Hybrid Lyapunov–Vision Framework for Chaos Identification in Fractional-Order Nonlinear Dynamics

Abstract

This study presents a unified computational framework for detecting chaotic behavior in a fractional-order Lorenz system by combining Lyapunov-based dynamical analysis with modern machine learning methods. The fractional-order system is simulated over a wide range of the control parameter, and the corresponding Lyapunov spectrum is computed to identify chaotic and non-chaotic regimes. These time-domain trajectories are transformed into high-resolution wavelet scalogram images, enabling a vision-based representation of fractional-order dynamics. The resulting image dataset is classified using both a Vision Transformer (ViT) model and a Support Vector Machine (SVM) classifier built on ViT-extracted feature embeddings. Experimental results demonstrate that the ViT model achieves near-perfect discrimination between chaotic and non-chaotic patterns, with an accuracy of 0.9627, a Cohen’s kappa of 0.920, and an MCC of 0.949. The SVM classifier yields even higher performance, achieving an accuracy of 0.9776, a kappa coefficient of 0.955, and an MCC of 0.955. ROC analyses confirm that both models reach an AUC of 1.00, indicating excellent separability between the two classes. The findings show that wavelet-based image encoding combined with transformer architectures provides a powerful and generalizable approach for chaos detection in fractional-order nonlinear systems. This integrated methodology offers a scalable solution for automated analysis of complex dynamical behavior and establishes a bridge between classical chaos theory and state-of-the-art deep learning models.

1. Introduction

The study of chaotic behavior in nonlinear dynamical systems has gained substantial importance across mathematics, physics, engineering, and computational sciences, owing to the profound implications of chaos for prediction, stability, control, and signal interpretation. Chaotic systems exhibit extreme sensitivity to initial conditions, meaning that arbitrarily small perturbations in the starting state can produce exponentially diverging trajectories over time. This intrinsic unpredictability, despite the deterministic nature of the underlying equations, has led to the development of various theoretical and computational tools for detecting and characterizing chaos, among which Lyapunov exponents play a central role. By quantifying the average exponential divergence or convergence of neighboring trajectories, the Lyapunov spectrum provides a rigorous mechanism for distinguishing between stable and chaotic regimes in nonlinear systems. A positive largest Lyapunov exponent is widely accepted as the hallmark of chaotic dynamics, enabling mathematically grounded classification of system behavior. Chaotic and hyperchaotic systems have been investigated through a wide range of analytical, numerical, and experimental approaches, leading to diverse models and metrics that deepen our understanding of nonlinear behavior. The following studies collectively illustrate how theoretical indicators, novel system designs, and advanced detection methods converge to characterize and utilize chaos across different dynamical frameworks. Sahin and colleagues designed a hyperchaotic circuit by integrating a flux-controlled memristor into a Wien bridge oscillator. The system exhibits non-clipping amplitudes, simplified implementation, and rich hyperchaotic behavior, confirmed through simulations and experiments [1]. Yang et al. proposed a four-dimensional hyperchaotic system with no equilibrium points and enhanced it using Julia fractal mappings, producing novel multi-scroll attractors. Complexity analysis and DSP implementation demonstrate suitability for cryptography and secure communication [2]. This study introduced AEADV, an efficient divergence-based metric for identifying chaos in nonuniform dissipative and conservative systems. Tested on Rössler, Hopfield networks, Sprott-D, and a biological model, the metric outperformed classical Lyapunov and bifurcation methods in computational efficiency [3]. Kuate and collaborators developed a simplified Chua-diode-driven Rössler variant capable of generating up to eight coexisting attractors. Recent studies show that hyperchaotic memristive neural networks enable secure and efficient image encryption for IoV and IoT applications. Multistability, multiscroll formation, and FPGA-based hardware realization further confirm the theoretical richness and practical applicability of these modern chaotic systems [4,5,6]. This work analyzed oval-like billiards through conservative generalized bifurcation diagrams (CGBD), revealing fractal structures, period-adding sequences, and chaotic layers near saddle points—demonstrating how geometric deformation governs conservative chaotic dynamics [7].

Although classical chaos theory typically focuses on integer-order differential equations, recent advances have demonstrated that fractional-order dynamical systems—systems whose derivatives have non-integer orders—can exhibit rich dynamical behavior, including chaos, even when the total system order is lower than three. Fractional calculus naturally incorporates memory and hereditary effects into system dynamics, offering a more flexible modeling framework that aligns closely with real-world physical and biological processes. As a result, fractional-order chaotic systems have become an active research area with applications in secure communications, nonlinear control, biosignal modeling, and electronic circuit design. Despite this growing interest, accurately detecting chaotic behavior in such systems remains a nontrivial task, especially when parameter spaces are large and system responses must be evaluated across many configurations. Fractional-order chaotic systems, enriched by memory-dependent dynamics and non-integer differentiation, have inspired a diverse body of research spanning theoretical analysis, model construction, control design, and data-driven detection. The following studies collectively demonstrate how different mathematical formalisms, control strategies, and machine-learning frameworks advance the characterization, stabilization, and practical exploitation of chaos in fractional-order environments. Kartal introduces a Caputo fractional-order Cournot duopoly model with relative profit delegation, discretized via piecewise constant arguments. The study establishes comprehensive stability criteria, analytically proves flip bifurcations, and confirms chaotic behavior through Lyapunov exponents and the 0–1 chaos test. Additionally, a Pyragas controller is shown to effectively suppress chaos in the economic system [8]. Nabil and Tayeb propose a 4D conformable fractional-order supply chain finance model, solved using the Conformable Adomian Decomposition Method. The system exhibits complex chaotic dynamics validated by Lyapunov spectra, fractal dimensions, bifurcation diagrams, and multiscale spectral entropy. A nonlinear controller achieves chaos synchronization, enabling the development of an RGB image encryption scheme, whose security is demonstrated through simulations [9]. This study investigates a chaotic financial model of fractional order and applies a washout filter-based control strategy to regulate unstable chaotic oscillations. The controller effectively reduces chaotic fluctuations without altering system equilibria, demonstrating applicability for stabilizing economic models influenced by memory-dependent dynamics [10]. Kavuran develops a deep BiLSTM classification framework to detect parameter-driven dynamical variations in the fractional-order Chen system. Synthetic time-series data are generated using the Adams–Bashforth–Moulton (ABM) predictor–corrector method, while chaoticity is confirmed via Lyapunov exponents. The BiLSTM model achieves 98–100% accuracy, showing strong capability in identifying subtle structural changes in fractional-order chaotic signals, with implications for secure communication and cryptology [11].

