Analyzing the fractional Chen attractor via controlled interpolative metric contractions

The present study advances the idea of controlled $(\alpha ,c)$–interpolative metric spaces as a broader framework that extends interpolative and extended interpolative frameworks and examines their analytical properties in conjunction with Banach-type fixed-point results. The proposed metric structure is then employed to study the fractional-order Chen attractor model defined via the Atangana–Baleanu–Caputo ( $\mathcal{A}\mathcal{B}\mathcal{C}$) derivative. Using the newly established metric framework, a suitable fixed-point operator is constructed in the Banach space $C([0,T];{\mathbb{R}}^3)$, and sufficient contraction conditions ensuring the existence and uniqueness of solutions are derived. The consistency and accuracy of the proposed numerical formulation are demonstrated through simulation results, highlighting improved sensitivity and richer chaotic behavior compared to integer-order models. The obtained results confirm that the controlled interpolative metric approach provides a flexible analytical tool for exploring nonlinear fractional systems and offers new insights into the stability analysis of chaotic dynamical models such as the Chen attractor.