Search Results (339)

In this study, an analysis of fractional-order Lü systems is performed through a framework approach consisting of analytical solution strategies in combination with numerical methods. On the analytical methodology front, the recently developed form of the new generalized differential transform method (NGDTM) is adopted for its efficiency in providing an approximate solution with high capability in tracking the behavior of these systems. On the other hand, the Grünwald–Letnikov via Riemann–Liouville scheme (GLNS) is adopted within this study as one of its tools in confirming whether chaos exists within these systems. The performance and accuracy of the proposed method are also rigorously tested, and comparisons are made numerically with the Adams–Bashforth–Moulton method, which is used here as a standard method for validation purposes. It is clear from the results that the combination of analytical and numerical methods can greatly enhance both the speed of computation and the accuracy of results. Additionally, the proposed method or approach is found to be quite robust and accurate and can thus be employed for analyzing various fractional dynamical systems that display chaotic attractors. The proposed method can also be expanded upon in the future for analyzing complex models in science and engineering. Full article

An investigation on a discrete-time infectious disease model that incorporating vertical transmission is presented in this paper. Departing from prior research centered on continuous-time frameworks, our study adopts a discrete-time formulation to better capture the complex epidemiological dynamics. We establish a model and conduct a bifurcation analysis of its equilibrium points. In particular, sufficient conditions for the local stability and the emergence of Neimark–Sacker and flip bifurcations are rigorously derived and analytically verified. As anticipated, variations in the bifurcation parameter give rise to distinct periodic regimes in the system response. To mitigate the instabilities and chaotic behaviors resulting from these bifurcations, we propose and validate two control strategies, which are Hybrid Control Method and State Feedback Control. Numerical simulations futher substantiated the analytical results, demonstrating that appropriate parameter adjustments can shift the system behavior from chaotic attractors and limit cycles toward stable equilibria. Our results show that by dynamically adjusting the intensity of prevention and control measures to mitigate unstable factors such as vertical transmission and high infection rates, or reducing the frequency of system updates to slow down the growth of infections, the epidemic can be transitioned from repeated outbreaks to a stable and manageable state. Full article

Sliding bearings–rotor systems are widely present in rotating machinery structures. The dynamic behavior triggered by friction and rub-impact faults is a key factor restricting the safe and stable operation of a rotor system. Existing studies mainly focus on analyzing dynamic characteristics but rarely explore the degree of friction and rub-impact in the system. This paper takes the sliding bearing–double-disk rotor system with friction and rub-impact as the research model, and defines the concept of the rubbing ratio. It analyzes the influence of relevant structural parameters on the system. The results reveal that the system exhibits rich nonlinear dynamics. Specifically, increasing either the rotor–stator clearance or the lubricant viscosity can drive the system into a broader regime of chaotic motion, while simultaneously reducing the extent of the rub-impact contact region. As the stator stiffness increases from 10⁷ N/m to 9 × 10⁷ N/m, the number of chaotic windows in the bifurcation diagram increases from one to three, while the maximum rubbing force rises by approximately 58% and the rubbing ratio increases from 50% to 56%. The phenomenon of coexisting attractors in the system is also revealed and analyzed. The above research results help to reveal the motion laws of this type of rotor system and have certain guiding significance for parameter matching and optimization design of the system dynamics. Full article

Stochastically forced nonlinear wave systems are commonly associated with complex dynamical behavior, although little is known about the general interaction of nonlinear dispersion, irrational forcing frequencies, and multiplicative noise. To fill this gap, we consider a generalized stochastic SIdV equation and examine the effects of deterministic and stochastic influences on the long-term behavior of the equation. The PDE was modeled using a stochastic traveling-wave transformation that simplifies it into a planar system, which was studied using Darboux-seeded constructions, Poincaré maps, bifurcation patterns, Lyapunov exponents, recurrence plots, and sensitivity diagnostics. We discovered that natural, implicit, and unique seeds produce highly diverse transformed wave fields exhibiting both irrational and golden-ratio forcing, controlling the transition from quasi-periodicity to chaos. Stochastic perturbation is demonstrated to suppress as well as to amplify chaotic states, based on noise levels, altering attractor geometry, predictability, and multistability. Meanwhile, OGY control is demonstrated to be able to stabilize chosen unstable periodic orbits of the double-well regime. A stochastic bifurcation analysis was performed with respect to noise strength $\sigma$, revealing that the attractor structure of the system remains robust under stochastic excitation, with noise inducing only bounded fluctuations rather than qualitative dynamical transitions within the investigated parameter regime. These findings demonstrate that the emergence, deformation, and controllability of complex oscillatory patterns of stochastic nonlinear wave models are jointly controlled by nonlinear structure, external forcing, and noise. Full article

