A New No Equilibrium Fractional Order Chaotic System, Dynamical Investigation, Synchronization and Its Digital Implementation

Abstract

In this paper, a new fractional order chaotic system without equilibrium is proposed, analyti-cally and numerically investigated, and numerically and experimentally tested. The analytical and numerical investigation were used to describe the system dynamical behaviors including, the system equilibria, the chaotic attractors, the bifurcation diagrams and the Lyapunov expo-nents. Based on the obtained dynamical behaviors, the system can excite hidden chaotic attrac-tors since it has no equilibrium. Then, a synchronization mechanism based on the adaptive con-trol theory has been developed between two identical new systems (master and slave). The adaptive control laws are derived based on synchronization error dynamics of the state varia-bles for the master and slave. Consequently, the update laws of the slave parameters are ob-tained, where the slave parameters are assumed to be uncertain and estimate corresponding to the master parameters by the synchronization process. Furthermore, Arduino Due boards were used to implement the proposed system in order to demonstrate its practicality in real-world applications. The simulation experimental results are obtained by MATLAB and the Arduino Due boards respectively, where a good consistent between the simulation results and the ex-perimental results. indicating that the new fractional order chaotic system is capable of being employed in real-world applications.