedicated to studying the global dynamics of a three dimensional circuit differential system defined by the equations$ \dot x = 1-x-y, \; \dot y = lx+(a+s)y+dyz+cyz^2+b y^3, \ \; \dot z = (cz^2+dz+s)y^2-\epsilon z, $which we refer to the generalized MCG system, where all parameters are real and $ bcl\varepsilon\ne0 $. Restrictedto an invariant plane $ z = 0 $ if $ s = 0 $, we obtain the fine focus conditions and the number of small-amplitude limit cycles by calculating Lyapunov constants and applying Hopf bifurcation theory and the averaging method. Then we analyze the dynamics near infinity on the Poincaré disc by Poincaré compactifications and the method of generalized normal sector. Furthermore, applying Poincaré compactifications in $ \mathbb{R}^3 $, we can characterize the dynamics of the generalized MCG system on the Poincaré sphere at infinity.