This paper presents the control design framework for the hybrid synchronization (HS) and parameter identification of the 3-Cell Cellular Neural Network. The cellular neural network (CNN) of this kind has increasing practical importance but due to its strong chaotic behavior and the presence of uncertain parameters make it difficult to design a smooth control framework. Sliding mode control (SMC) is very helpful for this kind of environment where the systems are nonlinear and have uncertain parameters and bounded disturbances. However, conventional SMC offers a dangerous chattering phenomenon, which is not acceptable in this scenario. To get chattering-free control, smooth higher-order SMC formulated on the smooth super twisting algorithm (SSTA) is proposed in this article. The stability of the sliding surface is ensured by the Lyapunov stability theory. The convergence of the error system to zero yields hybrid synchronization and the unknown parameters are computed adaptively. Finally, the results of the proposed control technique are compared with the adaptive integral sliding mode control (AISMC). Numerical simulation results validate the performance of the proposed algorithm.