Combining the advantages of both integer-order and fractional-order complex chaotic systems, we propose a hybrid-order complex Lorenz system. We demonstrate its abundant chaotic characteristics, including symmetry and dissipation, fixed points and their stability and Lyapunov exponents, with 0-1 test. Then we show that, as the initial value, parameters and the order are varying, the system exhibits diverse dynamical behaviors, with fixed points, limit cycles and chaotic attractors. We further show that the system has coexisting attractors and parametric attractors. In addition, we find that the system generates different chaotic attractors as the system hybrid order varies, referred to as order attractors. Finally, we examine the dynamic transport of the hybrid-order complex Lorenz system and design a piecewise continuous controller to realize offset boosting control. By varying the initial value, parameters or orders, we realize the dynamic transport of the system. Our simulation results confirm the dynamic transport of the hybrid-order complex Lorenz system.
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