Stability Analysis of Rössler Chaotic Attractor via the Nabla Discrete Fractional Operator: Existence, Uniqueness, Ulam–Hyers Stability, and Numerical Simulation

This research presents a fractional-order formulation and mathematical analysis of the Rössler chaotic attractor. By utilizing the Nabla discrete Atangana–Baleanu fractional difference derivative in the Caputo sense, the classical integer-order attractor is extended into the fractional domain. The existence and uniqueness of solutions for the resulting fractional system are established via the fixed-point theorem, thereby ensuring that the recommended attractor is well-posed. Furthermore, the Ulam–Hyers stability is investigated within the Nabla discrete Atangana–Baleanu fractional difference derivative in the Caputo sense framework. For numerical investigations, an Euler numerical scheme adapted to the fractional difference derivative is developed and implemented, yielding high-quality phase portraits of a chaotic attractor. The results highlight the effectiveness of fractional-order modeling and numerical methods in capturing the dynamics and stability of the Rössler chaotic system.