Fractional synchronization involving fractional derivatives with nonsingular kernels: Application to chaotic systems

We present a novel master-slave fractional synchronization in chaotic systems by using fractional derivatives with nonlocal and nonsingular kernel. The master system is the fractional-order chaotic system, and then, we designed a fractional-order high gain observer, which is the slave system, based on the chaotic system model to achieve the synchronization. We used three different oscillator systems, a novel Chen-Burke-Shaw chaotic attractor, a novel fractional-order chaotic system, and the fractional-order model of a simple autonomous Jerk circuit. Our research followed to test the performance of the fractional synchronization. We use two performance indices, the integral of the square error (ISE) and the integral of the square error multiplied by time (ITSE). We showed that by using the fractional-order approach, we can reduce the values on the ISE and ITSE indices; hence, we guaranteed a full synchronization between the master and slave system. Our analysis shows that when using fractional derivatives with variable order, the ISE and ITSE indices have a lower value than when using the classical derivatives. We used the definition of Atangana-Baleanu-Caputo and Liouville-Caputo derivatives for the three examples mentioned in this work. We numerically solved the equations using the Adams method. Our results show that when using the fractional-order approach, the ISE and ITSE indices are lower than the integer-order case.