Machine learning methods, and deep learning models in particular, have recently emerged as powerful tools for analyzing complex signals due to their ability to learn high-dimensional representations directly from data. When combined with time–frequency transformations such as continuous wavelet scalograms, deep neural architectures can effectively distinguish between chaotic and non-chaotic patterns in nonlinear systems without relying solely on classical analytical indicators. Vision-based models—including Vision Transformers (ViT) and traditional feature-based classifiers such as Support Vector Machines (SVM)—are especially promising for this purpose, as they can leverage rich image-encoded information extracted from system trajectories. Liu et al. propose a knowledge-embedded Vision Transformer framework aimed at improving tool performance evaluation (TPE) under varying machining conditions. By integrating prior machining parameters with multisensor signals, the method mitigates challenges related to noise, data distribution shifts, and limited training samples. The approach demonstrates enhanced generalization and robustness compared to conventional direct and indirect TPE strategies [12]. Gross et al. investigate the emergence of spontaneous symmetry breaking in large-scale multi-head attention (LS-MHA), revealing that each head forms specialized label clusters analogous to single-filter mechanisms in CNNs. They introduce a coarse-grained metric—single-head performance (SHP)—to characterize head specialization in compact convolutional transformer architectures. The study shows that replacing early transformer blocks with convolutional layers reduces latency without compromising accuracy, providing insights into unified ViT–CNN learning behaviors grounded in statistical physics [13]. Jia et al. conduct a comprehensive evaluation of classic ML models (CART, RF) and deep models (3D-CNN, Hamida, ViT, RVT) across hyperspectral datasets with systematically varied spatial resolution, spectral resolution, and signal-to-noise ratio. They show that (i) coarser spectral resolution degrades performance for most models except RVT, (ii) spatial-resolution effects depend heavily on scene homogeneity and target size, and (iii) SNR strongly impacts classical ML methods while ViT/RVT models remain largely unaffected. These findings bridge instrumentation constraints with downstream AI model selection and performance [14]. Ren et al. introduce a high-precision fault-location method relying on panoramic voltage features and lightweight ViT variants. Utilizing only 2.5 ms of single-ended voltage data, the method extracts Gramian angular fields, recurrence plots, Markov transition fields, and wavelet scalograms to characterize transient responses. Experiments on real-time digital simulators demonstrate strong resilience against noise, high transition resistance, and low sampling frequencies, illustrating the viability of ViT-based regression in ultra-fast HVDC protection environments [15]. Noh et al. present BiViT, a transformer-based multimodal architecture integrating driver bio-signals (ECG, EEG, PPG, SpO₂) with facial video features to detect risk behaviors across six distraction categories. The system leverages a ViT-based visual encoder, a time-series transformer for physiological data, and a joint fusion module. Experimental results demonstrate 0.91 accuracy and F1-score, significantly outperforming image-only systems and establishing the relevance of bio-signal integration for robust driver monitoring [16]. Zhang et al. develop ViT-LLMR, a motion recognition framework that applies self-attention to enhance MMG and IMU channels and feeds them into a ViT architecture. The model achieves 94.62% accuracy on eight lower-limb movements across six subjects, outperforming CNN-, RNN-, and ML-based feature-engineering approaches. The study highlights the importance of multimodal fusion and temporal–spatial attention mechanisms for wearable HCI applications such as rehabilitation, prosthetics, and robotics [17]. As machine learning and deep learning techniques increasingly shape the analysis of nonlinear and high-dimensional signals, a wide range of recent studies has explored how advanced architectures—particularly transformer-based models—can extract, fuse, and interpret complex spatiotemporal information beyond the capabilities of traditional analytical tools. The following works collectively demonstrate how these data-driven methods enhance robustness, improve representational power, and enable new applications across machining diagnostics, hyperspectral imaging, power system protection, multimodal driver monitoring, and human–computer interaction.

Motivated by these considerations, this study investigates the chaotic behavior of a fractional-order Lorenz system by integrating Lyapunov-based chaos identification with a modern computer-vision-driven classification approach. Time series generated from the fractional-order model are transformed into wavelet-based scalogram images and subsequently classified using both a Vision Transformer and an SVM classifier. Furthermore, conventional machine learning classifiers, including Random Forest (RF) and K-Nearest Neighbors (KNN), were incorporated as baseline comparison models to evaluate the effectiveness of the proposed ViT-based framework. This dual framework enables rigorous comparison between deep feature extraction architectures and classical kernel-based machine learning methods. By combining analytical chaos detection with supervised visual classification, the present work provides a comprehensive pipeline for detecting chaotic regimes in nonlinear fractional-order dynamical systems, while also demonstrating the effectiveness of transformer-based models for chaos recognition tasks. The results highlight the capacity of contemporary machine learning techniques to complement traditional dynamical systems theory and offer new pathways for automated analysis of complex nonlinear behavior. In this study, we present the following contributions:

• A unified physics-informed framework combining Lyapunov analysis, Continuous Wavelet Transform (CWT), and transformer-based deep learning is proposed for chaos identification in fractional-order nonlinear systems.
• The proposed approach transforms nonlinear time-series signals into wavelet scalogram images, reformulating chaos detection as an image classification problem.
• A Lyapunov exponent-based labeled scalogram dataset is constructed to ensure consistency between classical nonlinear dynamics and learning-based representations.
• A comparative evaluation using ViT, SVM, RF, and KNN classifiers is performed to analyze the effectiveness of deep learning and conventional machine learning methods for chaos classification.
• The proposed hybrid ViT–SVM framework achieves highly discriminative performance with near-perfect separability between chaotic and non-chaotic regimes.
• The proposed methodology provides a scalable and generalizable framework for analyzing memory-dependent chaotic behavior in fractional-order systems with potential applications in nonlinear control, signal processing, and secure communications.

2. Materials and Methods

2.1. Chaotic Systems and the Concept of Lyapunov Exponents

Chaos theory examines deterministic yet unpredictable behavior in dynamical systems that exhibit extreme sensitivity to initial conditions. Even minute differences in the initial state can lead to exponentially divergent trajectories over time, a phenomenon commonly known as the “butterfly effect.” One of the key quantitative measures of chaos is the Lyapunov exponent, which quantifies the exponential rates of divergence or convergence of neighboring trajectories in the phase space. Specifically, if the largest Lyapunov exponent is positive, the system is classified as chaotic. For example, in the Lorenz system, a positive largest Lyapunov exponent indicates the presence of chaos at specific parameter values. The Lyapunov exponent framework allows for the quantitative characterization of dynamical stability and chaotic behavior.

Lyapunov exponent computation is essential for detecting and characterizing chaotic behavior. In an n-dimensional system, there exist Lyapunov exponents, and the system is considered chaotic if the largest exponent is positive. Several numerical methods exist for computing Lyapunov exponents, including Wolf’s algorithm and Jacobian-based linearization techniques. The QR decomposition method and Gram-Schmidt reorthonormalization are commonly used to track exponential divergence rates. However, computing Lyapunov exponents in fractional-order systems is more challenging due to memory effects, requiring adapted numerical approaches such as the Benettin–Wolf algorithm or ABM fractional integration methods.

2.2. The Importance of Fractional-Order Differential Equations

Classically, for a dynamical system to exhibit chaotic behavior, it must be at least third-order, meaning it requires three independent variables and at least one nonlinear term. However, fractional-order differential equations (FODEs) extend this understanding by allowing the derivative order to take non-integer values, thereby incorporating memory effects into system dynamics. The concept of fractional derivatives was first considered by Leibniz and later formalized by Liouville and Riemann, but its engineering and physics applications have gained prominence in recent years [18]. Fractional-order models provide greater flexibility and more accurate representations compared to integer-order models. Notably, research has shown that chaos can emerge in systems where the total system order is less than three, with chaos observed in systems with an effective order in the range of 2.4–2.9. This suggests that chaos conditions can be realized at lower orders than previously expected. Fractional-order systems have been successfully applied in fields such as control theory, oscillator design, signal processing, and magnetic resonance imaging, garnering significant academic interest.

Fractional calculus generalizes classical Newtonian differentiation, allowing differentiation orders to take fractional values. Mathematically, fractional derivatives can be defined using the Caputo or Riemann–Liouville formulations, introducing memory effects into system dynamics [19]. This property enhances the accuracy of modeling viscoelastic materials, electrochemical processes, and biological systems. One of the most significant findings in chaotic systems is fractional-order chaos, where chaos emerges even in systems with total orders lower than three. Studies by Petráš demonstrated that various chaotic oscillators retain chaotic properties when expressed as fractional-order systems [20]. This has significantly increased academic interest in fractional-order systems for their potential applications in nonlinear control, cryptography, and secure communication [21,22,23,24,25,26].

2.3. Fundamental Properties of the Lorenz System

The Lorenz system, introduced by Edward Lorenz in 1963 in the study of atmospheric circulation, remains one of the most well-known examples of chaotic behavior [27,28]. This three-dimensional system is defined by a set of nonlinear differential equations, which, under specific parameter values, yield solutions exhibiting a strange attractor, a fractal structure. The classical Lorenz equations are given as follows:

$$\begin{array}{l}{D}_t^{1}x\left(t\right)=\sigma \left(y\left(t\right)-x\left(t\right)\right),\\ D_t^{1}y\left(t\right)=\rho x\left(t\right)-y\left(t\right)-x\left(t\right)z\left(t\right),\\ D_t^{1}z\left(t\right)=x\left(t\right)y\left(t\right)-\beta z\left(t\right),\end{array}$$

(1)

where $\sigma$, $\rho ,\beta$ are system parameters. Lorenz originally set $\sigma$ = 10, $\rho$ = 28, and $\beta$ = 8/3, under which the system exhibited a chaotic attractor due to its sensitivity to initial conditions and lack of periodicity. The Lorenz attractor, recognized for its butterfly-like shape is a classic representation of chaos theory. The system undergoes a bifurcation transition, where increasing the parameter $\rho$ beyond a critical threshold shifts the system from stable behavior to a chaotic regime. The system’s phase-space trajectories eventually evolve into broad-band, unpredictable patterns, making long-term forecasting impossible despite its deterministic nature.

