A Reservoir Computing Approach for Synchronizing Discrete-Time 3D Chaotic Systems

Reservoir computing (RC) is an efficient framework for processing time-series data. This work investigates the synchronization of two independently trained reservoir computers that, after training, operate without external input from the chaotic system and interact solely through symmetric linear coupling. This approach addresses a gap in existing reservoir computing-based synchronization studies, which predominantly rely on master–slave or system-driven configurations. In this work, we first build and train two reservoir computing models based on 3D nonlinear chaotic maps and hyperchaotic systems and then introduce a symmetric linear coupling mechanism between them. This study demonstrates that reservoir computing can accurately reproduce the short-term dynamics of chaotic systems and provides insight into the interactions between learned dynamical models, while also helping us understand how complex systems connect and operate collectively. We use this systematic approach to establish a framework for understanding how two trained reservoir computers interact under varying coupling strengths, enabling a detailed investigation of their synchronization behavior. To demonstrate the adaptability of the proposed framework to diverse dynamical behaviors, we systematically investigated three discrete chaotic and hyperchaotic systems: (1) discrete 3D sinusoidal map with discrete Lorenz attractor, (2) 3D sinusoidal map with conjoined Lorenz twin attractor, and (3) 3D quadratic hyperchaotic map. For performance evaluation, we trained coupled RCs and computed the synchronization error for different coupling strengths. We also present phase portraits and time-series plots of the attractors and RCs, along with the synchronization error as a function of the coupling strength, thereby demonstrating the possibility of synchronization of two linearly coupled RCs, which are independently trained on discrete, three-dimensional chaotic and hyperchaotic systems.