TL;DR
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.
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.
Four-dimensional chaotic systems have garnered significant attention due to their complex nonlinear dynamics, high-dimensional complexity, and wide range of applications in science and engineering. This paper proposes and investigates a novel four-dimensional chaotic system in both constant- and variable-order frameworks to reveal the influence of memory effects on its dynamical behavior. The variable-order formulation is established using the Liouville–Caputo fractional derivative, while an efficient numerical scheme based on Lagrange interpolation is developed to approximate the variable-order derivative accurately. A rigorous local stability analysis is first conducted to characterize the equilibrium points and establish their instability and non-hyperbolic nature under the different derivative formulations. The nonlinear dynamics of the proposed system are then comprehensively examined through phase portraits, time series, bifurcation diagrams, and Lyapunov exponent analysis. The results demonstrate that the variable-order model preserves the fundamental topological characteristics of the chaotic attractors while introducing adaptive transient responses and significantly richer dynamical behaviors than the corresponding constant fractional-order model. Furthermore, the proposed system generates previously unreported chaotic attractors and phase-space patterns, enriching the class of known four-dimensional chaotic systems and demonstrating the variable-order framework’s enhanced capability to produce diverse nonlinear phenomena through adaptive memory effects.
Reliable finite-horizon forecasting of chaotic dynamics is challenging because small approximation errors grow rapidly during recursive prediction. This study presents a controlled comparison of data-driven and physics-regularized forecasting methods for the Lorenz and Rössler systems. The proposed Hybrid Physics-Informed Feedforward Neural Network (Hybrid PI-FNN) learns a discrete state-transition map from a ten-state observation window through a five-step recursive rollout. Unlike conventional continuous-coordinate physics-informed neural networks, physical consistency is imposed using fourth-order Runge–Kutta transition targets derived from the known governing equations. The physics weight is selected using chronological recursive validation and evaluated against an architecturally identical multi-step FNN with λ=0. Conventional FNN, LSTM, Echo State Network (ESN), Autoregressive AR(10), and Dynamic Mode Decomposition baselines are also evaluated using untouched test trajectories. For the 1000-step Lorenz test rollout, the ESN achieved the lowest mean squared error (MSE) of 0.1701, followed by the LSTM with 22.9962. The Hybrid PI-FNN produced an MSE of 95.8157, compared with 87.9260 for its λ=0 ablation; therefore, physics regularization did not improve Lorenz test MSE, although their forecast horizons at a 10% normalized-error threshold were similar (273 and 272 steps, respectively). For the Rössler system, the Hybrid PI-FNN reduced recursive MSE from 0.2202 for the λ=0 ablation to 0.0834, corresponding to a 62.11% reduction, while both models completed the maximum evaluated 1000-step forecast horizon. Nevertheless, the ESN again achieved the lowest Rössler MSE of approximately 8.04×10−5. Finite-horizon correlation-dimension analysis, exact governing-equation Lyapunov spectra, computational-cost comparisons, and five-seed paired experiments were additionally conducted. The exact spectra confirmed one positive, one approximately neutral, and one negative exponent for each system, indicating chaotic but not hyperchaotic behavior. The paired multi-seed analysis did not establish a statistically significant forecasting advantage from physics regularization. These findings show that higher-order physics consistency can benefit particular systems and configurations, but it does not guarantee universal superiority in recursive chaotic forecasting.
In this study, we develop a novel variable-order fractional extension of an eight-dimensional (8D) hyperchaotic differential equation system modeled via the Liouville–Caputo operator. Moving beyond constant fractional-order models, our system implements time-variable orders, which enable its historical memory structure to evolve dynamically over time. To numerically approximate the trajectories of this complex 8D system, a second-order Lagrange numerical integration approach is formulated. An extensive dynamic analysis explores the behavior of this variable-order framework under two distinct configurations: a slowly periodic memory function and a smooth, monotonic hyperbolic tangent function. Topological complexity and multidimensional chaos are characterized using parameter-dependent bifurcation diagrams, phase portraits, Kaplan–Yorke fractal dimensions, and Kolmogorov–Sinai metric entropy. Numerical results show that both variable-order configurations display robust hyperchaotic dynamics characterized by four positive Lyapunov exponents. Crucially, the proposed variable-order extension enhances the phase space footprint of the baseline system, achieving a maximum Kaplan–Yorke dimension of 7.100, thereby offering excellent topological density for secure cryptographic applications.
This study investigates the capability of deep neural networks to infer the time-evolution of the Rössler system a canonical chaotic oscillator by leveraging initial conditions and forcing parameters as input variables and underscores the need for hybrid approaches to address long-term instability.
We use cookies to run the site and, with your consent, for analytics and to show ads. See our Cookie Policy.
