Memristor is the fourth basic electronic element discovered in addition to resistor, capacitor, and inductor. It is a nonlinear gadget with memory features which can be used for realizing chaotic, memory, neural network, and other similar circuits and systems. In this paper, a novel memristor-based fractional-order chaotic system is presented, and this chaotic system is taken as an example to analyze its dynamic characteristics. First, we used Adomian algorithm to solve the proposed fractional-order chaotic system and yield a chaotic phase (...) diagram. Then, we examined the Lyapunov exponent spectrum, bifurcation, SE complexity, and basin of attraction of this system. We used the resulting Lyapunov exponent to describe the state of the basin of attraction of this fractional-order chaotic system. As the local minimum point of Lyapunov exponential function is the stable point in phase space, when this stable point in phase space comes into the lowest region of the basin of attraction, the solution of the chaotic system is yielded. In the analysis, we yielded the solution of the system equation with the same method used to solve the local minimum of Lyapunov exponential function. Our system analysis also revealed the multistability of this system. (shrink)