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

# Prediction of Nonlinear Oscillations in a Jumping Quarter-Car Model Using Reservoir Computing
Source: [https://arxiv.org/html/2609.22205](https://arxiv.org/html/2609.22205)
Masahisa Watanabe††thanks:Corresponding author: masahisa\-watanabe@go\.tuat\.ac\.jpAffiliation:Division of Environmental and Agricultural Engineering, Institute of Agriculture, Tokyo University of Agriculture and Technology, Tokyo, JapanShiva DixitAffiliation:Amity Institute of Integrative Sciences and Health, Amity University Haryana \- Gurugram, Manesar, 122 413 Haryana, IndiaNirmal PunethaAffiliation:Amity Institute of Integrative Sciences and Health, Amity University Haryana \- Gurugram, Manesar, 122 413 Haryana, IndiaSwati ChauhanManish Dev ShirimaliAffiliation: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\.

Keywords:Quarter\-car model; Reservoir computing; Chaos prediction; Echo state network; Bifurcation Analysis

Acknowledgment

This work was supported by Japan Society for the Promotion of Science Grants\-in\-Aid \(26K14043\)\.

## Nomenclature and Abbreviations

## 1Introduction

Prediction of vehicle dynamics is significantly important to achieve smart driving, such as autonomous driving or advanced driver\-assistance systems\. Thus far, several studies have been conducted on the prediction of vehicle behavior\[[1](https://arxiv.org/html/2609.22205#bib.bib1),[2](https://arxiv.org/html/2609.22205#bib.bib2)\]\. These studies mostly assume a linear vehicle model\. In reality, vehicle dynamics can demonstrate rich nonlinearities, such as nonlinear axle suspensions\[[3](https://arxiv.org/html/2609.22205#bib.bib3)\], steering systems\[[4](https://arxiv.org/html/2609.22205#bib.bib4)\], and traction\[[5](https://arxiv.org/html/2609.22205#bib.bib5)\]\. These nonlinearities induce significant dynamic instabilities, such as bifurcation and chaos, which can deteriorate the stability and controllability of the vehicle system\. Especially in off\-road vehicle operation, such as agricultural and construction applications, dynamic instabilities can substantially reduce operational efficiency and, in severe cases, lead to hazardous accidents\[[6](https://arxiv.org/html/2609.22205#bib.bib6),[7](https://arxiv.org/html/2609.22205#bib.bib7)\]\. Therefore, prediction of nonlinear vehicle behavior, especially chaos prediction, is necessary for smart driving\.

Considerable research has been conducted on chaos prediction\[[8](https://arxiv.org/html/2609.22205#bib.bib8),[9](https://arxiv.org/html/2609.22205#bib.bib9),[10](https://arxiv.org/html/2609.22205#bib.bib10)\]\. Classical approaches to this problem generally rely on local linear approximations of the underlying dynamics, which can be sensitive to model mismatch and require accurate knowledge of the governing equations\. Owing to advances in computational resources and learning theory, machine learning techniques have increasingly been used for chaos prediction\[[11](https://arxiv.org/html/2609.22205#bib.bib11)\]as a data\-driven alternative that does not require an explicit dynamical model\. In particular, reservoir computing has been actively investigated and applied to the prediction of nonlinear dynamical systems\[[12](https://arxiv.org/html/2609.22205#bib.bib12),[13](https://arxiv.org/html/2609.22205#bib.bib13),[14](https://arxiv.org/html/2609.22205#bib.bib14),[15](https://arxiv.org/html/2609.22205#bib.bib15)\]\. However, applications of reservoir computing to chaos prediction have mainly focused on theoretical dynamical systems, and few studies have examined its use for practical industrial systems that combine smooth nonlinearities with discontinuous, contact\-induced switching\. The objective of this short report is to demonstrate, as a feasibility study, whether reservoir computing can predict the bifurcation structure, attractor topology, and time\-domain behavior of a jumping quarter\-car model – a practical vehicle system exhibiting exactly this type of non\-smooth nonlinearity\.

