Effects of global warming, time delay and chaos control on the dynamics of a chaotic atmospheric propagation model within the frame of Caputo fractional operator.

The Lorenz-84 climate model is a simplified mathematical model that describes the chaotic behavior of atmospheric convection and its impact on global climate patterns. It captures the presence of chaotic behavior in the motion of westerly wind and helps understand the concept of sensitivity to initial conditions. But as observed over the years, the westerlies are gaining more strength due to the rise in atmospheric temperature. In this work, we have modified the old model to observe the changes in the behavior of the system due to global warming and time delay. The modified model has been generalized using Caputo fractional derivative to provide a more accurate representation of the system with memory effect and non-local behavior. The stability of the new model has been tested at all the equilibrium points. Using Picard's operator and Banach's Fixed Point theorem, it has been shown that there exists a unique and bounded solution for the new model. It has been observed that the sole effect of global warming makes the system gradually unstable from chaotic as the fractional order α is decreased from 0.80 to 0. 50. Also, a shift in the bifurcation point has been noticed for the new model. All three Lyapunov exponents have been calculated for different fractional orders to confirm the presence of chaos in the modified model as well. A chaos control law has been constructed for the modified chaotic model using the sliding mode control theory. Interestingly, the chaos disappears completely when the effect of time-delay is considered in the modified model. Since our proposed time-delayed modified model shows an asymptotically stable nature for all fractional orders α less than 0. 85 , it is better suited to make more accurate predictions about the strength of the westerlies. • The Lorenz-84 climate model has been modified by incorporating the radiative forcing of CO 2 and time-delay. • The stability analysis, existence, uniqueness and boundedness are presented for the modified model. • Bifurcation analysis and Lyapunov exponent analysis have been performed to check the presence of chaos in the modified model. • A chaos control law has been constructed for the modified model. • The efficient predictor–corrector numerical method is applied to the new systems to demonstrate their nature. [ABSTRACT FROM AUTHOR]