Interest in fractional-order dynamical systems has increased significantly in recent years with the emergence of Lorenz-like behavior in a wide variety of engineering and scientific applications, such as electrical circuits, cryptography, and chemical reactions. The literature has shown that fractional dynamical systems exhibit unique chaotic properties not observed in integer-order systems. Therefore, reliable numerical analysis of fractional-order chaotic models is of great importance. The fractional-order Lorenz chaotic system discussed in this study [20,29,30]

$$\begin{array}{l}{D}_t^{{\alpha }_1}x\left(t\right)=\sigma \left(y\left(t\right)-x\left(t\right)\right),\\ D_t^{{\alpha }_2}y\left(t\right)=\rho x\left(t\right)-y\left(t\right)-x\left(t\right)z\left(t\right),\\ D_t^{{\alpha }_3}z\left(t\right)=x\left(t\right)y\left(t\right)-\beta z\left(t\right),\end{array}$$

(2)

with the initial conditions $x\left(0\right)=c_1,y\left(0\right)=c_2,z\left(0\right)=c_3$.

Since it is practically impossible to obtain an analytical solution for fractional-order Lorenz systems, researchers resort to numerical methods to investigate long-term behavior. Indeed, in recent years, numerous studies have been conducted on the existence and uniqueness of solutions for these systems [31], dynamical regime differences between classical and fractional systems, phase portraits, chaos measurements based on the largest Lyapunov exponent, and bifurcation diagrams [32]. Furthermore, detailed analyses of the dynamical properties of the system have been presented [33,34,35]. While each study suggests that the proposed method can represent chaotic systems with high precision, it is difficult to determine which method provides the most accurate results due to the lack of exact solutions for chaotic systems. Therefore, methods that can accurately capture long-time behavior, especially in fractional-order chaotic models, are needed.

On the other hand, the field of fractal and fractional computation has undergone significant development over the last three decades, and many different and non-equivalent definitions for fractional derivatives have been proposed during this period. The most widely used of these definitions are the Riemann–Liouville, Grünwald–Letnikov (GLFD), and Caputo derivatives. These derivatives are considered basic tools in modeling fractional differential equations.

Definition 1.

Riemann–Liouville fractional derivative of order $\alpha$ of a function $y\left(t\right)$ about $t$ on the interval ($t_0,t$) is defined as

$${}_{t_0}{}^{RL}D_t^{\alpha }y(t)=\left\{\begin{array}{l}{\displaystyle \frac{1}{\mathsf{\Gamma }\left(n-\alpha \right)}}{\displaystyle \frac{d^n}{d^n}}{\int }_{t_0}^t {\displaystyle \frac{y\left(\tau \right)}{(t-\tau {)}^{\alpha -n+1}}}d\tau ,0\le n-1<\alpha <n.\\ {\displaystyle \frac{d^{n}y\left(n\right)}{d{t}^n}},\alpha =n\in N.\end{array}\right.$$

(3)

Definition 2.

Caputo fractional derivative of order $\alpha$ of a function $y\left(t\right)\in C^n\left[t_0,t\right]$ about $t$ on the interval ($t_0,t$) is defined as

$${}_{t_0}{}^{C}D_t^{\alpha }y(t)=\left\{\begin{array}{l}{\displaystyle \frac{1}{\Gamma \left(n-\alpha \right)}}{\int }_{t_0}^t {\displaystyle \frac{y^{\left(n\right)}\left(\tau \right)}{(t-\tau {)}^{\alpha -n+1}}}d\tau ,0\le n-1<\alpha <n.\\ {\displaystyle \frac{d^{n}y\left(t\right)}{d{t}^n}},\alpha =n\in N.\end{array}\right.$$

(4)

Definition 3.

The Grünwald–Letnikov fractional derivative of $\alpha$-order on the interval ($t_0,t$) is defined as

$${}_{t_0}{}^{GL}D_t^{\alpha }y(t)=\underset{h\to 0}{lim} {\displaystyle \frac{1}{h^{\alpha }}}{\displaystyle {\sum }_{j=0}^{\left[\left(t-t_0\right)/h\right]}} (-1{)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{j}}\right)y(t-jh),$$

(5)

where$(-1{)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{j}}\right)={\displaystyle \frac{(-1{)}^j\Gamma (\alpha +1)}{\Gamma \left(j+1\right)\Gamma \left(\alpha -j+1\right)}}$.

Two methodologies are present in the literature for determining the numerical response of a fractional-order system: the time domain and the frequency domain. An enhanced variant of the ABM algorithm is one of the premier time-domain approaches utilizing the discretization of fractional differential equations [36]. The predictor–corrector scheme has been proposed. In frequency-domain methodologies, the Laplace transform of the fractional operator is substituted with a rational approximation through integer-order approximation. The approximation equations are subsequently transformed back into the time domain, contingent upon the application context. Numerical solutions of FOC oscillators derived using the ABM approach, which necessitates greater computing complexity and hardware resources, exhibit superior accuracy compared to Grünwald-Letnikov. Nonetheless, as this study lacks a hardware-based application, the predictive-corrective ABM algorithm, a recognized time-domain approximation method, was employed to simulate the fractional-order oscillator [37,38]. Consequently, the ABM technique is generated as seen in Equations (6)–(11), wherein the fractional derivative presented in Equation (5) aligns with Volterra’s integral equation in Equation (7) [21,39,40].

$$D_t^{\alpha }y\left(t\right)=f\left(y\left(t\right),t\right),y^{\left(k\right)}\left(0\right)=y_0^{\left(k\right)},k=0,1,\dots ,m-1$$

(6)

$$y(t)={\displaystyle {\sum }_{k=0}^{\lceil \alpha \rceil -1}} y_0^{(k)}{\displaystyle \frac{t^k}{k!}}+{\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha )}}{\int }_0^t (t-\tau {)}^{\alpha -1}f(\tau ,y(\tau ))d\tau$$

(7)

$$y_h\left(t_{n+1}\right)={\displaystyle \sum _{k=0}^{m-1}} {\displaystyle \frac{t_{n+1}^k}{k!}}{y}_0^{\left(k\right)}+{\displaystyle \frac{h^{\alpha }}{\mathsf{\Gamma }(\alpha +2)}}f\left(t_{n+1},y_h^p\left(t_{n+1}\right)\right)+{\displaystyle \frac{h^{\alpha }}{\mathsf{\Gamma }(\alpha +2)}}{\displaystyle \sum _{j=0}^n} a_{j,n+1}f\left(t_j,y_h\left(t_j\right)\right)$$

(8)

$$a_{j,n+1}=\left\{\begin{array}{ll}{n}^{\alpha +1}-(n-\alpha )(n+1{)}^{\alpha },& j=0,\\ (n-j+2{)}^{\alpha +1}+(n-j{)}^{\alpha +1}-2(n-j+1{)}^{\alpha +1},& 1\le j\le n,\\ 1,& j=n+1.\end{array}\right.$$

(9)

$$y_h^p\left(t_{n+1}\right)={\displaystyle {\sum }_{k=0}^{m-1}} {\displaystyle \frac{t_{n+1}^k}{k!}}{y}_0^{\left(k\right)}+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\displaystyle {\sum }_{j=0}^n} b_{j,n+1}f\left(t_j,y_h\left(t_j\right)\right)$$

(10)

$$b_{j,n+1}={\displaystyle \frac{h^{\alpha }}{\alpha }}\left[(n+1-j{)}^{\alpha }-(n-j{)}^{\alpha }\right]$$

(11)

