Design and implementation of a sliding mode controller and adaptive sliding mode controller for a novel fractional chaotic class of equations

In this article, we investigate the dynamics of a three-dimensional class of systems using Caputo fractional derivative. To regulate the chaos of this fractional order system, we propose a novel approach involving a sliding mode controller and an adaptive sliding mode controller. The study includes the existence and Ulam-Hyers stability results, the derivation of conditions ensuring global stability for the controlled system, even in the presence of novel uncertainties and outer interruptions. Our main contribution lies in introducing the adaptive law combined with sliding mode control, which effectively illustrates the chaotic behavior and dissipative nature of the chaotic system. This innovation has promising implications for various real-world applications where the control of chaotic dynamics is critical, such as in secure communication systems or ecological modeling Fitting the Nose-Hoover chaotic system to the defined class of equations, we have analyzed the impact of fractional order derivatives in the system and differentiate between chaotic and hyper chaotic behaviors by employing Lyapunov exponents. This differentiation has significant implications for understanding the underlying dynamics of complex systems in science and engineering. Furthermore, we have performed the bifurcation analysis to depict the chaotic behavior in the novel Nose-Hoover class, shedding light on the rich and intricate behaviors that fractional order dynamics can exhibit. To validate the proposed control scheme, we conduct numerical simulations and demonstrate its efficiency through the imposition of exponential and logarithmic uncertainties and disturbances. The model simulation results are provided.