Nonlinear dynamics, adaptive control and synchronization of a system modeled by a chemical reaction with integer- and fractional-order derivatives

This work addresses the influence of the fractional derivative on the bifurcation and route to chaos of a system modeled by a chemical reaction subjected to an external periodic force on the one hand, and the adaptive control and the synchronization of the same system on the other hand. The mathematical model which governs the dynamics of the system has been proposed. The equilibrium points have been determined and their stabilities are analyzed in the commensurable case. Based on Lyapunov's stability theory, an adaptive control law has been designed to asymptotically stabilize the system state variables at the origin. Similarly, an adaptive synchronization law has been established in order to perform the identical synchronization of the system. Numerical simulations based on appropriate algorithms were used to plot phase portraits, times stories, bifurcation diagrams, Lyapunov exponent, route to chaos and also to show the effectiveness of the theoretical results. The study pointed out that the system can be controlled by acting on the parameters in presence or the order of the derivative. The decrease in the order of the derivative makes it possible to widen the zone of stability of the system.