### 1\.1Vehicle model

In this section, the nonlinear vehicle model used in this study is explained\. A quarter\-car model with jumping nonlinearity is used because jumping is quite common in off\-road vehicle dynamics\[[16](https://arxiv.org/html/2609.22205#bib.bib17),[17](https://arxiv.org/html/2609.22205#bib.bib18),[18](https://arxiv.org/html/2609.22205#bib.bib19)\]\. Figure[1](https://arxiv.org/html/2609.22205#S1.F1)shows a schematic of the model when the vehicle maintains and loses contact with the supporting road\.

![Refer to caption](https://arxiv.org/html/2609.22205v1/Figures/fig1.png)Figure 1:Schematic representation of the quarter\-car model incorporating a jumping \(contact\-loss\) nonlinearity\. The sprung massmsm\_\{s\}and unsprung massmum\_\{u\}are coupled through a linear suspension spring–damper pair \(ksk\_\{s\},csc\_\{s\}\), while the unsprung mass interacts with the road profiled⁡\(t\)=d0​sin⁡ω​td\(t\)=d\_\{0\}\\sin\\omega tthrough a tire modeled as a unilateral spring of stiffnessktk\_\{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\.The equations of motion are as follows:

ms​z¨s=fs−ms​g,m\_\{s\}\\ddot\{z\}\_\{s\}=f\_\{s\}\-m\_\{s\}g,\(1\)
mu​z¨u=ft−fs−mu​g,m\_\{u\}\\ddot\{z\}\_\{u\}=f\_\{t\}\-f\_\{s\}\-m\_\{u\}g,\(2\)
where the dot denotes differentiation with respect to timett, andfsf\_\{s\}andftf\_\{t\}denote the passive suspension force and the vertical tire force, respectively\. These forces are expressed as follows:

fs=−ks​\(zs−zu\)−cs​\(z˙s−z˙u\),f\_\{s\}=\-k\_\{s\}\\left\(z\_\{s\}\-z\_\{u\}\\right\)\-c\_\{s\}\\left\(\\dot\{z\}\_\{s\}\-\\dot\{z\}\_\{u\}\\right\),\(3\)
ft=−kt​\(zu−d\),f\_\{t\}=\-k\_\{t\}\(z\_\{u\}\-d\),\(4\)
whereksk\_\{s\},ktk\_\{t\}, andcsc\_\{s\}denote the suspension stiffness, tire stiffness, and suspension damping coefficient, respectively\. If the relative displacementzu−dz\_\{u\}\-dbecomes positive, the tire loses contact with the ground\. In this case, the tire no longer acts as a spring, and the tire force is zero\. To describe this nonlinearity, similar to bouncing\-ball dynamics, the tire stiffness is switched depending on the sign ofzu−dz\_\{u\}\-d, as follows:

kt=\{kt​o,zu−d<00,zu−d≥0k\_\{t\}=\\begin\{cases\}k\_\{to\},&z\_\{u\}\-d<0\\\\ 0,&z\_\{u\}\-d\\geq 0\\end\{cases\}\(5\)
A sinusoidal road profile is used:

d⁡\(t\)=d0​sin⁡ω​t\.d\(t\)=d\_\{0\}\\sin\{\\omega t\}\.\(6\)
The following dimensionless variables and parameters are introduced to derive the nondimensional form of the model:

