Prediction of Nonlinear Oscillations in a Jumping Quarter-Car Model Using Reservoir Computing

Masahisa Watanabe; Shiva Dixit; Nirmal Punetha; Swati Chauhan; Manish Dev Shirimali

cs.LGSep 1, 2026

Prediction of Nonlinear Oscillations in a Jumping Quarter-Car Model Using Reservoir Computing

Authors: Masahisa Watanabe, Shiva Dixit, Nirmal Punetha, Swati Chauhan, Manish Dev Shirimali

Organizations: Division of Environmental and Agricultural Engineering, Institute of Agriculture, Tokyo University of Agriculture and Technology, Tokyo, Japan · Amity Institute of Integrative Sciences and Health, Amity University Haryana - Gurugram, Manesar, 122 413 Haryana, India · Graduate School of Engineering, Nagoya Institute of Technology, Nagoya 466 8555, Japan · Department of Physics, Central University of Rajasthan, Ajmer, 305 817 Rajasthan, India

Abstract

Reliable prediction of vehicle dynamics is essential for smart driving applications such as autonomous control and advanced driver-assistance systems. Off-road vehicles used in agricultural and construction settings are particularly prone to nonlinear behavior, including bifurcations and chaotic motion arising from intermittent loss of tire--road contact. Predicting such dynamics is challenging because it requires resolving both smooth nonlinearities and the discontinuous switching associated with contact loss. In this work, we investigate the feasibility of reservoir computing (RC) -- specifically an echo state network (ESN) -- for data-driven prediction of a jumping quarter-car model. The reservoir is trained on time-series data from a small number of points and evaluated on its ability to reconstruct bifurcation diagrams, phase-space attractors, and time trajectories across periodic and chaotic regimes. The trained reservoir qualitatively reproduces the period-doubling route to chaos, captures the geometric structure of periodic and chaotic attractors. These results demonstrate that reservoir computing is a feasible data-driven predictor of nonlinear dynamics in a practical, non-smooth vehicle system.

Figures & tables

Figure 1: Schematic representation of the quarter-car model incorporating a jumping (contact-loss) nonlinearity. The sprung mass

m_{s}

and unsprung mass

m_{u}

are coupled through a linear suspension spring–damper pair (

k_{s}

c_{s}

), while the unsprung mass interacts with the road profile

d(t)=d_{0}\sin\omega t

through a tire modeled as a unilateral spring of stiffness

k_{t}

. The tire stiffness is switched to zero whenever the wheel loses contact with the road surface, giving rise to the piecewise-smooth governing dynamics.

Figure 2: Schematic diagram of the echo state network (ESN) architecture employed as the reservoir-computing predictor. The input layer supplies the instantaneous state vector of the quarter-car model together with the road excitation amplitude $d_{0}$ to a fixed, sparsely and randomly connected reservoir of nonlinear units. Only the linear readout mapping from the reservoir state to the predicted output is trained, via ridge regression, while the internal reservoir weights remain untrained throughout.

Figure 3: Bifurcation diagrams of the tire deflection $Z_{u}$ as a function of the road excitation amplitude $d_{0}$ , comparing (a) the ground-truth response obtained by direct numerical integration of the governing equations with (b) the response reconstructed by the trained echo state network. Vertical dashed lines in panel (a) mark the three operating points ( $d_{0}=0.0165$ , $0.0185$ , and $0.0205$ ) used to train the reservoir. The predicted diagram in panel (b) qualitatively reproduces the period-doubling route to chaos observed in the ground truth, including extrapolation beyond the training range into the chaotic regime.

Figure 4: Comparison of ground-truth and reservoir-predicted phase-space attractors, projected onto the $(Z_{s},\dot{Z}_{s})$ and $(Z_{u},\dot{Z}_{u})$ planes, for three representative dynamical regimes: period-1 motion at $d_{0}=0.015$ (a, d), period-2 motion at $d_{0}=0.02$ (b, e), and chaotic motion at $d_{0}=0.025$ (c, f). Panels (a)–(c) show the ground-truth attractors obtained from direct simulation; panels (d)–(f) show the corresponding attractors reconstructed from the trained reservoir’s autonomous output. Close qualitative agreement in loop structure and, for the chaotic case, folded-sheet topology, indicates that the reservoir captures the underlying geometric organization of the flow rather than merely its instantaneous amplitude.

Figure 5: Time-domain forecasts of the suspension stroke $Z_{s}$ generated by the trained echo state network (blue) superimposed on the corresponding ground-truth trajectories (orange), for (a) period-1 motion at $d_{0}=0.015$ , (b) period-2 motion at $d_{0}=0.02$ , and (c) chaotic motion at $d_{0}=0.025$ . The reservoir sustains accurate long-horizon tracking of the periodic waveforms in (a) and (b), whereas in the chaotic case (c) the predicted and ground-truth trajectories diverge after a comparatively short horizon, consistent with the exponential error growth expected from sensitive dependence on initial conditions.

Chaotic Time-Series Prediction

Chaotic time-series prediction studies how to forecast nonlinear systems whose sensitive dependence on initial conditions limits long-range accuracy.

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Tailored minimal reservoir computing: on the bidirectional connection between nonlinearities in the reservoir and in data

Davide Prosperino, Haochun Ma, Christoph Räth

Ludwig-Maximilians-Universität München, Faculty of Physics, Geschwister-Scholl-Platz 1, 80539 Munich, Germany · Deutsches Zentrum für Luft- und Raumfahrt (DLR), Institute for Frontier Materials on Earth and in Space, Linder Höhe, 51147 Cologne, Germany

Nonlinear System Identification Reservoir Computing Chaotic Time-Series Prediction

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We study how the degree of nonlinearity in the input data affects the optimal design of reservoir computers, focusing on how closely the model's nonlinearity should align with that of the data. By reducing minimal RCs to a single tunable nonlinearity parameter, we explore how the predictive performance varies with the degree of nonlinearity in the reservoir. To provide controlled testbeds, we generalize to the fractional Halvorsen system, a novel chaotic system with fractional exponents. Our experiments reveal that the prediction performance is maximized when the reservoir's nonlinearity matches the nonlinearity present in the data. In cases where multiple nonlinearities are present in the data, we find that the correlation dimension of the predicted signal is reconstructed correctly when the smallest nonlinearity is matched. We use this observation to propose a method for estimating the minimal nonlinearity in unknown time series by sweeping the reservoir exponent and identifying the transition to a successful reconstruction. Applying this method to both synthetic and real-world datasets, including financial time series, we demonstrate its practical viability. Finally, we transfer these insights to classical RC by augmenting traditional architectures with fractional, generalized reservoir states. This yields performance gains, particularly in resource-constrained scenarios such as physical reservoirs, where increasing reservoir size is impractical or economically unviable. Our work provides a principled route toward tailoring RCs to the intrinsic complexity of the systems they aim to model.

Abstract

Chaotic Time-Series Prediction

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