In this paper, a fractional-order discrete map with a hyperbolic sine function has been proposed and studied. Firstly, the basic characteristics of the map in integer-order case are studied theoretically and numerically. Secondly, dynamics of the map are investigated via numerical simulations. Attractors and bifurcation diagram spectrums are given when a parameter is varied. Furthermore, the map with the Caputo fractional difference operator has been studied. The chaotic attractors in commensurate-order and incommensurate-order cases are shown. For the characteristics of hyperbolic sine function, the chaotic attractors with different structures for the map can be obtained. It can be concluded that the map has rich dynamics in integer-order and fractional-order cases. Finally, stabilization and adaptive synchronization of the fractional-order map are realized by designing suitable controllers, respectively. Numerical results are used to demonstrate the effectiveness of the controllers for the map. Full article

Circuit implementation is a widely accepted method for validating theoretical insights observed in chaotic systems. It also serves as a basis for numerous chaos-based engineering applications, including data encryption, random number generation, secure communication, neuromorphic computing, and so forth. To get feasible, compact, and cost-effective circuit implementations of chaotic systems, the underlying mathematical model may be simplified while preserving all rich nonlinear behaviors. In this framework, this manuscript presents a simplified Hopfield Neural Network (HNN) capable of generating a broad spectrum of complex behaviors using a minimal number of electronic elements. Based on Shil’nikov’s theorem for heteroclinic orbits, the number of non-zero synaptic connections in the matrix weights is reduced, while simultaneously using only one nonlinear activation function. As a result of these simplifications, we obtain the most compact electronic implementation of a tri-neuron HNN with the lowest component count but retaining complex dynamics. Comprehensive theoretical and numerical analyses by equilibrium points, density-colored continuation diagrams, basin of attraction, and Lyapunov exponents, confirm the presence of periodic oscillations, spiking, bursting, and chaos. Such chaotic dynamics range from single-scroll chaotic attractors to double-scroll chaotic attractors, as well as coexisting attractors to transient chaos. A brief security application of an S-Box utilizing the presented HNN is also given. Finally, a physical implementation of the HNN is given to confirm the proposed approach. Experimental observations are in good agreement with numerical results, demonstrating the usefulness of the proposed approach. Full article

The theory underlying non-linear dynamical systems remains essential for understanding complex behaviors in science and engineering. In this study, we propose a new chaotic dynamical system that exhibits a butterfly-shaped attractor without any equilibrium point. Despite its compact structure comprising only five terms, the system demonstrates rich chaotic behavior distinct from conventional oscillator models. Detailed modeling and dynamical analyses are conducted to confirm the presence of chaos and to characterize the system’s sensitivity to initial conditions. Furthermore, synchronization of the proposed dynamical system is investigated using both identical and non-identical control algorithms. In the identical case, the activation function of the neural network is governed by the butterfly oscillator dynamics, whereas in the non-identical case, a sigmoidal activation function is employed. The proposed synchronization algorithms enable faster convergence by pinning a subset of nodes in the network. Finally, a practical implementation of the conceived dynamical system in an encryption framework is presented, with the aim to demonstrate its feasibility and potential application in secure communication systems. The results highlight the effectiveness of the proposed approach for both theoretical exploration and engineering applications involving chaotic dynamical systems. Full article

