Numerical Analysis of a Normalized Time-Fractional Lorenz System

In this paper, we present numerical analysis of a normalized time-fractional Lorenz system. To accurately determine which factor influences the outcome in an experiment, the effects of all other variables must be eliminated. To enable a fair and interpretable comparison across varying fractional orders, we apply a normalized time-fractional derivative that keeps the total integral of the weight function normalized to one. Based on this normalization, comparisons of the dynamical influence of the fractional order [Formula: see text] can be conducted fairly because only the distribution of memory weights changes while the overall scale remains consistent. The proposed model allows us to distinguish whether the observed behavior arises from changes in [Formula: see text] itself or from variations in the accumulated memory strength, and thus it significantly enhances the interpretability of the results. This method enables a detailed exploration of dynamical behaviors — including both stability and chaos — by adjusting the [Formula: see text] value. Computational experiments confirm that the model offers rich dynamical features for modeling nonlinear systems with memory effects.