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Memelements play an important part in the design of high density memory systems and low power memory. In this paper, we created a novel fractional-order chaotic circuit with a memristor and a memcapacitor with a linear inductor. The various dynamical properties of the fractional-order system are investigated by using some dynamic analysis methods like Lyapunov exponents and bifurcation after the numerical solution for the fractional-order system. In addition, to show the applicative advantages of the proposed chaotic system, we have realized about the synchronization of fractional-order chaotic systems and used it in secure communication systems first time in the literature according to our knowledge. The results of the theoretical analysis and simulation show that the simple fractional-order chaotic system has very rich dynamic properties and can be used in different engineering applications.