Heteroclinic Loop and Homoclinic Loop in a Controlled Chen System

The simulation of the Heteroclinic Loop and Homoclinic Loop in a controlled Chen system is finished. The controlled Chen system is $Z_2$ symmetric, and the limit cycle loop or the attractor is observed as a varying free parameter. As the loop becomes the boundary of the unstable manifold of the equilibrium solution, the heteroclinic orbit from the unstable equilibrium solution to the loop is formed. Usually, the twins’ unstable manifold appears due to $Z_2$ symmetry. The Generalized Hopf point brings forth the limit point cycle bifurcation, and nearby, the homoclinic bifurcation is observed. The homoclinic bifurcation arises since the equilibrium solution undergoes a Bogdanov-Takens bifurcation of codimension two. A novel phenomena of multi-loop coexistence are observed near the intersection point of the homoclinic bifurcation curve and Hopf line. Later, homoclinic curve tangents to the Bautin bifurcation line, the stable limit cycle expands into a homoclinic solution, which is called a limit point homoclinic solution.