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This study investigates an invariant three-torus (IT 3 ) and related quasi-periodic bifurcations generated in a three-coupled delayed logistic map. The IT 3 generated in this map corresponds to a four-dimensional torus in vector fields. First, to numerically calculate a clear Lyapunov diagram for quasi-periodic oscillations, we demonstrate that it is necessary that the number of iterations deleted as a transient state is large and similar to the number of iterations to be averaged as a stationary state. For example, it is necessary to delete 10,000,000 transient iterations to appropriately evaluate the Lyapunov exponents if they are averaged for 10,000,000 stationary state iterations in this higher-dimensional discrete-time dynamical system even if the parameter values are not chosen near the bifurcation boundaries. Second, by analyzing a graph of the Lyapunov exponents, we show that the local bifurcation transition from an invariant two-torus (IT 2 ) to an IT 3 is caused by a Neimark-Sacker bifurcation, which can be a onedimension-higher quasi-periodic Hopf (QH) bifurcation. In addition, we confirm that another bifurcation route from an IT 2 to an IT 3 can be generated by a saddle-node bifurcation, which can be denoted as a one-dimension-higher quasi-periodic saddle-node bifurcation. Furthermore, we find a codimension-two bifurcation point at which the curves of a QH bifurcation and a quasi-periodic saddle-node bifurcation intersect.

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