Search Results (501)

A Lyapunov-based framework for stability analysis and synthesis of adaptive neural-network (NN) controllers for a class of uncertain second-order nonlinear systems (SNS) with bounded external perturbations and unmodelled dynamics is presented. Online learning is employed for the reconstruction of the plant nonlinearity with the use of a radial-basis-function (RBF) network whose weights are adapted using a direct adaptation law deduced from a single composite Lyapunov function. The proposed controller couples the weight update to a persistent robustifying action, while the closed-loop stability is guaranteed throughout the learning transient, in contrast to schemes that guarantee stability after learning has converged. Using a composite Lyapunov function in the filtered tracking error and the weight-estimation error, we prove that all closed-loop signals are uniformly ultimately bounded (UUB) and that the tracking error converges to an explicitly characterized residual set whose radius is governed by the network reconstruction accuracy, the disturbance bound and the design gains. A $\sigma$-modification ensures parameter boundedness without persistency of excitation, and a robustness theorem shows that bounded parametric perturbations of the plant preserve stability and enlarge the ultimate bound only gradually (a graceful degradation, rather than a loss of the guarantee). The open-loop plant (a forced double-well Duffing oscillator) is characterized by means of equilibrium and Jacobian analyses. A bifurcation diagram and the largest Lyapunov exponent are presented, which show a chaotic regime (with ${\lambda }_1\approx 0.17$). Numerical experiments indicate that the proposed controller is able to suppress the chaotic motion with a small value of the ultimate bound, and maintain a smooth reference motion with a small and constant RMS error of order ${10}^{-3}$, which is approximately 26 times less than the RMS error obtained with a tuned fixed-gain baseline, and the theoretical dependence of the ultimate bound on the disturbance and the design gains is confirmed by sensitivity sweeps. Full article

Nonlinear electromagnetic wave propagation in complex plasma environments has attracted considerable attention due to its important applications in nonlinear optics, plasma physics, space science, and communication technologies. In the present study, a time-fractional Drinfel’d–Sokolov–Wilson equation (DSWE) is investigated under the influence of electromagnetic wave perturbations. The fractional-order formulation incorporates memory and hereditary effects, providing a more realistic description of wave propagation in nonlinear dispersive media. By employing an appropriate fractional traveling-wave transformation, the governing nonlinear fractional partial differential equation is reduced to a nonlinear ordinary differential equation. Exact solitary wave solutions are subsequently constructed using the ${\displaystyle \frac{G^{\prime }}{G^2}}$-expansion technique. Furthermore, the nonlinear dynamical behavior of the reduced system is examined through phase portraits, bifurcation diagrams, Lyapunov exponents, sensitivity analysis, and multistability investigations. Particular attention is devoted to understanding the emergence of chaotic dynamics induced by electromagnetic wave effects and fractional-order interactions. The obtained results reveal that the fractional-order parameter significantly influences the stability, propagation characteristics, and dynamical evolution of nonlinear wave structures. The coexistence of multiple attractors, transitions between stable states, and chaotic regimes is identified for various parameter configurations. These findings provide deeper insight into the complex dynamics governed by the time-fractional DSWE and contribute to the understanding of nonlinear electromagnetic wave propagation in plasma and other nonlinear dispersive media. Full article

Constructing scalable, non-degenerate, and intensity-tunable high-dimensional hyperchaotic maps is a key challenge for chaotic cryptography. Existing n-dimensional real-valued chaotic systems frequently suffer from dynamical degradation, limited adjustability of Lyapunov exponents, and reduced complexity under high-dimensional settings. To mitigate these drawbacks, this paper proposes an n-dimensional non-degenerate hyperchaotic system (nD-NDHS). Rigorous theoretical derivations demonstrate that all Lyapunov exponents can be continuously adjusted to positive values using a single global control parameter, which guarantees stable hyperchaotic behavior for different tested dimensions. Four evaluation metrics including Lyapunov exponents, correlation dimension, sample entropy, and Kolmogorov entropy are adopted for comprehensive assessment, alongside comparisons with state-of-the-art n-dimensional chaotic maps. Bifurcation diagrams, phase trajectories and Lyapunov exponent spectra are employed to analyze multiple instantiations and validate the generality of the presented framework. A 4D instantiation is physically realized on an STM32 embedded platform, and the corresponding pseudorandom number generator is subjected to the complete NIST SP800-22 and TestU01 test suites. Experimental results reveal that nD-NDHS exhibits compelling chaotic properties and improved dimensional robustness, where all statistical tests are passed to confirm favorable statistical randomness. The proposed model provides a novel complexity-controllable and degradation-resistant hyperchaotic paradigm, which is well adapted to high-dimensional encryption and lightweight hardware-oriented pseudorandom number generation. Full article

Four-dimensional chaotic systems have garnered significant attention due to their complex nonlinear dynamics, high-dimensional complexity, and wide range of applications in science and engineering. This paper proposes and investigates a novel four-dimensional chaotic system in both constant- and variable-order frameworks to reveal the influence of memory effects on its dynamical behavior. The variable-order formulation is established using the Liouville–Caputo fractional derivative, while an efficient numerical scheme based on Lagrange interpolation is developed to approximate the variable-order derivative accurately. A rigorous local stability analysis is first conducted to characterize the equilibrium points and establish their instability and non-hyperbolic nature under the different derivative formulations. The nonlinear dynamics of the proposed system are then comprehensively examined through phase portraits, time series, bifurcation diagrams, and Lyapunov exponent analysis. The results demonstrate that the variable-order model preserves the fundamental topological characteristics of the chaotic attractors while introducing adaptive transient responses and significantly richer dynamical behaviors than the corresponding constant fractional-order model. Furthermore, the proposed system generates previously unreported chaotic attractors and phase-space patterns, enriching the class of known four-dimensional chaotic systems and demonstrating the variable-order framework’s enhanced capability to produce diverse nonlinear phenomena through adaptive memory effects. Full article