2.4. Dataset Creation

The analysis developed in this study provides a complete computational framework for investigating the chaotic behavior of the fractional-order Lorenz system by means of Lyapunov exponents and by transforming the resulting time series into an image-based dataset suitable for deep learning. The Lyapunov exponents were computed using the MATLAB R2025b-based routine for commensurate Caputo-type fractional-order systems. The extended system was solved using the predictor–corrector Adams–Bashforth–Moulton (ABM) method implemented in the fde12 solver, while periodic Gram–Schmidt orthonormalization was applied to ensure numerical stability during exponent estimation. The primary objective is to examine how the Lyapunov spectrum varies as the control parameter $p$ (equivalently, the Rayleigh number $\rho$) is swept over a prescribed interval, and to determine the corresponding parameter ranges in which the system exhibits regular or chaotic dynamics. The system dimension is three, corresponding to the three state variables $x,y$, and $z$. The initial condition vector is set to $x_{\mathrm{start}}=[0.1,0.1,0.1{]}^T$, and the fractional derivative order is chosen as ${\alpha }_1={\alpha }_2={\alpha }_3=0.995$, so that the model incorporates memory effects characteristic of fractional-order dynamics rather than classical integer-order behavior. The temporal domain is defined on the interval $\left[0,100\right]$ and is discretized using a time step of $h=0.005$, which yields 20,001 discrete time points for each trajectory. For the computation of Lyapunov exponents, a normalization interval $h_{\mathrm{norm}}=5$ is adopted. The control parameter is varied from $p_{\mathrm{m}\mathrm{i}\mathrm{n}}=0$ to $p_{\mathrm{m}\mathrm{a}\mathrm{x}}=250$ with $n=250$ increments, leading to a uniform parameter step size. So that the system is evaluated at 251 distinct parameter values over the interval $\left[0,250\right]$. The fractional-order Lorenz system considered in this study is governed by the classical Lorenz equations with fixed parameters $\sigma =10$ and $\beta ={\displaystyle \frac{8}{3}}$, and a variable control parameter $p$, namely

$$\begin{array}{l}{D}_t^{0.995}x\left(t\right)=10\left(y\left(t\right)-x\left(t\right)\right),\\ D_t^{0.995}y\left(t\right)=\rho x\left(t\right)-y\left(t\right)-x\left(t\right)z\left(t\right),\\ D_t^{0.995}z\left(t\right)=x\left(t\right)y\left(t\right)-{\displaystyle \frac{8}{3}}z\left(t\right),\end{array}$$

(12)

These equations are integrated in their fractional-order form with a derivative order $\alpha =0.995$ using a numerical solver specifically designed for fractional differential equations (such as a predictor–corrector-based method for Caputo derivatives). For each parameter value $p$, the algorithm simultaneously computes the full Lyapunov spectrum $\left[L{E}_1,L{E}_2,L{E}_3\right]$ via a dedicated fractional-order Lyapunov exponent routine, and integrates the corresponding trajectory of the Lorenz system over the entire time interval. The numerical results are organized in two main data structures: a matrix in which each row contains the current parameter value $p$ and the associated triplet of Lyapunov exponents, and a cell-based repository that stores the complete time-domain solution $Y\left(t\right)\in {\mathbb{R}}^{3\times \mathrm{20,001}}$ for each parameter value. In this way, the procedure yields both the parametric evolution of the Lyapunov spectrum and the underlying state trajectories for all sampled values of $p$. After completing the parameter sweep, the dependence of the Lyapunov exponents on $p$ is visualized to reveal chaotic and non-chaotic regions within the parameter space. The three Lyapunov exponents are plotted as functions of $p$, and a horizontal reference line at zero is superimposed to separate stable regimes from unstable or chaotic regimes. To mitigate numerical fluctuations and obtain a clearer view of the overall dynamical trends, one of the exponents is smoothed using a suitable smoothing operation. This smoothed Lyapunov exponent profile provides a more robust indicator of the onset of chaos and facilitates the identification of parameter intervals where the system transitions from regular to chaotic behavior. Additionally, representative trajectories corresponding to particular parameter values can be examined in the time domain to illustrate typical behaviors in regular and chaotic regimes. Figure 1 illustrates the Lyapunov exponent spectrum of the fractional-order Lorenz system as a function of the control parameter $p$.

Figure 2a–c illustrate the time-domain responses of the fractional-order Lorenz system for a representative chaotic parameter value $p=250$ and fractional derivative order $\alpha =0.995$. The presented signals correspond to the 29th parameter sample within the total parameter sweep dataset and represent the temporal evolution of the state variables x(t), y(t), and z(t), respectively. The trajectories of the state variables $x\left(t\right),y\left(t\right)$, and $z\left(t\right)$ exhibit pronounced aperiodic oscillations, irregular amplitude modulations, and intermittent bursts, which collectively indicate the breakdown of periodic motion and the emergence of chaotic dynamics. In particular, the absence of repeating patterns and the strong variability in peak amplitudes suggest a high sensitivity to initial conditions, a fundamental property of deterministic chaos. Among the three state variables, $z\left(t\right)$ demonstrates comparatively larger amplitude excursions, reflecting the nonlinear energy exchange and coupling structure inherent in the Lorenz system.

The corresponding three-dimensional phase-space representation is depicted in Figure 3. The reconstructed trajectory in the $\left(x\right(t),y(t),z(t\left)\right)$ space forms a dense, butterfly-shaped attractor, which is a classical geometric signature of the Lorenz chaotic regime. The bounded yet non-periodic nature of the orbit confirms that the system evolves within a strange attractor, where trajectories remain confined while never intersecting or repeating. The observed stretching and folding mechanisms in the phase portrait provide a geometric interpretation of the positive Lyapunov exponent identified for this parameter value. Moreover, the smooth but intricate structure of the attractor highlights the influence of fractional-order dynamics, where memory effects slightly modify the classical Lorenz attractor while preserving its essential chaotic characteristics.

Together, the time-domain signals and the phase-space portrait provide complementary evidence of chaotic behavior in the fractional-order Lorenz system. These visual representations not only validate the Lyapunov-based classification of the dynamical regime but also serve as a basis for transforming the system trajectories into image-based datasets for subsequent deep learning-based chaos detection and classification tasks.

To derive an explicit binary classification of chaotic versus non-chaotic behavior, the smoothed Lyapunov exponent vector is processed through a threshold-based decision mechanism. As illustrated in Figure 4, the smoothed Lyapunov exponent profile and the corresponding binary indicator function jointly highlight the parameter intervals associated with chaotic and regular dynamics. For each parameter sample $i$, if the smoothed Lyapunov exponent satisfies $LE\left(i\right)>0.001$, the system is marked as chaotic at that parameter, whereas values below this threshold are interpreted as indicative of regular (non-chaotic) dynamics. The threshold value of 0.001 is chosen to suppress the influence of low-level numerical noise around zero, while still reliably detecting genuinely positive exponents. The indices of the nonzero entries of $x$ directly indicate the parameter values $p$ for which the system enters a chaotic regime. This analysis provides a clear segmentation of the parameter axis into chaotic and non-chaotic regions, offering a concise summary of the system’s global bifurcation structure in terms of Lyapunov exponents.

The next step consists of transforming this indicator-based characterization into a structured categorical labeling suitable for dataset construction. Samples with indicator values greater than 0.001 were classified as “chaotic,” whereas those with values within the interval ([0, 0.001]) were classified as “nonchaotic.” Subsequently, these categorical labels were numerically encoded as 1 for chaotic and 0 for nonchaotic samples to facilitate dataset construction and machine learning-based analysis. This procedure associates each parameter value $p$ with a categorical label derived directly from the Lyapunov exponent analysis. The dataset is then reorganized so that each trajectory is decomposed into its three component signals, corresponding to the $x,y$, and $z$ state variables, and these components are stacked in consecutive rows of a new data matrix. The categorical label associated with each parameter value is replicated across the three component signals, yielding a label vector aligned with this expanded representation. A second discretization step with broader interval bounds is used to ensure that the labels remain consistent after this restructuring, thereby producing a final label set that is compatible with the new data format.

