This paper presents a novel four-dimensional (4D) memristive system, notable for its simplicity and unique dynamic behaviors. Comprising only seven terms and devoid of equilibrium points, this system is capable of generating hidden attractors. A comprehensive analysis of its dynamic properties is conducted, including Lyapunov exponents, Kaplan–Yorke dimensions, temporal diagrams and phase portraits, multistability, and offset boosting. Numerical simulations over a sufficiently long time interval show that the Lyapunov function remains bounded, thereby confirming the dissipative nature and global stability of the newly proposed 4D hyperchaotic system. The theoretical model is further validated through electronic simulations of the chaotic system using Multisim software. Additionally, the paper explores synchronization between two identical 4D hyperchaotic systems. The proposed system, despite its structural simplicity, exhibits intricate chaotic dynamics, making it suitable for various practical applications. In particular, a method for chaotic encryption and decryption of an information signal was developed and validated through numerical testing. Based on the method of complete synchronization of chaotic systems, the possibility of detecting a weak signal is demonstrated. The proposed system is also implemented on an Arduino UNO to demonstrate its practical applicability for real-time chaotic signal generation and image encryption.