Robustness of topological conjugacy in autonomous reservoirs

Ascertain whether and under what conditions the learned autonomous reservoir dynamics Γ, constructed via the mapping Φ and its approximate inverse in the readout, remain robustly topologically conjugate to the original system f in the presence of small approximation errors, and characterize the persistence of this conjugacy over time during prediction.

Background

The paper explains that, under ESP and GS, the driven reservoir induces a homeomorphism Φ mapping the original system’s attractor to the reservoir’s attractor, yielding conjugate dynamics Γ = Φ ∘ f ∘ Φ{-1}.

In practice, the learned autonomous reservoir only approximates Φ{-1} and Γ, so exact conjugacy can break after the initial prediction steps due to small errors. Understanding whether conjugacy is robust to such imperfections is identified as an open question.

References

The robustness of the autonomous reservoir dynamics as a topological conjugate remains an open question with active and ongoing research.

— Contraction and Synchronization in Reservoir Systems  (2408.04058 - Wong et al., 2024) in Section 4.3 (Topological Conjugacy)

So, for the Hénon map, we therefore conjecture that the error vectors connecting the ground truth and the different RCs' forecasts are inside the unstable manifold as $k$ increases.

— Understanding the superiority of multi-model ensemble forecasts through reservoir computing  (2608.20017 - Moya et al., 20 Aug 2026) in Section 7, Section “Higher-dimensional systems,” subsection “Folded towel map”