Background: Glioblastoma (GBM) is among the most aggressive and morphologically heterogeneous brain tumors. Beyond static imaging biomarkers, its structural organization can be viewed as a nonlinear dynamical system. Characterizing morphodynamic attractors within such a system may reveal latent stability patterns of morphological change and potential indicators of morphodynamic organization. Methods: We analyzed 494 subjects from the multi-institutional BraTS 2020 dataset using a fully automated computational pipeline. Each multimodal MRI volume was encoded into a 16-dimensional latent space using a 3D convolutional autoencoder. Synthetic morphological trajectories, generated through bidirectional growth–shrinkage transformations of tumor masks, enabled training of a contraction-regularized Neural Ordinary Differential Equation (Neural ODE) to model continuous-time latent morphodynamics. Morphological complexity was quantified using fractal dimension (DF), and local dynamical stability was measured via a Lyapunov-like exponent (λ). Robustness analyses assessed the stability of DF–λ regimes under multi-scale perturbations, synthetic-order reversal (directionality; sign-aware comparison) and stochastic noise, including cross-generator generalization against a time-shuffled negative control. Results: The DF–λ morphodynamic phase map revealed three characteristic regimes: (1) stable morphodynamics (λ < 0), associated with compact, smoother boundaries; (2) metastable dynamics (λ ≈ 0), reflecting weakly stable or transitional behavior; and (3) unstable or chaotic dynamics (λ > 0), associated with divergent latent trajectories. Latent-space flow fields exhibited contraction-induced attractor-like basins and smoothly diverging directions. Kernel-density estimation of DF–λ distributions revealed a prominent population cluster within the metastable regime, characterized by moderate-to-high geometric irregularity (DF ≈ 1.85–2.00) and near-neutral dynamical stability (λ ≈ −0.02 to +0.01). Exploratory clinical overlays showed that fractal dimension exhibited a modest negative association with survival, whereas λ did not correlate with clinical outcome, suggesting that the two descriptors capture complementary and clinically distinct aspects of tumor morphology. Conclusions: Glioblastoma morphology can be represented as a continuous dynamical process within a learned latent manifold. Combining Neural ODE–based dynamics, fractal morphometry, and Lyapunov stability provides a principled framework for dynamic radiomics, offering interpretable morphodynamic descriptors that bridge fractal geometry, nonlinear dynamics, and deep learning. Because BraTS is cross-sectional and the synthetic step index does not represent biological time, any clinical interpretation is hypothesis-generating; validation in longitudinal and covariate-rich cohorts is required before prognostic or treatment-monitoring use. The resulting DF–λ morphodynamic map provides a hypothesis-generating morphodynamic representation that should be evaluated in covariate-rich and longitudinal cohorts before any prognostic or treatment-monitoring use. Full article

This paper explores the dynamic analogy between the discrete Lorenzian attractor and a modified Kropotov–Pakhomov neural network (MRNN). A one-dimensional peak map is used to extract the successive maxima of the Lorenzian system and preserve the basic properties of the chaotic flow. The MRNN, governed by the Bogdanov–Hebb learning rule with dissipative feedback, is formulated as a discrete nonlinear operator whose parameters can reproduce the same hierarchy of modes as the peak map. It is theoretically shown that the map multiplier and the spectral radius of the monodromy matrix of the MRNN provide equivalent stability conditions. Numerical diagrams confirm the correspondence between the control parameters of the Lorenz model and the network parameters. The results establish the MRNN as a neural emulator of the Lorenz attractor and offer an analysis of self-organization and stability in adaptive neural systems. Full article

Chaotic dynamical systems are ubiquitous in nature and modern technology, with applications ranging from secure communications and cryptography to the design of chaos-based sensors and modeling biological phenomena such as arrhythmias and neuronal behavior. Given their complexity, precise analysis of these systems is crucial for both theoretical understanding and practical implementation. The characterization of chaotic dynamical systems typically relies on conventional measures such as Lyapunov exponents and fractal dimensions. While these metrics are fundamental for describing dynamical behavior, they are often computationally expensive and may fail to capture subtle changes in the overall geometry of the attractor, limiting comparisons between systems with topologically similar structures and similar values in common chaos metrics such as the Lyapunov exponent. To address this limitation, this work proposes a geometric framework that treats chaotic attractors as spatial objects, using topological tools—specifically the $\alpha$-sphere—to quantify their shape and spatial extent. The proposed method was validated using Chua’s system (including two reported variations), the Rössler system (standard and piecewise-linear), and a fractional-order multi-scroll system. A parametric characterization of the Rössler system was also performed by varying parameter b. Experimental results show that this geometric approach successfully distinguishes between attractors where classical metrics reveal no perceptible differences, in addition to being computationally simpler. Notably, we observed geometric variations of up to 80% among attractors with similar dynamics and introduced a specific index to quantify these global discrepancies. Although this geometric analysis serves as a complement rather than a substitute for chaos detection, it provides a reliable and interpretable metric for differentiating systems and selecting attractors based on their spatial properties. Full article

