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The cosmological dynamics are rigorously investigated through the systematicapplication of autonomous system analysis to the gravitational field equationsin non-metricity gravity. The systematic procedure to analyze the late-timecosmic acceleration in higher-order non-metricity gravity is demonstrated byexploring non-hyperbolic critical points with the center manifold theory. Thestability properties of these critical points are also evaluated based on theanalysis of eigenvalues and phase portraits. It is explicitly shown that thestable node can be realized. The critical points of each model are individuallyanalyzed, and their corresponding cosmological implications are derived. Thestability properties of these critical points are evaluated based on theanalysis of eigenvalues and phase portraits, revealing that each model includesat least one stable node. Furthermore, the evolution plots of the cosmologicalparameters confirm the models capacity to exhibit accelerated expansion.

Autonomous systems and attractor behaviors in non-metricity gravity: stability analysis and cosmic acceleration