Chaos-Free Networks Are Stable Recurrent Neural Networks

Gated Recurrent Neural Networks (RNNs) are widely used for nonlinear system identification due to their high accuracy, although they often exhibit complex, chaotic dynamics that are difficult to analyze. This letter investigates the system-theoretic properties of the Chaos-Free Network (CFN), an architecture originally proposed to eliminate the chaotic behavior found in standard gated RNNs. First, we formally prove that the CFN satisfies Input-to-State Stability (ISS) by design. However, we demonstrate that the CFN architecture does not intrinsically guarantee Incremental ISS (δISS), as ensuring this property relies on specific parametric constraints. To address this, we introduce the Decoupled-Gate Network (DGN), a novel structural variant of the CFN that removes internal state connections in the gating mechanisms. Finally, we prove that the DGN unconditionally satisfies the δISS property, providing an incrementally stable architecture for identifying nonlinear dynamical systems without requiring complex network training modifications. Numerical results confirm that the DGN maintains the modeling capabilities of standard architectures while adhering to these rigorous stability guarantees.