This paper presents a chaotic complex system with a fractional-order derivative. The dynamical behaviors of the proposed system such as phase portraits, bifurcation diagrams, and the Lyapunov exponents are investigated. The main contribution of this effort is an implementation of Mittag-Leffler boundedness. The global attractive sets (GASs) and positive invariant sets (PISs) for the fractional chaotic complex system are derived based on the Lyapunov stability theory and the Mittag-Leffler function. Furthermore, an effective control strategy is also designed to achieve the global synchronization of two fractional chaotic systems. The corresponding boundedness is numerically verified to show the effectiveness of the theoretical analysis.
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