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.

这项工作介绍了一个混沌卫星系统，包括 Caputo-Prabhakar 分数导数，并研究了该系统的特征和复杂动力学。通过在复特征值平面上评估系统的雅可比矩阵的特征值来分析非线性Caputo-Prabhakar分数系统的渐近稳定性。为了数值求解Caputo-Prabhakar分数混沌卫星系统并展示系统的动态特性，我们提出了一种数值算法。接下来，我们证明了系统解的存在性和唯一性，并分析了系统在平衡点附近的动力学行为。然后，我们通过反馈控制程序和Lyapunov第二方法控制系统的混沌振动。更远，通过设计控制律在两个相同的 Caputo-Prabhakar 分数混沌卫星系统之间实现混沌同步。此外，我们说明了 Caputo-Prabhakar 分数导数参数对系统动态行为的影响。选择合适的上述导数参数值能够成功地增加稳定区域并在没有任何控制器的情况下实现混沌控制。通过数值模拟，我们表明，当卫星姿态从该点倾斜时，Caputo-Prabhakar分数卫星系统中适当的导数参数值能够将卫星的姿态返回到其平衡点。