Homoclinic-doubling and homoclinic-gluing bifurcations in the Takens-Bogdanov normal form with D4 symmetry

In this paper, we investigate the dynamics of a fourth-order normal form near a double Takens-Bogdanov bifurcation. The reduced system of this normal form possesses eight pairs of homoclinic orbits for certain parameter values. The nonlinear time transformation method is applied to obtain an analytical approximation of the homoclinic orbit in the perturbed system and to construct the homoclinic bifurcation curve as well. Using numerical continuation, period-doubling and homoclinic-doubling cascades emanating from a codimension-2 bifurcation point are found. A codimension-2 homoclinic-gluing bifurcation point at which several homoclinic orbits concerning the origin glue together to form a new homoclinic orbit is also obtained. It is shown that in the vicinity of these bifurcation points, the system may exhibit chaos and chaotic attractors.