Deep Koopman–LQR synchronization of chaotic systems with application to secure communication: Resolving the vanishing-input pathology

Synchronization of chaotic systems is a key challenge in nonlinear dynamics, especially when optimality and stability certification are needed. Although Koopman-based control offers a promising approach for linearizing nonlinear systems in lifted coordinates, its use in chaotic synchronization is fundamentally constrained by a scale-separation issue. With fine temporal discretization, the control influence becomes numerically insignificant relative to the intrinsic nonlinear evolution, leading to the lifted input operator collapsing and controllability being lost. This work addresses the vanishing-input problem using a multi-step control-affine Koopman approach with explicit paired supervision to maintain input identifiability in the lifted space. This method restores controllability and enables discrete-time LQR synthesis, with a clear spectral radius stability certificate for chaotic synchronization. When applied to the Rössler system, the framework achieves rapid convergence (0.09 s) and low synchronization error (ISE = 0.38), outperforming neural, sliding-mode, adaptive, and proportional–integral (PI) synchronization methods under the same constraints. The learned Koopman eigenfunctions also reveal a geometric decomposition of the chaotic attractor into amplitude and phase modes, offering interpretability beyond numerical performance. These findings establish a certifiable, optimal-by-design framework for chaotic synchronization and clarify the structural requirements for Koopman-based control in strongly nonlinear regimes