Anif N. Shikder | Searcharxiv

arXiv2026

Exact expression for maximum Lyapunov exponent during transients in computationally powerful dynamical networks

We study a network whose rich spatiotemporal dynamics have recently been shown to enable dynamics-based computation, including logic gates, short-term memory, and simple encryption. The network's time dynamics can be exactly solved through a nonlinear coordinate transformation. Here, we derive an exact analytical expression for the network's time-dependent maximum Lyapunov exponent (MLE). We demonstrate, both numerically and analytically, that the network exhibits positive MLEs during the transients that are useful for computation. Our framework enables algebraic manipulation of transient lifetimes through network connectivity and initial conditions, providing a rigorous theoretical foundation for understanding and controlling computation with transients.

nlin.CD↗

arXiv2026

An explicit operator explains end-to-end computation in the modern neural networks used for sequence and language modeling

We establish a mathematical correspondence between state space models, a state-of-the-art architecture for capturing long-range dependencies in data, and an exactly solvable nonlinear oscillator network. As a specific example of this general correspondence, we analyze the diagonal linear time-invariant implementation of the Structured State Space Sequence model (S4). The correspondence embeds S4D, a specific implementation of S4, into a ring network topology, in which recent inputs are encoded, as waves of activity traveling over the one-dimensional spatial layout of the network. We then derive an exact operator expression for the full forward pass of S4D, yielding an analytical characterization of its complete input-output map. This expression reveals that the nonlinear decoder in the system induces interactions between these information-carrying waves that enable classifying real-world sequences. These results generalize across modern SSM architectures, and show that they admit an exact mathematical description with a clear physical interpretation. These insights enable a new level of interpretability for these systems in terms of nonlinear oscillator networks.

cs.NE↗