Bifurcations of invariant torus and knotted periodic orbits for the generalized Hopf-Langford system

In this paper, we study the bifurcations of invariant torus and knotted periodic orbits for the generalized Hopf-Langford system. By using bifurcation theory of dynamical systems, we obtain the exact explicit form of the heteroclinic orbits and knot periodic orbits. Moreover, under small perturbation, we prove that the perturbed planar system has two symmetric stable limit cycles created by Poincare bifurcations. Therefore, the corresponding three-dimensional perturbed system has an attractive invariant rotation torus.