Abstract
Compared with fractional-order chaotic systems with a large number of dimensions, three-dimensional or integer-order chaotic systems exhibit low complexity. In this paper, two novel four-dimensional, continuous, fractional-order, autonomous, and dissipative chaotic system models with higher complexity are revised. Numerical simulation of the two systems was used to verify that the two new fractional-order chaotic systems exhibit very rich dynamic behavior. Moreover, the synchronization method for fractional-order chaotic systems is also an issue that demands attention. In order to apply the Lyapunov stability theory, it is often necessary to design complicated functions to achieve the synchronization of fractional-order systems. Based on the fractional Mittag–Leffler stability theory, an adaptive, large-scale, and asymptotic synchronization control method is studied in this paper. The proposed scheme realizes the synchronization of two different fractional-order chaotic systems under the conditions of determined parameters and uncertain parameters. The synchronization theory and its proof are given in this paper. Finally, the model simulation results prove that the designed adaptive controller has good reliability, which contributes to the theoretical research into, and practical engineering applications of, chaos.
Highlights
Research on chaotic systems has not ceased over the past fifty years
It has been found that fractional-order chaotic systems have higher nonlinearity and a higher spreading power spectrum compared to integral ones [23]
Motivated by the above discussion, this paper investigates two novel four-dimensional fractional-order chaotic systems with different complexities, and establishes a universal adaptive law based on the Mittag–Leffler [48,49] fractional stability theory
Summary
Research on chaotic systems has not ceased over the past fifty years. Scholars have discovered a large number of three-dimensional integer-order chaotic systems, such as the Lorenz system [1,2,3], the. It has been found that fractional-order chaotic systems have higher nonlinearity and a higher spreading power spectrum compared to integral ones [23] These systems have a very broad range of potential applications in the field of secure communication and other related sciences where chaotic synchronization is the key technology. Feki Moez combined the Lyapunov stability theory with an adaptive law to realize the synchronization of an integer-order Lorenz system and applied it to secure communication [31]. Motivated by the above discussion, this paper investigates two novel four-dimensional fractional-order chaotic systems with different complexities, and establishes a universal adaptive law based on the Mittag–Leffler [48,49] fractional stability theory.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
