Efficient Controller design for optimal stabilization and synchronization of incommensurate non-integer order chaotic systems

Given the wide-ranging presence of chaotic systems in nature and technology , it is crucial for researchers to develop versatile and effective control designs to ensure the stability and synchronization of these non-linear complex systems. For this reason, this study proposes a novel efficient control using the Fractional-Order backstepping technique tuned by Genetic Algorithm optimization method to guarantee the stability and synchronization of incommensurate non-integer order chaotic systems. The genetic algorithm aims to minimize the control energy for both stability and synchronization of the two studied non-identical non-integer order chaotic systems configured in a master-slave form. The stability analysis in this study is based on the Mittag-Leffler principle to stabilize each subsystem, using chosen candidate functions that include fractional-order derivative terms. This makes the approach well-suited for Lyapunov stability analysis of the whole system. The results of applying the proposed control approach on non-integer form of Duffing-Holmes system and Van-der Pol oscillator show that the desired stability around the equilibrium point of these simulated non-identical chaotic systems was successfully achieved for different initial conditions. Simultaneously , zero synchronization error (e=0) between these two chaotic systems was ensured, thereby fulfilling the synchronization condition. This paper confirmed the effectiveness of the proposed control strategy, which integrates the fractional-order backstepping technique with a genetic algorithm, in achieving both stabilization and synchronization of incommensurate non-integer order chaotic systems under varying initial conditions.