This paper constructs a new five–dimensional hyper–chaotic system with complex dynamic behaviors. It also analyzes the chaotic attractor, bifurcation diagram, equilibrium points, Poincare map, Kaplan–Yorke dimension and Lyapunov exponent behaviors. We prove that the introduced new hyper-chaotic system has complex and nonlinear behaviors. Next, the work describes fast terminal sliding mode control scheme for the control and synchronization of the new hyper–chaotic system. Stability analysis is performed using the Lyapunov stability theory. For the synchronization, both master and slave systems are perturbed by different parameter and model uncertainties. Both steps of the sliding mode controller have finite–time convergence properties. Subsequently, it has been shown that the state paths of both master–slave systems can reach each other in a finite time. One of the main features of the proposed controller is the finite time stability of the terminal sliding surface designed with high–order power function of error and derivative of error. Finally, using the MATLAB simulation, the results are confirmed for the new hyper–chaotic system.