Synchronization and anti-synchronization of a novel fractional order chaotic system with a quadratic term

This paper introduces a new three-dimensional chaos system that is different from any existing chaotic systems. The basic dynamic properties of the suggested system have been analyzed through theoretical and numerical studies and the rich chaotic behavior has been confirmed using phase portraits, Lyapunov analysis and bifurcation diagram. A simple model with a nonlinear quadratic term, unstable saddle equilibrium points and broadband chaotic behavior are some of the interesting features of the suggested system. Then, due to higher accuracy of fractional order models than integer order models, a novel fractional chaotic system has been extracted based on the suggested chaotic system and new nonlinear methods planned to achieve the synchronization and anti-synchronization goals for the system. A backstepping approach is designed to synchronize and ensure stability in Lyapunov's concept. Also, the anti-synchronization between the two novel fractional systems is achieved by applying the active control technique, and subsequently Lyapunov stability is shown under the proposed scheme. The simulation results in MATLAB environment show the synchronization and anti-synchronization efficiency for the proposed innovative fractional order turbulent system.