This work introduces a chaotic satellite system including the Caputo–Prabhakar fractional derivative and investigates the characteristics and complex dynamics of the system. The asymptotic stability of the nonlinear Caputo–Prabhakar fractional system is analyzed by assessing the eigenvalues of the Jacobian matrix of system in the complex eigenvalues plane. To solve the Caputo–Prabhakar fractional chaotic satellite system numerically and exhibit the dynamic characteristics of the system, we state a numerical algorithm. Next, we prove the existence and uniqueness of the solution to the system and analyze the dynamical behaviors of the system around the equilibria. Then, we control the chaotic vibration of the system by means of the feedback control procedure and the Lyapunov second method. Further, chaos synchronization is achieved between two identical Caputo–Prabhakar fractional chaotic satellite systems by designing control laws. Furthermore, we illustrate the effect of the parameters of Caputo–Prabhakar fractional derivative on the dynamic behaviors of the system. Choosing the suitable values of the mentioned derivative parameters is able to successfully increase the stability region and achieve chaos control without any controllers. By numerical simulations, we show that the appropriate values of the derivative parameters in the Caputo–Prabhakar fractional satellite system are able to return back the satellite’s attitude to its equilibrium point, when the satellite attitude is tilted of this point. This is despite the fact that the integer-order form of the system and even the fractional system with the Caputo fractional derivative remain chaotic.
