This work demonstrates that the NMSE from an RC-based prediction model can be used as an indicator for system dynamical characterization, and provides an alternative approach for probing dynamical regimes from finite time series, paving the way for embedded neuromorphic tools dedicated to the experimental analysis of complex nonlinear systems.
Abstract
The complexity of nonlinear dynamical systems can be analyzed using indicators such as Lyapunov exponents or bifurcation diagrams, whose estimation remains challenging and data-intensive in experimental contexts with constraints. At the same time, reservoir computing (RC) has emerged as a powerful tool for chaotic time series prediction, and the associated metric, the normalized mean square error (NMSE), is solely used to evaluate predictive performance. In this work, we demonstrate that the NMSE from an RC-based prediction model can be used as an indicator for system dynamical characterization. We propose a unified, numerical, and experimental approach based on a single-node time-delay reservoir computing architecture implemented around a Mackey-Glass oscillator and a low-cost hardware platform using an Arduino Due board. The approach is evaluated on discrete and continuous dynamical systems. The variations of NMSE as a function of control parameters are systematically compared with those of Lyapunov exponents and bifurcation diagrams. The results show that different transition regimes (periodic, chaotic, and hyperchaotic) can be reproducible in the variation of the NMSE, both numerically and experimentally. Viewed in this dynamical order, our results also reveal a sigmoidal relationship between the NMSE and the largest Lyapunov exponent. Therefore, the NMSE, beyond its classical role as a prediction metric, can be used as a data-driven indicator reflecting the complexity of a target system’s dynamics. This study thus provides an alternative approach for probing dynamical regimes from finite time series, paving the way for embedded neuromorphic tools dedicated to the experimental analysis of complex nonlinear systems.
Using ordinary least squares regression on high-degree polynomial features with 512-bit arithmetic, a system-agnostic method is introduced that matches the accuracy of standard 64-bit numerical ODE solvers using the systems’ governing equations, suggesting that forecasting low-dimensional chaotic systems from noise-free data is effectively a solved problem.
We study the impact of the regularization coefficient λ on Reservoir Computer (RC) performance for chaotic time series prediction of the Lorenz system. We find that a larger λ results in a larger error of the first prediction step of the RC. The RC error initially evolves according to a rapid, nonautonomous expansion at very early times followed by the expected exponential, Lyapunov growth of the error. The maximum Lyapunov exponent of the prediction is close to the maximum Lyapunov exponent of the Lorenz system regardless of λ, whereas the error of the first prediction step and the rate of the nonautonomous growth are λ-dependent and are responsible for the λ dependence of the Valid Prediction Time (VPT) of the RC. Interestingly, we find that the VPT can exceed 30 Lyapunov times for vanishingly small λ, as we are predicting a noiseless system with a small sampling step and spectral radius. Moreover, we emphasize the importance of the numerical solver used to generate the Lorenz dataset and define a Valid Ground Truth Time (VGTT) during which the outputs of several common solvers agree. A VPT exceeding the VGTT is not meaningful, as a different solver could produce a different result.
L. Hurley, S. E. Shaheen· AIP Advances· 0 citations
This work constructs a novel two-dimensional discrete fractional-order chaotic map and establishes a corresponding TDF-TDR framework with the proposed map as the driving module, and deduces the complete mathematical formulation and conducts systematic dynamical analysis.
Bin Huo, Yufei Chen, Baoxiang Du· Nonlinear dynamics· 0 citations
The numerical investigation of fractional-order chaotic systems is computationally demanding due to the long-memory effect of fractional derivatives and the need for small integration step sizes. In particular, computing bifurcation diagrams and Lyapunov exponent spectra over wide parameter ranges becomes time-consuming when conventional serial for-loop implementations are used. In this study, a parallel computational approach is proposed for the simultaneous extraction of bifurcation diagrams and Lyapunov exponents of fractional-order chaotic systems. For each parameter value, the system equations and the corresponding variational equations are integrated simultaneously, while periodic Gram–Schmidt reorthonormalization is applied for Lyapunov exponent estimation. MATLAB’s parfor structure is employed to distribute independent parameter realizations across multiple workers. The proposed approach is validated using the commensurate fractional-order Lorenz system with q1 = q2 = q3 = 0.99, where the bifurcation parameter β is varied over [1, 10]. The obtained bifurcation structures are consistent with the sign variation of the largest Lyapunov exponent, confirming the transition between chaotic and regular regimes. The serial and parallel implementations produce numerically identical Lyapunov spectra within the reported precision, indicating that the proposed parallelization strategy preserves the dynamical outputs of the corresponding serial computation. In terms of computational performance, the serial implementation required 6617.57 s, whereas the parallel implementation completed the same analysis in 1473.20 s, corresponding to a speed-up of approximately 4.49× with a parallel efficiency of 74.87%. In addition, an incommensurate fractional-order case with q1 = 0.99, q2 = 0.98, and q3 = 0.97 was tested, where the parallel implementation produced the same largest Lyapunov exponent curve as the serial implementation while reducing the computation time from 6843.45 s to 1974.79 s. These results show that the proposed parfor-based approach is a numerically consistent and computationally efficient tool for high-resolution bifurcation and Lyapunov exponent analyses of fractional-order chaotic systems.
Haris Çalgan, Han Li· ADBA Computer Science· 0 citations
Maximizing the dominant Lyapunov exponent λ1 of an incommensurate fractional-order chaotic system, while respecting the dynamical conditions for a strange attractor, is a non-convex, gradient-free problem on a history-dependent landscape. Existing metaheuristic studies typically use hard-cutoff penalties that distort the fitness landscape and integer-order Lyapunov estimators that can be biased for strongly fractional regimes. This paper presents a constraint-faithful optimization framework combining (i) subtractive-hinge penalties that vanish on the feasible set, (ii) a memory-consistent Grünwald–Letnikov variational Lyapunov estimator with adaptive tail-sum truncation, (iii) joint search over parameters and incommensurate orders by the Marine Predators Algorithm, and (iv) a fractional conditional Lyapunov exponent (FCLE) that recovers the integer-order limit. Applied with a fixed configuration to the fractional-order Lorenz, Ma–Chen financial, Iqbal–Wang, and Hyper–Chen systems, the framework converges to feasible attractors with enlarged Lyapunov spectra. Dissipativity is rigorously verified; all selected optima have strictly negative Lyapunov trace at the reported precision. FCLE analysis on the optimized Lorenz attractor recovers the integer-order identity cmin=λ1 under full-state coupling, and shows that single-state x-coupling raises the threshold to ≈9λ1*. The optimized fractional-order Lorenz attractor is employed as the random-number generator of a recent chaos-based image-encryption scheme, where it yields strong statistical results across standard benchmarks.
Massoud M. Aboukhalaf, Mohamed A. El-Beltagy, A. Radwan et al.· Mathematical and Computation...· 0 citations
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