Finally, the time-domain dataset is converted into an image-based dataset through a time–frequency representation obtained by the continuous wavelet transform (CWT). For each one-dimensional signal in the reorganized dataset, a wavelet filter bank is constructed for the given signal length, with 12 voices per octave to ensure high-resolution coverage of the frequency axis. The magnitude of the complex CWT coefficients is then computed, providing a two-dimensional time–frequency map (scalogram) for each signal. These coefficient matrices are normalized to the interval [0, 1], quantized to 8 -bit grayscale values, and then mapped to RGB color images using a standard colormap (such as “jet” with 256 colors). Each resulting image is resized to $224\times 224$ pixels, which matches the typical input dimensionality required by modern deep learning architectures, including Vision Transformers and convolutional neural networks. The images are saved to disk in a directory structure organized by class, with separate subfolders for “chaotic” and “nonchaotic” samples. The complete dataset comprises 748 scalogram images in JPEG format, each with dimensions 224 × 224 × 3 (height × width × color channels). These images are systematically organized into two subdirectories (“chaotic” and “nonchaotic”) within the specified root directory with filenames incorporating class labels and sequential indices for unambiguous identification. The final dataset consisted of 669 valid CWT scalogram images, including 361 chaotic and 308 non-chaotic samples. Although the initial dataset generation process produced a larger number of scalogram images, some ambiguous or low-quality scalograms located near transitional dynamical regions were manually excluded to improve label consistency and dataset reliability. An 80:20 stratified train–test split strategy was then adopted, resulting in approximately 535 training samples and 134 test samples. As illustrated in Figure 5, the CWT scalograms exhibit distinct patterns for chaotic and non-chaotic dynamical regimes of the fractional-order Lorenz system.

To avoid potential data leakage between the training and test subsets, dataset partitioning was performed at the parameter level rather than at the individual image level. Specifically, all x(t), y(t), and z(t) subtrajectory scalogram images generated from the same control parameter value p(i) were assigned exclusively either to the training set or to the test set. Therefore, correlated representations originating from the same dynamical trajectory could not simultaneously appear in both subsets, ensuring strict independence between training and testing data.

This image-based representation transforms the time series classification problem into a computer vision task, enabling the application of state-of-the-art deep learning architectures such as Vision Transformers for automated chaos detection and classification in fractional-order dynamical systems.

Through this pipeline, the original problem of chaos detection in a fractional-order Lorenz system is transformed into a supervised image classification task. The Lyapunov-based classification provides a principled ground truth for labeling each time series, while the wavelet-based scalogram images supply rich time–frequency information that can be exploited by deep learning models. As a result, the methodology not only enables systematic exploration of chaotic regimes in the Lorenz parameter space but also produces a high-quality, image-based dataset that can be used to train and evaluate state-of-the-art deep learning architectures for automated detection and classification of chaotic behavior in fractional-order dynamical systems.

3. Results

3.1. Vision Transformer (ViT) Model Architecture

The Vision Transformer (ViT) is a deep neural network architecture that applies the Transformer originally developed for natural language processing directly to image data. Instead of relying on convolutional layers like CNNs, ViT splits an input image into multiple fixed-size patches and treats each patch as a “token” (analogous to a word in a sentence). These patch tokens are fed into a Transformer encoder, which uses self-attention mechanisms to learn both local and global relationships between different parts of the image. This design allows ViTs to capture long-range dependencies in images more effectively than CNNs, while still achieving competitive performance on image classification tasks [41].

Images are represented as tensors of size $H\times W\times C$ (Height × Width × Channels). For instance, a typical input to a Vision Transformer (ViT) model is a $224\times 224$ RGB image with three channels, resulting in an input tensor of dimension $224\times 224\times 3$. Since ViT architectures require fixed-size inputs, images are commonly resized to a standard resolution (e.g., $224\times 224$) prior to processing. This resized image tensor serves as the starting point of the ViT architecture. Rather than processing the entire image holistically, ViT partitions the input into a set of non-overlapping patches of size $P\times P$ pixels (e.g., $16\times 16$ pixels per patch), which collectively tile the image. Assuming that $H$ and $W$ are divisible by the patch size $P$, the total number of patches $N$ is given by

$$N={\displaystyle \frac{H}{P}}\times {\displaystyle \frac{W}{P}}$$

(13)

As illustrated in Figure 6, the input image is first resized and then divided into non-overlapping $P\times P$ patches to generate the token sequence used by the Vision Transformer (ViT) architecture. For example, a $224\times 224$ image with $P=16$ produces $(224/16)\times (224/16)=14\times 14=196$ patches. Each patch can be regarded as a small image of dimension $P\times P\times C$ (with $C=3$ for RGB images). These patches are then flattened into one-dimensional vectors of length $P\times P\times C$; for a $16\times 16$ RGB patch, this corresponds to $16\times 16\times 3=768$ elements. Consequently, the input image is transformed into a sequence of $N$ patch vectors, each encoding the local visual content of a specific region of the image. Analogous to word tokens in natural language processing, these flattened patch vectors act as “visual tokens,” yielding, in the above example, a sequence of 196 tokens that can be processed by the transformer encoder.

After extraction and flattening, each patch vector is passed through a learnable linear projection (a fully connected layer) to map it to a desired embedding dimension $D$. In other words, for each flattened patch $x\in {\mathbb{R}}^{P^{2}C}$, we compute an embedded vector $z\in {\mathbb{R}}^D$ via a linear transformation:

$$z=W\hspace{0.17em}x+b, W\in {\mathbb{R}}^{\left(P^{2}C\right)\times D},$$

(14)

where $D$ is a hyperparameter (projection dimension). This Dense layer can either reduce or increase dimensions. After this patch embedding step, we obtain $N$ vectors each of size $D$. These form an $N\times D$ sequence matrix that will be the input to the Transformer. Let $E_{\mathrm{pos}}$ be an $N\times D$ matrix of positional embeddings. We then add these to the patch embeddings matrix $Z$ element-wise:

$$Z_{\mathrm{input}}=Z_{\mathrm{patch}}+E_{\mathrm{pos}},$$

(15)

where $Z_{\mathrm{patch}}\in {\mathbb{R}}^{N\times D}$ is the matrix of patch embeddings and $Z_{\mathrm{input}}$ is the final input to the Transformer encoder. This ensures each token carries information about where its patch was located in the image.

The architecture follows the ViT framework proposed by Alexey Dosovitskiy [21], configured as a reduced-scale model for limited-data conditions. Input images of size $224\times 224\times 3$ are divided into nonoverlapping $16\times 16$ patches, producing $N=196$ tokens. Each patch is projected into a $D=64$ dimensional embedding with positional encoding and processed through $L=8$ Pre-LN Transformer encoder blocks with $h=4$ attention heads. The feed-forward dimension was reduced to $D_f=128$ to decrease model complexity under limited-data conditions. GELU activation, Layer Normalization $(ϵ=1\times {10}^{-6}$), and residual connections were employed throughout the network.

The classification head consists of two fully connected GELU-activated layers with 256 and 128 units, followed by a softmax output layer. Dropout rates of 0.1 and 0.5 were applied within the encoder and classification head, respectively, to reduce overfitting. The model was trained using the AdamW optimizer with learning rate and weight decay set to $1\times {10}^{-4}$ for 100 epochs and a batch size of 32. The dataset was split into 80:20 training and testing subsets using stratified sampling, images were normalized to the [0, 1] range, and labels were one-hot encoded. Reproducibility was ensured using fixed random seeds and deterministic TensorFlow operations on an NVIDIA TPU.

The Vision Transformer (ViT) architecture is composed of a stack of Transformer encoder layers that operate on a sequence of $N$ image patch embeddings, each of dimension $D$. Each encoder layer begins with layer normalization, followed by multi-head self-attention (MHSA), which enables global interaction among all patch tokens. For each token, query, key, and value matrices are computed via learned linear projections, and the scaled dot-product attention is defined as

$$\mathrm{A}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}(Q,K,V)=\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{m}\mathrm{a}\mathrm{x}\left({\displaystyle \frac{Q{K}^T}{\sqrt{d_k}}}\right)V,$$

(16)

where $Q,K,V\in {\mathbb{R}}^{N\times d_k}$ and $d_k$ denotes the dimension of each attention head. Multiple attention heads operate in parallel, and their outputs are concatenated to form an $N\times D$ representation.

