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In this paper, we propose a new three-dimensional fractional-order chaotic system with four nonlinear terms. The continuous fractional-order system is solved as a discrete map by employing Adomian Decomposition Method (ADM). Corresponding phase portraits for commensurate system are investigated and all Lyapunov characteristic exponents are calculated based on the QR decomposition method. It is found that the proposed fractional-order system exhibits chaotic behavior with fractional order less than three. Moreover, the problem of master-slave synchronization for commensurate system is considered and Lyapunov-based and modified versions of Laplacian-based synchronization methods are employed for this purpose. Simulation results for synchronization of the new system with these methods are presented. Effectiveness of these synchronization methods are compared against each other based on different performance criteria which shows the superiority of the modified Laplacian-based [ABSTRACT FROM AUTHOR]