Computational sensitivity in the analysis of Torus and its bifurcations

In this study, we investigate an invariant three-torus (IT3) and the related bifurcations generated in a three-coupled delayed logistic map. Here, IT3 in this map corresponds to a four-torus in vector fields. First, we reveal that, to observe a clear Lyapunov diagrams for quasi-periodic oscillations, it is necessary and important to remove a large number of transient iterations, for example, 10,000,000 transient iteration count as well as 10,000,000 stationary iteration count to evaluate Lyapunov exponents in this higher dimensional discrete-time dynamical system even if the parameter values are not chosen near the bifurcation boundaries. Second, by observing the graph of Lyapunov exponents, the global transition from an invariant two-torus (IT₂) to an IT₃ is a Neimark-Sacker type bifurcation that should be called a one-dimensional higher quasi-periodic Hopf bifurcation of an IT₂. In addition, we confirm that another bifurcation route from an IT2 to an IT3 would be caused by a saddle-node type bifurcation that should be called a one dimensional higher quasi-periodic saddle-node bifurcation of an IT₂.