Exploring the origins of switching dynamics in a multifunctional reservoir computer

TL;DR

This paper investigates the origins of metastable switching dynamics in multifunctional reservoir computers using the seeing double problem, where two equal-radius circular orbits with opposite rotations must be reconstructed with a single readout. By varying the spectral radius ρ and the center separation x_cen, the authors perform attractor continuation and reveal a sequence in which one reconstructed attractor becomes unstable via collision with a saddle, a new attractor arises, and the system exhibits metastable switching between regions corresponding to the original orbits. The key finding is that a new attractor born during the bifurcation sequence facilitates switching, producing diverse patterns from chaotic to periodic, with residence times showing a sawtooth pattern across multiple exponential branches. These results highlight the critical role of memory and saddle dynamics in RCs, offer a framework to study metastability and chaotic itinerancy in data-driven dynamical systems, and have implications for designing multifunctional RCs that can robustly reconstruct multiple attractors.

Abstract

The concept of multifunctionality has enabled reservoir computers (RCs), a type of dynamical system that is typically realised as an artificial neural network, to reconstruct multiple attractors simultaneously using the same set of trained weights. However there are many additional phenomena that arise when training a RC to reconstruct more than one attractor. Previous studies have found that, in certain cases, if the RC fails to reconstruct a coexistence of attractors then it exhibits a form of metastability whereby, without any external input, the state of the RC switches between different modes of behaviour that resemble properties of the attractors it failed to reconstruct. In this paper we explore the origins of these switching dynamics in a paradigmatic setting via the `seeing double' problem.

Figures (8)

• Figure 1: Illustrating the result of training the RC to reconstruct a coexistence of $\mathcal{C}_{A}$ and $\mathcal{C}_{B}$ when $\rho=0.2$ for $x_{cen}=8.0$ in panel (a) and for $x_{cen}=6.5$ in panel (b). Black arrows indicate direction of rotation on both orbits. Dynamics of the closed-loop RC are illustrated in solid curves, training data by dashed curves.
• Figure 2: Illustrating the result of tracking the changes in $\hat{\mathcal{C}}_{A}$ and $\hat{\mathcal{C}}_{B}$ with respect to changes in $\rho$ for $x_{cen}=6.5$. Panel (e) describes how the local maxima of the corresponding attractors that are tracked, $x_{m}$, changes with respect to $\rho$. Panels (a)-(d) highlight some of the most significant changes in the dynamics of $\hat{\mathcal{C}}_{A}$ and $\hat{\mathcal{C}}_{B}$ at certain values of $\rho$ from the perspective of $\mathbb{P}$, the prediction state space.
• Figure 3: Illustrating the result of tracking the changes in $\hat{\mathcal{C}}_{A}$ and $\hat{\mathcal{C}}_{B}$ with respect to changes in $\rho$ for $x_{cen}=5.0$. Panel (e) describes how the local maxima of the corresponding attractors that are tracked, $x_{m}$, changes with respect to $\rho$. Panels (a)-(d) highlight some of the most significant changes in the dynamics of $\hat{\mathcal{C}}_{A}$ and $\hat{\mathcal{C}}_{B}$ at certain values of $\rho$ from the perspective of $\mathbb{P}$, the prediction state space.
• Figure 4: Illustrating the result of tracking the changes in $\hat{\mathcal{C}}_{A}$ and $\hat{\mathcal{C}}_{B}$ with respect to changes in $\rho$ for $x_{cen}=3.5$. Panel (e) describes how the local maxima of the corresponding attractors that are tracked, $x_{m}$, changes with respect to $\rho$. Panels (a)-(d) highlight some of the most significant changes in the dynamics of $\hat{\mathcal{C}}_{A}$ and $\hat{\mathcal{C}}_{B}$ at certain values of $\rho$ from the perspective of $\mathbb{P}$, the prediction state space.
• Figure 5: Illustrating the result of tracking the changes in $\hat{\mathcal{C}}_{A}$ and $\hat{\mathcal{C}}_{B}$ with respect to changes in $\rho$ for $x_{cen}=2.0$. Panel (e) describes how the local maxima of the corresponding attractors that are tracked, $x_{m}$, changes with respect to $\rho$. Panels (a)-(d) highlight some of the most significant changes in the dynamics of $\hat{\mathcal{C}}_{A}$ and $\hat{\mathcal{C}}_{B}$ at certain values of $\rho$ from the perspective of $\mathbb{P}$, the prediction state space.