Residual connections are applied after the attention operation, yielding $X^{\prime }=X+MHSA\left(LN\right(X\left)\right)$, followed by layer normalization and a position-wise feed-forward network (FFN). The FFN is implemented as a two-layer multilayer perceptron with nonlinear activation and is given by

$$\mathrm{F}\mathrm{F}\mathrm{N}\left(x^{\prime }\right)=W_2\sigma \left(W_1{x}^{\prime }+b_1\right)+b_2,$$

(17)

where $W_1\in {\mathbb{R}}^{D\times D_f},W_2\in {\mathbb{R}}^{D_f\times D}$, and typically $D_f=4D$. A second residual connection completes the encoder block as $X^{″}=X^{\prime }+\mathrm{F}\mathrm{F}\mathrm{N}\left(X^{\prime }\right)$. Stacking $L$ such blocks enable the model to learn hierarchical and globally contextualized image representations.

After the final encoder layer, patch-level features are aggregated to obtain an image-level representation. Although the original ViT employs a learnable CLS token for classification, in this study, the patch-level features from the final encoder layer are flattened into a single vector and passed through fully connected layers for classification. The final classifier produces logits $z=W_{\mathrm{out}}h+b$, which are converted into class probabilities via the softmax function,

$${\widehat{y}}_c={\displaystyle \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(z_c\right)}{{\displaystyle {\sum }_{j=1}^K} \mathrm{e}\mathrm{x}\mathrm{p}\left(z_j\right)}},$$

(18)

where $K$ denotes the number of classes.

In summary, the Vision Transformer processes an image by breaking it into patches, embedding those patches (with positional information), and then applying a standard Transformer encoder to learn relationships between patches via self-attention. The final output either from a special token or a pooled feature is passed through a classifier (one or more dense layers) to predict the image’s class. This architecture leverages the Transformer’s strength in modeling sequence relationships for image recognition, capturing both local details (within patches) and global context (across patches) effectively. It has been shown to achieve excellent results on image classification tasks, especially when trained on large datasets, and demonstrates that an image is worth 16 × 16 words. The overall architecture of the proposed ViT–SVM classification pipeline is presented in Figure 7.

In binary classification problems, the performance of a model is evaluated using quantities derived from the confusion matrix, which contains true positives (TPs), true negatives (TNs), false positives (FPs), and false negatives (FNs). The overall correctness of the classifier is expressed through accuracy, defined as

$$\mathrm{Accuracy}={\displaystyle \frac{TP+TN}{TP+TN+FP+FN}},$$

(19)

which measures the proportion of correctly classified samples in the entire dataset. The reliability of positive predictions is quantified by precision,

$$\mathrm{Precision}={\displaystyle \frac{TP}{TP+FP}},$$

(20)

while the model’s ability to detect all actual positive instances is captured by recall (or sensitivity),

$$\mathrm{Recall}={\displaystyle \frac{TP}{TP+FN}}$$

(21)

A balanced indicator combining both characteristics is the F1-score, computed as the harmonic mean of precision and recall,

$$F1={\displaystyle \frac{2·\mathrm{Precision}·\mathrm{Recall}}{\mathrm{Precision}+\mathrm{Recall}}}$$

(22)

To account for class-wise behavior, macro-averaging computes the arithmetic mean of the class-specific metrics, such as

$$\text{Macro-Precision}={\displaystyle \frac{{\mathrm{Precision}}_1+{\mathrm{Precision}}_2}{2}},$$

(23)

whereas micro-averaging aggregates all TP, FP, and FN values across classes before computing the metrics, following

$$\text{Micro-Recall}={\displaystyle \frac{\sum TP}{\sum \left(TP+FN\right)}}$$

(24)

Beyond these measures, agreement between predictions and ground truth can be evaluated using Cohen’s kappa, which adjusts the observed accuracy for the agreement expected by chance. First, the observed agreement $p_o$ and expected agreement $p_e$ are computed as

$$p_o={\displaystyle \frac{TP+TN}{TP+TN+FP+FN}},p_e=\left({\displaystyle \frac{TP+FP}{N}}·{\displaystyle \frac{TP+FN}{N}}\right)+\left({\displaystyle \frac{FN+TN}{N}}·{\displaystyle \frac{FP+TN}{N}}\right),$$

(25)

and the kappa coefficient is then obtained as

$$\kappa ={\displaystyle \frac{p_o-p_e}{1-p_e}}$$

(26)

A more stringent scalar measure, the Matthews correlation coefficient (MCC), incorporates all four confusion matrix entries in a single expression and provides a balanced evaluation even under class imbalance. It is defined as

$$MCC={\displaystyle \frac{TP·TN-FP·FN}{\sqrt{\left(TP+FP\right)\left(TP+FN\right)\left(TN+FP\right)\left(TN+FN\right)}}},$$

(27)

with values ranging from −1 (complete disagreement) to +1 (perfect classification), while 0 corresponds to chance-level performance. Together, these metrics provide a comprehensive and statistically rigorous assessment of classifier performance in binary decision problems.

The quantitative evaluation reveals that the ViT-based classifier achieves a highly robust discriminative performance in distinguishing between chaotic and nonchaotic classes. The classification performance of the proposed ViT-based model was evaluated using a confusion matrix and receiver operating characteristic (ROC) analysis, as shown in Figure 8. The confusion matrix shows that the model correctly identifies 67 chaotic and 62 nonchaotic samples, while misclassifications are limited to only 5 false negatives in the chaotic class and no false positives in the nonchaotic class. This distribution indicates a conservative and highly reliable decision boundary for the negative class, accompanied by exceptionally precise positive-class predictions.

Class-wise precision and recall values further characterize this behavior. The chaotic class attains a precision of 1.000, demonstrating that every chaotic prediction corresponds to a true chaotic instance, whereas its recall of 0.9305 indicates that only a very small portion of chaotic samples is not successfully recovered by the model. Conversely, the nonchaotic class exhibits a recall of 1.000, confirming that the classifier correctly retrieves every nonchaotic sample, while its precision of 0.925 reflects a moderate degree of class overlap manifested in a small number of chaotic instances being predicted as nonchaotic. The corresponding F1-scores, 0.964 for chaotic and 0.961 for nonchaotic, illustrate that the classifier maintains a well-balanced trade-off between sensitivity and specificity.

As summarized in

Table 1. Precision, recall, and F1-score of the ViT-based model for chaotic and non-chaotic classification.

	Precision	Recall	F1-Score
Chaotic	1.000	0.9305	0.964
Nonchaotic	0.925	1.000	0.961
Macro Avg	0.9625	0.9653	0.9625
Micro Avg	1.000	0.9627	0.9810

and

Table 2. Overall performance metrics of the ViT-based model, including accuracy, Cohen’s kappa, and MCC.

Accuracy	Cohen’s Kappa	MCC
0.9627	0.920	0.949

, aggregated metrics also support this observation. The macro-averaged precision, recall, and F1-score (0.9625, 0.9653, and 0.9625, respectively) reveal that the model sustains consistently high performance across both classes without bias. Micro-averaged metrics, driven by the global count of true positives and false negatives, remain similarly strong, with micro precision 1.000, micro recall 0.9627, and a micro F1-score of 0.9810, demonstrating excellent overall discrimination across the dataset.

The ROC analysis provides additional evidence of the model’s separability characteristics. The ROC curve closely adheres to the ideal upper-left boundary, and the model achieves an AUC of 1.00, indicating perfect rank-order discrimination between the two classes. The dashed diagonal line in Figure 8b represents the performance of a random classifier (chance level), corresponding to an area under the curve (AUC) of 0.5. This result implies an exceptionally high signal-to-noise ratio in the learned representation space and demonstrates that the classifier assigns higher decision scores to positive instances in a strictly monotonic manner relative to negative ones.

Taken together, the confusion matrix patterns, ROC behavior, and numerical performance metrics collectively indicate that the proposed ViT architecture achieves near-optimal classification reliability for the chaotic–nonchaotic discrimination task. The high precision, strong recall balance, and perfect AUC suggest that the feature representations extracted by the ViT model are highly expressive and well-aligned with the intrinsic structure of the underlying dynamical behavior, enabling excellent generalization and robust operational performance.

