On the existence, stability and chaos analysis of a novel 4D atmospheric dynamical system in the context of the Caputo fractional derivatives

In this study, changes in westerly waves and their connections to increased global warming under the influence of greenhouse gases were investigated via a Caputo fractional four-dimensional atmospheric system. The idea of the existence of chaotic behavior in the westerly wind's motion was depicted. It has been noted that westerlies are becoming stronger due to rising air temperatures. An analysis of the existence, uniqueness, boundedness, stability of equilibrium points, and conservative behavior of the solutions was conducted. To prove the existence of chaos in the modified model, the Lyapunov exponents, Poincaré map, and bifurcation were computed. A sliding mode controller to control the chaos in this novel fractional-order system was designed, and conditions for the global stability of the controlled system with and without external disturbances and uncertainties were derived. The finite-time interval for the system to reach the sliding surface was computed. The developed controller's performance was evaluated with respect to both commensurate and non-commensurate fractional derivatives. In each scenario, the impact of fractional orders was investigated. Numerical simulations were used to support theoretical statements about how the controller affects the system.