# Beyond the Edge of Chaos: Stability-Expressivity Transfer in Reservoir Forecasting

arXiv: https://arxiv.org/abs/2607.17909 · HTML: https://mlsanity.com/paper/2607.17909/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 33 references that are in the map.
- First posted on arXiv 20 Jul 2026.

## Builds on (most cited first, 9 of 9)

- [Forecasting Chaotic Systems with Very Low Connectivity Reservoir Computers](https://mlsanity.com/paper/1910.00659/) (2019)
- [Lyapunov spectra of chaotic recurrent neural networks](https://mlsanity.com/paper/2006.02427/) (2020)
- [Model-free prediction of spatiotemporal dynamical systems with recurrent neural networks: Role of network spectral radius](https://mlsanity.com/paper/1910.04426/) (2019)
- [Generative learning for nonlinear dynamics](https://mlsanity.com/paper/2311.04128/) (2023)
- [Do Reservoir Computers Work Best at the Edge of Chaos?](https://mlsanity.com/paper/2012.01409/) (2020)
- [Attractor reconstruction with reservoir computers: The effect of the reservoir's conditional Lyapunov exponents on faithful attractor reconstruction](https://mlsanity.com/paper/2401.00885/) (2023)
- [Local Dynamics in Trained Recurrent Neural Networks](https://mlsanity.com/paper/1511.05222/) (2015)
- [Detection of Generalized Synchronization using Echo State Networks](https://mlsanity.com/paper/1710.08286/) (2017)
- [Criticality in Reservoir Computer of Coupled Phase Oscillators](https://mlsanity.com/paper/2108.06395/) (2021)

## Abstract (arXiv)

The edge-of-chaos heuristic has long served as a guiding principle for designing reservoir computers, yet its relevance to machine performance remains elusive. Here, taking the spectral radius of the reservoir network as the control parameter, we show that the radius yielding the best forecasting performance does not coincide with the Lyapunov edge of the isolated, teacher-forced, or closed-loop generative reservoir. By analyzing the collective dynamics of the teacher-forced reservoir, we find that the target dynamics are represented mainly by stable Lyapunov modes whose finite-time stability is strongly modulated by the input. This finding motivates a stability-expressivity transfer index, which balances the stability of these modes against their expressivity in representing the target. Across chaotic and quasiperiodic targets, and for both asymmetric and symmetric reservoirs, this index accurately identifies the optimal spectral radius for autonomous forecasting.