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.