Nonlinear analysis of the fractional Lorenz-84 model with a Rabotnov exponential kernel law

The dynamic behaviors of climate models have been extensively studied since Edward Norton Lorenz introduced the Lorenz-84 model in 1984. This paper investigates the chaotic character of the nonlinear Lorenz-84 system using a fractional derivative approach, specifically the Yang-Abdel-Cattani fractional operator. To approximate the considered system, we use a hybrid technique that combines the generalized transform with He’s polynomial approach. Our simulations include Poincaré maps, bifurcation diagrams, phase portrait graphs, time series plots, and sensitivity studies, providing comprehensive insights into the system’s behavior. To verify the existence and uniqueness of the model result, we apply the Picard operator and Banach fixed point theorem. The outcomes are visualized using MATLAB R2016a. A novelty of this work is the successful application of the Yang-Abdel-Cattani fractional derivative to the Lorenz-84 model. This innovative approach not only enhances the accuracy of the model but also opens new avenues for the application of fractional calculus in climate modeling. The model’s sensitivity to initial conditions and parameter variations is analyzed to understand its implications for atmospheric dynamics. Our results demonstrate that as the radiative forcing parameter increases, the system transitions from stable to chaotic behavior. These findings show the importance of considering fractional derivatives to accurately capture the chaotic dynamics of climate systems. The results reveal new dynamic behaviors not observable in the integer-order model, enhancing our understanding of fractional chaotic systems. This work offers a powerful methodology for solving complex differential equations and contributes significantly to the field of dynamical systems.