This paper presents a comprehensive qualitative investigation into the discrete-time topological transitions and localized phase space dynamics of a three-dimensional T-system. Crucially, rather than employing a conventional Euler discretization scheme, which frequently induces numerical instabilities and unphysical divergence, the continuous-time vector field is systematically mapped into a discrete layout using a non-local Non-Standard Finite Difference framework. Following the formulation of the rational map, the local topological architecture of the system is rigorously investigated at both the trivial equilibrium and the non-trivial interior equilibrium zones.

The primary objective of this work is to establish the precise boundaries governing the emergence of Neimark-Sacker bifurcations. To achieve this, we implement the explicit algebraic Flip and Neimark-Sacker bifurcation criteria developed, which bypasses traditional approximation limits by acting directly on the transcendental characteristic polynomials of the system matrices. Under specific parameter configurations, a scheme-induced subcritical Neimark-Sacker bifurcation artifact is exposed at the origin, evaluated via three-dimensional center manifold projections and complex normal form operators. Conversely, under the physical regime, the stability boundaries shift to the interior manifold, revealing a non-degenerate subcritical Neimark-Sacker bifurcation.

All theoretical analyses are seamlessly supported by comprehensive numerical experiments. Finally, high-density bifurcation diagrams and detailed phase space portraits are provided to visually confirm the structural evolution of the trajectories, capturing the birth, contraction, and expansion of the bifurcated discrete manifolds and invariant closed curves.