<p>This paper investigates (predefined-)time-synchronized((P)TS) control for two chaotic systems with different dimensions. A (P)TS fast terminal sliding mode control scheme is developed to ensure simultaneous convergence of both chaotic systems to the origin within a predefined time window, distinguishing it from conventional predefined-time control methods. To achieve the goal of fast convergence, a novel predefined-time stability theorem is first established. Subsequently, the concept of (P)TS stability is formally proposed, with a rigorous mathematical proof procedure explicitly delineating the controller's compliance criteria. Based on the predefined-time theorem, (P)TS stability theory, and sliding mode control framework, terminal sliding surface and controller are designed, accompanied by sufficient conditions and mathematical proof. For comparison with traditional terminal sliding mode methods, a parameter-equivalent predefined-time stable sliding surface and controller are derived. Numerical simulation verifies the effectiveness of the proposed control scheme to achieve synchronized control of the controlled error system, which achieves same time convergence within the predefined time, and has a faster convergence speed.</p>