Zs=zs−zu,Z\_\{s\}=z\_\{s\}\-z\_\{u\},\(7\)
Zu=zu−d,Z\_\{u\}=z\_\{u\}\-d,\(8\)
ts=ωs​t,t\_\{s\}=\\omega\_\{s\}t,\(9\)
ωs=ksms,\\omega\_\{s\}=\\sqrt\{\\frac\{k\_\{s\}\}\{m\_\{s\}\}\},\(10\)
α=msmu,\\alpha=\\frac\{m\_\{s\}\}\{m\_\{u\}\},\(11\)
β=ktks,\\beta=\\frac\{k\_\{t\}\}\{k\_\{s\}\},\(12\)
ζ=cs2​ms​ks,\\zeta=\\frac\{c\_\{s\}\}\{2\\sqrt\{m\_\{s\}k\_\{s\}\}\},\(13\)
γ=gωs2=ms​gks,\\gamma=\\frac\{g\}\{\\omega\_\{s\}^\{2\}\}=\\frac\{m\_\{s\}g\}\{k\_\{s\}\},\(14\)
Ω=ωωs=ω​msks\.\\Omega=\\frac\{\\omega\}\{\\omega\_\{s\}\}=\\omega\\sqrt\{\\frac\{m\_\{s\}\}\{k\_\{s\}\}\}\.\(15\)
Using the variables and parameters above, the dimensionless equations of motion are derived as

Z¨s=−\(1\+α\)​Zs−2​ζ​\(1\+α\)​Z˙s\+α​β​Zu,\\ddot\{Z\}\_\{s\}=\-\(1\+\\alpha\)Z\_\{s\}\-2\\zeta\(1\+\\alpha\)\\dot\{Z\}\_\{s\}\+\\alpha\\beta Z\_\{u\},\(16\)
Z¨u=α​Zs\+2​ζ​α​Z˙s−α​β​Zu−γ\+d0​Ω2​sin⁡Ω​ts,\\ddot\{Z\}\_\{u\}=\\alpha Z\_\{s\}\+2\\zeta\\alpha\\dot\{Z\}\_\{s\}\-\\alpha\\beta Z\_\{u\}\-\\gamma\+d\_\{0\}\\Omega^\{2\}\\sin\{\\Omega t\_\{s\}\},\(17\)
where the dot denotes differentiation with respect totst\_\{s\}\. To describe the jumping nonlinearity of the vehicle in Eqs\. \([16](https://arxiv.org/html/2609.22205#S1.E16)\)–\([17](https://arxiv.org/html/2609.22205#S1.E17)\),β\\betais switched depending onZuZ\_\{u\}:

β=\{βj,Zu<00,Zu≥0\\beta=\\begin\{cases\}\\beta\_\{j\},&Z\_\{u\}<0\\\\ 0,&Z\_\{u\}\\geq 0\\end\{cases\}\(18\)
whereβj=kt​o/ks\\beta\_\{j\}=k\_\{to\}/k\_\{s\}is the stiffness ratio evaluated at the original \(in\-contact\) tire stiffness\.

### 1\.2Reservoir computing

Training a fully recurrent neural network \(RNN\) is not straightforward in practice\. The standard approach, backpropagation through time, tends to converge slowly and is prone to vanishing or exploding gradients when sequences are long\[[12](https://arxiv.org/html/2609.22205#bib.bib12)\]\. Reservoir computing avoids this problem by not training the recurrent part of the network at all\. Instead, a large recurrent network – the “reservoir” – is generated once, randomly, and then left fixed; only a simple linear readout from the reservoir states to the desired output is trained, reducing the learning problem to an ordinary linear regression\[[12](https://arxiv.org/html/2609.22205#bib.bib12),[13](https://arxiv.org/html/2609.22205#bib.bib13)\]\. This idea was introduced independently under different names – echo state networks \(ESN\) by Jaeger\[[12](https://arxiv.org/html/2609.22205#bib.bib12)\]and liquid state machines \(LSM\) by Maass et al\.\[[13](https://arxiv.org/html/2609.22205#bib.bib13)\]– and was shown early on to be remarkably effective: Jaeger and Haas demonstrated that an ESN could outperform previous methods on a standard chaotic time\-series benchmark by several orders of magnitude\[[19](https://arxiv.org/html/2609.22205#bib.bib16)\], which largely drew attention to the approach\.

For a reservoir to be useful, it must satisfy the echo state property: the influence of the initial reservoir state should die out over time, so that the current reservoir state depends only on recent input history rather than on initialization\[[12](https://arxiv.org/html/2609.22205#bib.bib12)\]\. In practice, this is achieved by scaling the reservoir’s recurrent weight matrix so that its spectral radiusρr​c\\rho\_\{rc\}remains close to or below one; ifρr​c\\rho\_\{rc\}is set too low the reservoir forgets useful history too quickly, while if it is set too high the reservoir state fails to settle and training becomes unstable\.

