Global dynamics of cubic Cherkas systems with <inline-formula><tex-math id="M1">$ \mathbb{Z}_2 $</tex-math></inline-formula>-equivariance (Ⅰ): The saddle case

In this paper, we investigate bifurcation diagrams and global phase portraits on the Poincaré discfor a cubic Cherkas system with symmetry. When the sum of indices of equilibria is $ -1 $, we obtain some interesting dynamical behaviors, including Hopf bifurcation, heteroclinic bifurcation, saddle-node bifurcation, and saddle connection bifurcation. These results greatly enrich the dynamics of planar nonlinear differential systems and demonstrate different features.