A Comprehensive Analysis of Complex Dynamics in the Fractional-Order Rössler System

This paper proposes a numerical technique to study dynamical systems and uncover new behaviors in chaotic fractional-order models, a field that continues to attract significant research interest due to its broad applicability and the ongoing development of innovative methods. Through various types of simulations, this approach is able to uncover novel dynamic behaviors that were previously undiscovered. The results guarantee that initial conditions and fractional-order derivatives have a significant contribution to system dynamics, thus distinguishing fractional systems from traditional integer-order models. The approach demonstrated has excellent consistency with traditional approaches for integer-order systems while offering higher accuracy for fractional orders. Consequently, this approach serves as a powerful and efficient tool for studying complex chaotic models. Fractional-order dynamical systems (FDSs) are particularly noteworthy for their ability to model memory and hereditary characteristics. The method identifies new complex phenomena, including new chaos, unusual attractors, and complex time-series patterns, not documented in the existing literature. We use Lyapunov exponents, bifurcation analysis, and Poincaré sections to thoroughly investigate the system dynamics, with particular emphasis on the effect of fractional-order and initial conditions. Compared to traditional integer-order approaches, our approach is more accurate and gives a more efficient device for facilitating research on fractional-order chaos.