Memristor is widely used to construct various memristive neural networks with complex dynamical behaviors. However, hidden multiple attractors have never been realized in memristive neural networks. This paper proposes a novel chaotic system based on a memristive Hopfield neural network (HNN) capable of generating hidden multiple attractors. A multi-segment memristor model with multistability is designed and serves as the core component in constructing the memristive Hopfield neural network. Dynamical analysis reveals that the proposed network exhibits various complex behaviors, including hidden multiple attractors and a super multi-stable phenomenon characterized by the coexistence of infinitely many double-chaotic attractors—these dynamical features are reported for the first time in the literature. This encryption process consists of three key steps. Firstly, the original chaotic sequence undergoes transformation to generate a pseudo-random keystream immediately. Subsequently, based on this keystream, a global permutation operation is performed on the image pixels. Then, their positions are disrupted through a permutation process. Finally, bit-level diffusion is applied using an Exclusive OR(XOR) operation. Relevant research shows that these phenomena indicate a high sensitivity to key changes and a high entropy level in the information system. The strong resistance to various attacks further proves the effectiveness of this design. Full article

This paper investigates the dynamical properties of the fractional-order Arneodo system using a Grünwald–Letnikov-based numerical discretization. Fractional-order operators introduce memory and hereditary effects, enabling a more realistic description than classical integer-order models. The local stability of equilibrium points is examined through eigenvalue analysis of the Jacobian matrix, along with dissipativity conditions and the emergence of complex attractors. A comprehensive dynamical investigation is presented through phase portraits, time series, Lyapunov exponents, and bifurcation diagrams for varying fractional orders. Numerical findings demonstrate the emergence of new chaotic and hyperchaotic attractors. The results confirm that the fractional order strongly influences the system’s stability, sensitivity, and complexity. Our results confirm the relevance of fractional-order modeling in applications, such as secure communication, random number generation, and complex system analysis. Full article

Background: Realistic simulation of ECG signals is essential for validating signal-processing algorithms and training artificial intelligence models in cardiology. Many existing approaches model either waveform morphology or heart rate variability (HRV), but few achieve both with high accuracy. This study proposes a hybrid method that combines morphological accuracy with physiological variability. Methods: We developed a mathematical model that integrates Gaussian mesa functions (GMF) for waveform generation and a chaotic Rössler attractor to simulate RR-interval variability. The GMF approach allows fine control over the amplitude, width, and slope of each ECG component (P, Q, R, S, T), while the Rössler system introduces dynamic modulation through the use of seven parameters. Spectral and statistical analyses were applied, including power spectral density (PSD) computed via the Lomb–Scargle, STFT, CWT, and histogram analyses. Results: The synthesized signals demonstrated physiological realism in both the time and frequency domains. The LF/HF ratio was 1.5–2.0 when simulating a normal rhythm and outside these limits in a simulated stress rhythm, consistent with typical HRV patterns. PSD analysis captured clear VLF (0.003–0.04 Hz), LF (0.04–0.15 Hz), and HF (0.15–0.4 Hz) bands. Histogram distributions showed amplitude ranges consistent with real ECGs. Conclusions: The hybrid GMF–Rössler approach enables large-scale ECG synthesis with controllable morphology and realistic HRV. It is computationally efficient and suitable for artificial intelligence training, diagnostic testing, and digital twin modeling in cardiovascular applications. Full article

In this article, we investigate the dynamical analysis and soliton solutions of the microtubule equation. First, the Lie symmetry method is applied to the considered model to reduce the governing partial differential equation into an ordinary differential equation. Next, the multivariate generalized exponential rational integral function method is employed to derive exact soliton solutions. Finally, the bifurcation analysis of the corresponding dynamical system is discussed to explore the qualitative behavior of the obtained solutions. When an external force influences the system, its behavior exhibits chaotic and quasi-periodic phenomena, which are detected using chaos detection tools. We detect the chaotic and quasi-periodic phenomena using 2D phase portrait, time analysis, fractal dimension, return map, chaotic attractor, power spectrum, and multistability. Phase portraits illustrating bifurcation and chaotic patterns are generated using the RK4 algorithm in Matlab version 24.2. These results offer a powerful mathematical framework for addressing various nonlinear wave phenomena. Finally, conservation laws are explored. Full article

This study introduces a novel robotic control paradigm, “chaos redirection,” which utilizes a single chaotic Hopfield Neural Network (HNN). We introduce “false attractors” synthetic trajectories created by applying controlled temporal shifts to the HNN’s state variables. This method allows a single chaotic source to be sculpted into distinct, task-specific behaviors for autonomous robots. We apply this framework to three applications: area cleaning, systematic search, and security patrol. Quantitative, statistically validated analysis demonstrates the successful generation of functionally distinct behaviors, including high-frequency, confined re-visitation for security patrols; maximized exploratory efficiency for search tasks; and high-entropy, non-repetitive paths for thorough cleaning. Our findings establish this as a robust and computationally efficient framework for applications requiring unpredictable, yet structured, behavior. Full article