3.2. Feature Extraction and SVM Classification Process

To construct a feature extractor model from the Vision Transformer architecture, the layer structure of the model is systematically scanned. During this scanning process, the position of the Flatten layer, which transforms multi-dimensional tensors into one-dimensional vectors, is identified. The Flatten layer undertakes the task of flattening the multi-dimensional outputs obtained from the transformer blocks before transmitting them to the classification head, typically forming a critical transition point in the architecture between the final feature representation and the classification layers.

Once the index of the Flatten layer is determined, the preceding layer is selected as the feature extraction layer. This selection is theoretically justified by the fact that the pre-Flatten layer is typically the final Layer Normalization and contains the highest-level semantic representation of the image. The feature vectors obtained at this stage have passed through all transformer blocks and encode the global contextual information of the image.

The construction of the feature extractor model is accomplished using the Keras Functional API. The new model shares the input layer of the original ViT model, but the output point is configured as the designated feature layer. This configuration excludes the classification head (fully connected layers) of the model, enabling it to perform only the feature extraction function. This approach is commonly preferred for transfer learning and downstream tasks.

In the event that the feature layer cannot be successfully identified, an error handling mechanism is activated, recommending that the user verify the model architecture. This situation may arise when the model structure deviates from the standard ViT architecture.

For validation of the constructed feature extractor model, a sample image is selected from the training dataset and tested. The image is resized to comply with the model’s batch processing requirement. The feature extraction process is executed through the predict method, and the dimensional characteristics of the obtained feature vector are reported. These extracted high-level features are suitable for use in image similarity analysis, clustering, classification, and other machine learning applications.

The performance profile of the SVM classifier demonstrates strong discriminative capability for separating chaotic and nonchaotic classes. The classification performance of the SVM model was evaluated using a confusion matrix and receiver operating characteristic (ROC) analysis, as shown in Figure 9. The confusion matrix indicates that the model correctly classifies 71 chaotic and 60 nonchaotic instances, with misclassifications limited to 1 false positive in the chaotic class and 2 false negatives in the nonchaotic class. This distribution reflects a highly stable decision boundary, with only marginal overlap between class manifolds, suggesting that the SVM kernel mapping preserves class separability effectively in the transformed feature space.

Class-level precision and recall metrics further elucidate this behavior. The chaotic class achieves a precision of 0.9736, indicating a very low rate of false alarms, while its recall of 0.9861 demonstrates that the classifier successfully retrieves nearly all chaotic samples. The nonchaotic class exhibits a precision of 0.9836 and a recall of 0.9677, revealing a similarly strong detection capability with only minor leakage into the opposite class. The corresponding F1-scores, 0.9798 for chaotic and 0.9756 for nonchaotic, confirm that the classifier maintains a balanced trade-off between sensitivity and specificity across both decision regions.

Macro-level averages, with a macro precision of 0.9786, macro recall of 0.9769, and macro F1 of 0.9777, indicate consistent class-independent performance without systematic bias. Micro-averaged metrics—driven by global TP, FP, and FN counts—are slightly higher, with micro precision and recall both yielding 0.9776, reflecting reliable performance across the full dataset. These aggregated measures highlight the SVM model’s effectiveness in maintaining balanced discrimination even when integrating all samples simultaneously.

The ROC analysis provides further validation of the classifier’s robustness. The ROC curve maintains a trajectory close to the ideal upper-left boundary, and the model achieves an AUC of 1.00, indicating essentially perfect rank-order separability between chaotic and nonchaotic patterns. This behavior suggests that the SVM decision function produces smoothly monotonic confidence scores that reliably separate the two classes, reflecting a high signal-to-noise ratio within the learned feature representation.

Overall, the evidence from the confusion matrix, class-level metrics, macro/micro aggregates, and ROC characteristics demonstrates that the SVM classifier delivers near-optimal performance in the chaotic–nonchaotic discrimination task. The minimal misclassification rate, strong precision–recall balance, and perfect AUC collectively imply that the SVM effectively captures the underlying structural differences between classes, offering a high-fidelity and generalizable decision function for dynamical behavior classification.

As shown in Figure 10a, the confusion matrix indicates that the Random Forest model correctly classified 65 chaotic and 57 non-chaotic samples, while only a limited number of instances were misclassified. These results demonstrate the model’s capability to distinguish between chaotic and non-chaotic dynamical regimes. Figure 10b further confirms this performance through the ROC curve, which achieves an AUC value of 0.98. Figure 11a presents the confusion matrix of the KNN classifier. The model correctly classified 64 chaotic and 60 non-chaotic samples, with only a limited number of misclassifications. Figure 11b shows the corresponding ROC curve, achieving an AUC value of 0.98, indicating effective discrimination between chaotic and non-chaotic dynamical regimes.

Table 3. Class-wise precision, recall, and F1-score comparison of Random Forest, KNN, and SVM classifiers for chaotic and non-chaotic classification.

Method	Class	Precision	Recall	F1-Score
Random Forest	Chaotic	1.000	0.931	0.964
	Non-chaotic	0.925	1.000	0.961
KNN	Chaotic	0.970	0.889	0.928
	Non-chaotic	0.882	0.968	0.923
SVM	Chaotic	0.973	0.986	0.979
	Non-chaotic	0.984	0.968	0.976

shows that SVM delivers the most balanced class-wise performance, achieving high precision, recall, and F1-scores for both chaotic and non-chaotic classes. Random Forest also performs strongly, with perfect precision for the chaotic class and perfect recall for the non-chaotic class, though small misclassifications reduce its overall balance. KNN yields the weakest results, with lower recall and F1-scores, indicating less robust discrimination of nonlinear dynamics.

Table 4. Overall performance comparison of Random Forest, KNN, and SVM classifiers in terms of accuracy, Cohen’s kappa, MCC, and AUC.

Method	Accuracy	Cohen’s Kappa	MCC	AUC
Random Forest	0.963	0.920	0.949	0.98
KNN	0.925	0.850	0.856	0.98
SVM	0.978	0.955	0.955	1.00

confirms these findings at the overall level. SVM attains the best performance with the highest accuracy (0.978), kappa (0.955), MCC (0.955), and an AUC of 1.00, demonstrating excellent separability and generalization. Random Forest remains competitive, with high accuracy (0.963) and strong kappa, MCC, and AUC values, but still falls slightly behind SVM. Overall, the results indicate that SVM is the most effective classifier for distinguishing chaotic and non-chaotic behaviors.

4. Discussion

Chaos detection seeks to determine whether an observed time series originates from a truly deterministic chaotic system or from random noise. Research employs both classical nonlinear time series tools and artificial intelligence-based methods that have emerged in recent years. The most fundamental criterion is that a positive largest Lyapunov exponent indicates chaos [42,43,44,45,46,47,48,49,50]. Algorithms for calculating the Lyapunov exponent (LE) from experimental data have been compared and it has been shown that the whole LE spectrum can be estimated by the neural network-based methods [47]. LE estimation is sensitive to data length and noise; the need for sufficient and high-quality data is emphasized [46,47,49,51,52]. The attractor is reconstructed using delayed coordinates; criteria such as mutual information for embedding dimension and delay time, false nearest neighbors, and correlation dimension are used [42,43,50,52,53,54,55]. It has been reported that the false nearest neighbor and complex autocorrelation methods yield the best results in parameter selection [55]. The correlation dimension, fractal dimension, and determinism tests are used to examine whether the chaotic attractor is low-dimensional and deterministic [42,50,52,53,54,56]. Common Tools for Chaos Detection are shown in

Table 5. Common Tools for Chaos Detection.

Tool/Test	Purpose	Typical Application	References
Largest Lyapunov exponent	Is chaos present or absent?	Physics, ecology, engineering	[42,44,45,46,47,48,49,52,53]
Bifurcation diagram	Chaotic/periodic parameter regions	Maps, encryption maps	[47,50]
Phase portrait/phase space	Visualizing the attractor	Continuous dynamical systems, RCS	[43,48,50,52,53]
0–1 chaos test	Binary (chaotic/ordered) classification	Noisy/moderately noisy series	[50,57,58]

.

In short and noisy time series, deterministic dynamics (including chaotic ones) can be statistically distinguished by comparing the predictive power of linear and nonlinear Volterra–Wiener models; this method has been found to be more accurate than classical methods [45]. In ecological and population dynamics, “noise-suppressing” and “noise-amplifying (chaotic)” systems have been distinguished using LE estimation in the presence of dynamic noise [44].

