Bifurcation of 3-Torus Attractor in Certain Continuous-Time Dynamical Systems*

Previously, we investigated the bifurcation of 2-torus attractor in continuous-time dynamical systems. The systems we investigated was a phase-locked loop with two inputs and a three-coupled van der Pol oscillators. The former example demonstrated saddle-node and double covering bifurcations and the latter a Neimark-Sacker bifurcation of 2-torus. Lyapunov exponents and Lyapunov bundles were used to analyse these bifurcations. In this paper, we investigate bifurcation of 3-torus in the phase-locked loop with three sinusoidal inputs and the four-coupled van der Pol oscillators. Our method to analyse bifurcation of 3-torus uses Poincare map and Poincare slice together with Lyapunov exponents and Lyapunov bundles.