In this study, an ESN is used as the reservoir\. At every time step, the current state of the quarter\-car model –ZsZ\_\{s\},ZuZ\_\{u\}, and their time derivatives – together with the road excitation amplituded0d\_\{0\}, is fed into the reservoir through a sparse, randomly weighted input connection with scalingWi​nW\_\{in\}\. The reservoir itself is a sparsely connected network \(average connectivity degreekr​ck\_\{rc\}\) ofNr​cN\_\{rc\}nonlinear \(tanh\) units with leak rateαr​c\\alpha\_\{rc\}, following the leaky\-integrator formulation commonly used in ESN implementations\. The readout weights connecting reservoir states to the predicted output are trained offline using ridge regression with regularization coefficientβr​c\\beta\_\{rc\}, chosen to avoid overfitting to the relatively short training trajectories used here\.

Formally, the reservoir state𝐫⁡\(t\)∈ℝNr​c\\mathbf\{r\}\(t\)\\in\\mathbb\{R\}^\{N\_\{rc\}\}is updated at each time step according to

𝐫⁡\(t\+Δ​t\)=\(1−αr​c\)​𝐫​\(t\)\+αr​c​tanh⁡\(Wr​e​s​𝐫​\(t\)\+Wi​n​𝐮​\(t\)\),\\mathbf\{r\}\(t\+\\Delta t\)=\(1\-\\alpha\_\{rc\}\)\\,\\mathbf\{r\}\(t\)\+\\alpha\_\{rc\}\\tanh\\left\(W\_\{res\}\\,\\mathbf\{r\}\(t\)\+W\_\{in\}\\,\\mathbf\{u\}\(t\)\\right\),\(19\)where𝐮⁡\(t\)=\[Zs,Zu,Z˙s,Z˙u,d0\]⊤\\mathbf\{u\}\(t\)=\\left\[Z\_\{s\},Z\_\{u\},\\dot\{Z\}\_\{s\},\\dot\{Z\}\_\{u\},d\_\{0\}\\right\]^\{\\top\}is the input vector,Wr​e​sW\_\{res\}is the fixed, sparse, randomly generated reservoir weight matrix with spectral radiusρr​c\\rho\_\{rc\}and connectivity degreekr​ck\_\{rc\}, andWi​nW\_\{in\}is the fixed, randomly generated input weight matrix whose elements are randomly generated from a uniform distribution\. The predicted output is obtained from a linear readout,

𝐲^​\(t\)=Wo​u​t​𝐫​\(t\),\\hat\{\\mathbf\{y\}\}\(t\)=W\_\{out\}\\,\\mathbf\{r\}\(t\),\(20\)where only the readout weight matrixWo​u​tW\_\{out\}is trained, via ridge regression with regularization coefficientβr​c\\beta\_\{rc\}:

Wo​u​t=𝐘𝐑⊤​\(𝐑𝐑⊤\+βr​c​𝐈\)−1,W\_\{out\}=\\mathbf\{Y\}\\mathbf\{R\}^\{\\top\}\\left\(\\mathbf\{R\}\\mathbf\{R\}^\{\\top\}\+\\beta\_\{rc\}\\mathbf\{I\}\\right\)^\{\-1\},\(21\)with𝐘\\mathbf\{Y\}and𝐑\\mathbf\{R\}denoting, respectively, the matrices of target outputs and corresponding reservoir states collected over the training trajectory\. Figure[2](https://arxiv.org/html/2609.22205#S1.F2)summarizes this input→\\toreservoir→\\tooutput structure\. The internal reservoir connections are kept fixed throughout training, and only the output weights are optimized\.