Gray-level images were generated from the p–q plane projections of the 0–1 test and summarized using persistence homology to distinguish between chaos and non-chaos; periodic and chaotic dynamics could be separated even under high noise [57]. Chaos/non-chaos classification from recurrence matrix images using a single CNN was achieved with over 92% accuracy [59]. Multi-modal deep networks combining time series, recurrence plots, and spectrograms can distinguish between different chaotic and non-chaotic systems in a multi-class manner even in the presence of colored noise [58]. These tools have been used for chaos detection in river flow [53], radar cross-section [48], radar target detection [51], and signal processing and engineering signals [51,54,56]. The R package integrates embedding, LE estimation via artificial neural networks, and a formal statistical test of the chaos hypothesis [49].

The literature highlights a combination of phase space reconstruction + Lyapunov exponents + dimension analysis as central to chaos detection. However, for noisy, short, or complex data, new methods such as 0–1 tests, topological data analysis, and deep learning-based classifiers are gaining prominence. Parameter selection, data length, and noise level appear to be decisive factors in determining the reliability of a given method.

The proposed study has many important methodological and scientific advantages over the current approaches to chaos detection and classification in the literature. Most traditional studies rely solely on classical nonlinear analysis tools such as Lyapunov exponents, phase space reconstruction, bifurcation diagrams, or the 0–1 chaos test. In contrast, this study proposes a unified and automated computational framework that integrates classical chaos theory with modern transformer-based deep learning methods. Thus, the study not only determines whether the system exhibits chaotic behavior but also enables the automatic learning and classification of chaotic patterns by artificial intelligence. In this regard, the study builds a strong bridge between classical dynamical systems theory and contemporary artificial intelligence methods.

One of the study’s most significant strengths is the physically and mathematically meaningful labeling of the dataset. While in many deep learning-based approaches in the literature, class labels are created intuitively or directly based on the dataset, in this study, chaotic and non-chaotic regimes are derived directly from the Lyapunov exponent spectrum. This ensures that the created dataset is grounded in physical reality and enables the model to learn not only visual patterns but also the system’s actual dynamic behavior.

Another important contribution of this work is the study of nonlinear fractional systems. Deep learning-based chaos studies in the literature have mostly been conducted on classical integer-order systems or chaotic maps. In contrast, the proposed approach was developed on a memory-dependent fractional-order Lorenz system modeled using Caputo derivatives and the ABM solution method. Fractional-order systems exhibit much more complex behavior than classical systems due to memory effects and past dependency. Therefore, the integration of fractional calculus, Lyapunov analysis, and transformer-based learning within the same framework constitutes a significant methodological innovation in the literature on nonlinear dynamics.

One of the key features distinguishing this study from many CNN, BiLSTM, or recurrence plot-based studies in the literature is the use of the Vision Transformer (ViT) architecture. The self-attention-based ViT architecture has the ability to learn long-range time–frequency relations in wavelet scalogram images better than traditional convolution-based structures, which is a major advantage, especially for representing multiscale, complex and irregular chaotic patterns. Thus, the study presents a state-of-the-art approach in the literature regarding the application of transformer-based global representation learning to chaos detection problems.

Additionally, the study offers an original contribution by transforming the chaos detection problem into a computer vision problem through continuous wavelet transform (CWT)-based scalogram images. Instead of directly analyzing raw time series, they are transformed into high-resolution time–frequency images that encode the system dynamics both in time and frequency. This approach allows deep learning models to learn richer discriminative features and facilitates the easy generalization of the framework to different fractional-order systems such as Chen, Chua, or Rössler.

Another significant advantage of the work is its hybrid ViT–SVM architecture. The Vision Transformer is a powerful feature extractor, while the Support Vector Machine acts as a classifier in the proposed model. This hybrid architecture reduces the risk of overfitting, especially with limited data, ensures more efficient use of the transformer embeddings, and improves the class distinction. The accuracy rate of about 97.76% and the AUC value of 1.00 strongly suggest the high discriminative ability of the proposed architecture.

The proposed work offers not only a high-performance chaos classification model but also a scalable and generalizable next-generation analysis paradigm that combines fractional-order nonlinear dynamics, Lyapunov-based chaos analysis, wavelet-based time–frequency imaging, and transformer-based representation learning into a single unified architecture. In this regard, the study goes beyond classical chaos detection methods to propose a modern, physics-based, AI-supported chaos analysis framework.

5. Conclusions

This study has presented a unified and high-fidelity computational framework for the characterization and classification of chaotic behavior in the fractional-order Lorenz system by integrating Lyapunov-spectrum-based nonlinear dynamical analysis with modern vision-driven machine learning architectures. Through the application of a high-precision predictor–corrector algorithm for fractional differential equations, long-time trajectories were computed with sufficient numerical accuracy to resolve the subtle bifurcation structures and memory-induced distortions that distinguish fractional-order dynamics from their integer-order counterparts. The resulting time series were mapped into wavelet-based scalogram representations, enabling a rich time–frequency encoding of the underlying dynamical evolution. These representations were subsequently classified using both a Vision Transformer (ViT) and an SVM classifier trained on ViT-derived embeddings, providing a robust supervised learning paradigm grounded in principled Lyapunov-based ground truth labels. The empirical results demonstrate that the ViT model achieves near-perfect separability between chaotic and nonchaotic regimes, with an accuracy of 0.9627 and an AUC of 1.00, while the SVM classifier surpasses this performance with an accuracy of 0.9776 and an AUC of 1.00. These findings underscore the capacity of the proposed hybrid method to capture the intrinsic structural differences associated with chaotic transitions, highlighting the discriminative power of transformer-based global attention mechanisms when coupled with the fine-grained spectral content extracted via continuous wavelet transforms. Furthermore, the detailed inspection of the Lyapunov spectrum across the control parameter space reveals that fractional-order Lorenz dynamics exhibit expanded, fragmented, and memory-dependent chaotic regions, reflecting a richer and more intricate dynamical landscape than that observed in the classical integer-order Lorenz model. This demonstrates that fractional-order systems, due to their inherent nonlocality, modify the conventional bifurcation and stability patterns in a manner that cannot be inferred through integer-order analysis alone. Overall, the study shows that combining analytical chaos indicators with transformer-based visual classification constitutes a powerful, scalable, and generalizable approach for the automated identification of complex dynamical behavior. Beyond offering strong numerical performance, this integrated methodology bridges nonlinear dynamics, wavelet-based signal analysis, and state-of-the-art deep learning models, thereby providing a foundation for future investigations aiming to decode and control chaotic behavior in fractional-order nonlinear systems.

While this study evaluates the proposed hybrid Lyapunov–Vision framework specifically on the fractional-order Lorenz system, the methodology is highly adaptable to other fractional-order nonlinear systems (e.g., fractional-order Chen, Rössler, and Chua systems, or memory-dependent economic and biological models). The framework’s generalizability relies on a modular, three-stage pipeline. First, ABM numerical integration and Lyapunov-based thresholding can systematically generate and label time-series data for multidimensional fractional differential equations. Second, the Continuous Wave Transform (CWT) operates independently of the underlying mathematical model by uniformly transforming the resulting 1D time-wavelength path into standardized 2D time–frequency scalogram images. Finally, the Vision Transformer (ViT) and SVM classification components are fundamentally system-independent; they learn chaotic features directly from image-based spatial and frequency representations, rather than from explicit system parameters. Consequently, by simply modifying the governing equations during the initial data generation phase, this automated pipeline can be directly deployed to identify chaotic regimes in various other fractional-order dynamics without requiring structural changes to the deep learning architecture.

Future studies may focus on real-time chaos detection and monitoring systems using embedded or IoT-based platforms. In addition, the proposed framework may offer promising applications in cryptology, including chaos-based image encryption, secure key generation and nonlinear cryptographic systems, owing to the memory-dependent and highly parameter-sensitive nature of fractional-order chaotic systems. Although the present study is based on simulated data, future research may incorporate experimentally measured real-world datasets and investigate alternative deep learning architectures to further improve classification performance and generalization capability.