Extrapolating learned dynamics across bifurcations based on generalized synchronization

Lingfeng Chen; Masanobu Inubushi

nlin.CDOct 25, 2026

Extrapolating learned dynamics across bifurcations based on generalized synchronization

Authors: Lingfeng Chen, Masanobu Inubushi

Organizations: Department of Applied Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku, Tokyo 162-8601, Japan · Graduate School of Engineering Science, The University of Osaka, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan

Abstract

Predicting dynamics across bifurcations by extrapolating trained models is crucial in diverse fields of physics. Generalized synchronization provides a framework for understanding dynamical reconstruction by neural networks, especially in the context of reservoir computing. Building on this framework, we derive a formula combining pretrained readouts to reconstruct unseen chaotic dynamics. Using classical Lagrange interpolation theory, we derive rigorous bounds on readout errors and conjugacy residuals. Numerical examples demonstrate accurate prediction of chaotic dynamics across bifurcations.

Figures & tables

Figure 1: Concept of readout interpolation and extrapolation with a linear combination of pretrained readout weights $W_{\theta}=(1-\theta)W_{\mu_{a}}+\theta W_{\mu_{b}}$ . The same fixed reservoir is used throughout; only the readout weights change. Weights $W_{\mu_{a}}$ and $W_{\mu_{b}}$ are trained at $\mu_{a}$ and $\mu_{b}$ . The mixing coefficient $\theta$ gives interpolation for $0\leq\theta\leq 1$ and extrapolation otherwise. The lower axis uses the Lorenz parameter $\mu=\rho$ , with $\mu_{a}=23$ , $\mu_{b}=24$ , and comparison value $\tilde{\mu}(2)=25$ under the parameter-correspondence ansatz in Eq. ( 11 ).

Figure 2: Reconstruction of Lorenz dynamics. (a) Error in the reconstructed fixed-point position, $d(\theta)$ [Eq. ( 22 )], plotted against $\tilde{\rho}(\theta)=23+\theta$ on the $\rho$ axis. The black curve does not encode stability. Interpolation corresponds to $23\leq\rho\leq 24$ ; the remaining range is extrapolation. (b) Predicted trajectory at $\theta=2$ (left, blue) and reference trajectory obtained by numerically integrating the Lorenz equations at $\rho=25$ (right, red). (c) Estimated marginal probability densities of $X$ from predicted (blue dashed) and reference (red solid) trajectories: (c-1) $(\theta,\rho)=(2,28)$ , (c-2) $(5,28)$ , (c-3) $(2,25)$ , and (c-4) $(5,25)$ , in reading order. (d) Kullback–Leibler divergence $D_{\mathrm{KL}}(P_{\theta}\|Q_{\rho})$ between the predicted and reference marginal distributions of $X$ , versus $\theta$ and $\rho$ . The circle, triangle, square, and diamond identify (c-1)–(c-4), respectively. Dark blue denotes smaller divergence and white larger divergence.

Figure 3: Reconstruction of fixed points of scalar normal forms. (a) Saddle-node, with readouts trained at $\rho_{a}=-0.01$ and $\rho_{b}=-0.007$ . (b) Supercritical pitchfork, with readouts trained at $\rho_{a}=-0.1$ and $\rho_{b}=-0.07$ . Solid/dashed black curves denote stable/unstable fixed-point branches of the target equations; blue/red curves show the readout values of stable/unstable fixed points of the autonomous map $\widehat{G}_{\theta}$ . Arrows mark the two parameter values at which training data are available. The target parameter $\rho$ and the reconstructed results labeled by $\tilde{\rho}(\theta)=\rho_{a}+\theta(\rho_{b}-\rho_{a})$ share the same horizontal axis.

Dynamical Systems

Dynamical systems studies how states evolve over time, including stability, feedback, chaos, and transitions in physical, biological, and engineered systems.

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Inferring bifurcation diagrams of two distinct chaotic systems by a single machine

Jianmin Guo, Yao Du, Yizhen Yu, Yong Zou+1

School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China · College of Physics and Electronic Information Engineering, Qinghai Normal University, Xining 810008, China · School of Physics and Electronic Science, East China Normal University, Shanghai 200062, China

Dynamical Systems Nonlinear System Identification Reservoir Computing Chaotic Dynamical Systems

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We propose a dual-channel reservoir-computing scheme for inferring the dynamics of two distinct chaotic systems with a single machine. By augmenting a standard reservoir with a system-label channel and a parameter-control channel, the machine can be trained from time series collected from a few sampled states of the two systems. We show that the trained machine not only predicts the short-time evolution of the sampled states, but also reproduces the long-term statistical properties of unseen states, thereby enabling reconstruction of the bifurcation diagrams of both systems from partial observations. The effectiveness of the scheme is demonstrated for the Lorenz and Rössler systems in numerical simulations and for the Chua and Rossler circuits in experiments. Functional-network analysis further shows that the two target systems are encoded by distinct dynamical patterns in the reservoir. These results extend multifunctional and parameter-aware reservoir computing, and provide a route to data-driven inference of multiple nonlinear systems using a single machine.

Abstract

Dynamical Systems

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