This architecture has previously been applied with good results to chaotic time\-series prediction\[[14](https://arxiv.org/html/2609.22205#bib.bib14)\]and, more recently, to model\-free forecasting of large, high\-dimensional spatiotemporally chaotic systems\[[15](https://arxiv.org/html/2609.22205#bib.bib15)\]\. What is less established is how well it performs on a real mechanical system with a non\-smooth, contact\-induced nonlinearity, such as the jumping quarter\-car model considered here, as opposed to a smooth textbook chaotic system such as the Lorenz or Rössler systems\. This is the question addressed in this report\.

![Refer to caption](https://arxiv.org/html/2609.22205v1/Figures/fig2.png)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 amplituded0d\_\{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\.

## 2Results and Discussion

Reservoir computing is applied here to predict three aspects of the model: bifurcation structure, attractor topology, and time\-series behavior\. The vehicle parameters are set toα=2\.333\\alpha=2\.333,βj=10\\beta\_\{j\}=10,ζ=0\.1552\\zeta=0\.1552,γ=0\.113\\gamma=0\.113, andΩ=6\.8\\Omega=6\.8\. The echo state network parameters are set toNr​c=1350N\_\{rc\}=1350,αr​c=0\.75\\alpha\_\{rc\}=0\.75,ρr​c=0\.547\\rho\_\{rc\}=0\.547,kr​c=24k\_\{rc\}=24,Wi​n=0\.185W\_\{in\}=0\.185, andβr​c=7×10−6\\beta\_\{rc\}=7\\times 10^\{\-6\}\. These values were determined by Bayesian optimization of the validation prediction error, followed by manual fine\-tuning\.

### 2\.1Prediction of bifurcation

Bifurcation is one of the major instabilities in nonlinear dynamics\. Figure[3](https://arxiv.org/html/2609.22205#S2.F3)\(a\) shows the ground\-truth bifurcation diagram ofZuZ\_\{u\}for the quarter\-car model with jumping nonlinearity with respect tod0d\_\{0\}\. Period\-1 motion \(P1P\_\{1\}\) is observed fromd0=0\.01d\_\{0\}=0\.01to0\.01880\.0188, after which period\-2 motion \(P2P\_\{2\}\) appears\. A discontinuous bifurcation occurs and chaotic motion \(C\) emerges atd0=0\.0225d\_\{0\}=0\.0225\. To predict the bifurcation, the echo state network is trained at three operating points:d0=0\.0165d\_\{0\}=0\.0165\(P1P\_\{1\}\),d0=0\.0185d\_\{0\}=0\.0185\(P1P\_\{1\}\), andd0=0\.0205d\_\{0\}=0\.0205\(P2P\_\{2\}\)\. Figure[3](https://arxiv.org/html/2609.22205#S2.F3)\(b\) shows the predicted bifurcation diagram\. The predicted diagram is qualitatively similar to the ground truth: period doubling occurs atd0=0\.0194d\_\{0\}=0\.0194, followed by period\-2 motion that becomes chaotic atd0=0\.022d\_\{0\}=0\.022\. This transition appears continuous in the prediction, whereas the ground\-truth bifurcation is discontinuous – likely because the reservoir’s smooth, continuous\-state dynamics tend to average across the abrupt switching behavior near the contact\-loss discontinuity, rather than reproducing it exactly\. Despite this discrepancy, the reservoir predicts the overall bifurcation structure qualitatively well, including extrapolation beyond the training range into the chaotic regime\.

![Refer to caption](https://arxiv.org/html/2609.22205v1/Figures/fig3.png)Figure 3:Bifurcation diagrams of the tire deflectionZuZ\_\{u\}as a function of the road excitation amplituded0d\_\{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 \(d0=0\.0165d\_\{0\}=0\.0165,0\.01850\.0185, and0\.02050\.0205\) 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\.
### 2\.2Prediction of attractors

