Firstly, a new fractional-order chaotic system is proposed. When the linear term x in the second formula of the system was replaced by its absolute value, the range of its unique parameter k that makes the wing of the original system doubled is explored in detail. Furthermore, the numerical simulation and the circuit simulation of the original system and its double-wing system are achieved via Matlab and Multisim software respectively. Finally, based on sliding mode control theory and stability theory in fractional calculus, a new sliding mode controller is designed to realize the synchronization of the new system and its double-wing system respectively. Simulation results are provided to illustrate the effectiveness of the proposed scheme.