To evaluate the topological accuracy of the prediction, Figure[4](https://arxiv.org/html/2609.22205#S2.F4)compares ground\-truth and predicted attractors\. Panels \(a\)–\(c\) and \(d\)–\(f\) show the ground\-truth and predicted attractors, respectively, forP1P\_\{1\}\(d0=0\.015d\_\{0\}=0\.015\),P2P\_\{2\}\(d0=0\.02d\_\{0\}=0\.02\), and C \(d0=0\.025d\_\{0\}=0\.025\)\. The predicted attractors closely reproduce the qualitative loop structure of the periodic cases and the folded\-sheet topology of the chaotic case\. For all training runs, ground\-truth trajectories were generated by numerical integration of Eqs\. \([16](https://arxiv.org/html/2609.22205#S1.E16)\)–\([17](https://arxiv.org/html/2609.22205#S1.E17)\) using a fourth\-order Runge–Kutta scheme with a fixed time step of0\.010\.01\. Training trajectories consisted of500500time units at each operating point, following a transient of10510^\{5\}time units that was discarded to remove initialization effects\.

![Refer to caption](https://arxiv.org/html/2609.22205v1/Figures/fig4.png)Figure 4:Comparison of ground\-truth and reservoir\-predicted phase\-space attractors, projected onto the\(Zs,Z˙s\)\(Z\_\{s\},\\dot\{Z\}\_\{s\}\)and\(Zu,Z˙u\)\(Z\_\{u\},\\dot\{Z\}\_\{u\}\)planes, for three representative dynamical regimes: period\-1 motion atd0=0\.015d\_\{0\}=0\.015\(a, d\), period\-2 motion atd0=0\.02d\_\{0\}=0\.02\(b, e\), and chaotic motion atd0=0\.025d\_\{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\.
### 2\.3Prediction of time series

Time\-series prediction is the most important outcome for practical use, since it can be directly applied to predictive control of the vehicle\. Predictions were generated using the trained reservoir for each type of motion\. Figure[5](https://arxiv.org/html/2609.22205#S2.F5)\(a\)–\(c\) shows predicted time series ofZsZ\_\{s\}forP1P\_\{1\}\(d0=0\.015d\_\{0\}=0\.015\),P2P\_\{2\}\(d0=0\.02d\_\{0\}=0\.02\), and C \(d0=0\.025d\_\{0\}=0\.025\), respectively\. Long\-term prediction is achieved forP1P\_\{1\}andP2P\_\{2\}, indicating that the reservoir accurately captures periodic dynamics\. For the chaotic case, only short\-term prediction is possible due to sensitive dependence on initial conditions – a defining characteristic of chaotic motion – after which the predicted and ground\-truth trajectories diverge exponentially, consistent with expected behavior near a positive leading Lyapunov exponent\.

![Refer to caption](https://arxiv.org/html/2609.22205v1/Figures/fig5.png)Figure 5:Time\-domain forecasts of the suspension strokeZsZ\_\{s\}generated by the trained echo state network \(blue\) superimposed on the corresponding ground\-truth trajectories \(orange\), for \(a\) period\-1 motion atd0=0\.015d\_\{0\}=0\.015, \(b\) period\-2 motion atd0=0\.02d\_\{0\}=0\.02, and \(c\) chaotic motion atd0=0\.025d\_\{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\.

## 3Conclusion

In this work, the feasibility of reservoir computing for prediction of a nonlinear vehicle model was investigated through numerical simulations\. Trained on data from only three operating points, the reservoir qualitatively predicted the bifurcation structure, attractor topology, and time\-series behavior of a jumping quarter\-car model across period\-1, period\-2, and chaotic regimes, including reasonable extrapolation beyond the training range\. These results support reservoir computing as a viable data\-driven predictor for practical mechanical systems with non\-smooth, contact\-induced nonlinearities\. Future work should include quantitative validation via Lyapunov analysis and valid prediction time, as well as extension to predictive control applications\.

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Funding sources

This work was supported by Japan Society for the Promotion of Science Grant\-in\-Aid \(Nos\. 26K18165\)\.

Declaration of Competing Interest

The authors have no relevant financial or non\-financial interests to disclose\.

Data Availability

The datasets generated during the current study are available from the corresponding author on reasonable request\.