Vortex-induced vibration (VIV) is one of the main causes of fatigue failure in subsea pipelines and has recently attracted significant attention from researchers. Previous studies have mainly focused on idealized straight pipes, with limited consideration of gravity-induced slight curvature in free-spanning fluid-conveying pipes. In reality, the deformation configuration of a free-spanning fluid-conveying pipe is not fixed but varies with parameters such as internal flow velocity and tension, which in turn affect its dynamic behavior. A theoretical model, taking into account the axial stretching effect and the gravity-induced initial slight curvature, is developed to predict the VIV responses of free-spanning fluid-conveying pipes. The governing equations are derived based on Hamilton’s principle. The interaction between the external flow and the pipe structure is simulated using the van der Pol equation. By combining the Galerkin method and the Runge–Kutta method, the vibration responses of the pipe are obtained. The accuracy of the proposed model is validated by comparing the predicted VIV response curves and bifurcation diagrams with those reported in previous studies. The initial static deformation of the structure under different tensions and internal velocities is obtained through numerical calculations. It is found that the gravity-induced slight curvature leads to a reduction in the VIV response mode. The static deformation of the pipe decreases with increasing axial tension, while it increases with increasing internal flow velocity. Under the same external flow velocity, the gravity-induced initial deformation reduces the dominant vibration frequency and causes the vibration response to transition from quasi-periodic to periodic motion. Full article

Demand, inventory, and production continuously influence one another in supply chains. Sudden demand changes, inaccurate records, production disruptions, and external factors can create irregular fluctuations. Although demand or sales data are often available, inventory and production information may be incomplete or delayed. This study applies the Extended Kalman Filter (EKF) to jointly estimate inventory, production quantity, and three model parameters using demand as the only measured variable. Its main contribution is a recursive structure that updates the state and parameter estimates with each new measurement. The method is tested on a three-variable supply chain model exhibiting chaotic behavior. Performance is assessed through numerical errors, convergence, time series, phase portraits, and bifurcation diagrams. Robustness to different initial estimates, sensitivity to process- and measurement-noise covariance settings, and performance under chaotic, semi-chaotic, and ordered regimes are also examined. Parameter errors range from 1.5% to 2.4%, while normalized errors for demand, inventory, and production remain below 3%. A larger assumed measurement-noise covariance slows convergence. Under the selected conditions, the closest agreement occurs in the chaotic regime, but this is not interpreted as a general superiority of the EKF. With real-data validation, the framework could support estimation of missing or delayed information, short-term monitoring, early warning, and decision support. Full article

This research examines the behavioral shifts within a discrete-time predator–prey framework, constructed by applying the forward Euler discretization to a continuous model. The system incorporates Smith’s growth dynamics for the prey population alongside a Holling type III functional response to characterize predator behavior. Through bifurcation analysis, it is demonstrated that the interior fixed point undergoes stability loss via Neimark–Sacker and period-doubling transitions, leading to the emergence of quasiperiodic oscillations and chaos. Furthermore, the application of normal-form theory verifies the nondegeneracy of these bifurcations and establishes the direction of the resulting orbits. We use phase portraits, Lyapunov exponents, and bifurcation diagrams to confirm the model’s rich dynamics. These numerical tools demonstrate how the system moves from stable equilibria to more intricate behaviors. The application of partial rank correlation coefficients reveals the most influential parameters governing the system’s asymptotic population levels, providing a global perspective on parameter sensitivity. The Ott–Grebogi–Yorke (OGY) chaos control strategy is employed to suppress unwanted bifurcations and stabilize chaotic oscillations within the system. These results underscore the role of nonlinear interactions and discrete-time frameworks in precipitating unpredictable population fluctuations while simultaneously offering a suite of mechanisms for enhancing the stability of ecological networks. Full article

Modeling and stability analysis of a PV–energy storage AC/DC integrated three-port grid-connected power electronic device is investigated in this paper for low-voltage single-phase renewable energy applications. The device consists of a PV Boost converter port, a battery-side bidirectional DC-DC converter port, and a single-phase full-bridge grid-connected inverter. To analyze the coupling-induced stability characteristics of this multi-converter system, mathematical models of the PV array, battery, and AC/DC integrated device are established. Considering the periodic time-varying nature introduced by the single-phase grid voltage, a phase-angle-based simplified discrete model is developed to transform the system into a discrete model evaluated at a fixed grid-voltage phase angle. Based on this model, eigenvalue sensitivity, eigenvalue trajectories, and bifurcation diagrams are used to identify the influence of key control parameters on system stability. The results show that excessive proportional gains in the inverter current loop and energy storage control loop reduce the stability margin and may lead to period-doubling bifurcation, Hopf bifurcation, or unstable grid current operation. The period-doubling and Hopf stability boundaries are identified at k_p4 ≈ 1.40 and k_p2 ≈ 1.75, respectively. In simulation, the grid-current THD increases from 2.07% to 3.60% as k_p4 rises from 1.3 to 1.6 and from 2.07% to 2.27% as k_p2 rises from 1.7 to 1.8. MATLAB/Simulink simulations and hardware-in-the-loop experiments further verify that the identified stability boundaries are consistent with the degradation of grid current quality and the increase in total harmonic distortion. The proposed modeling and analysis method provides a reference for parameter tuning and stable operation of single-phase PV–energy storage three-port grid-connected devices. Full article

During the operation of full-ceramic bearings, defects such as cracks and spalls inevitably develop on the bearing balls. These defects reduce bearing service life and compromise the stable operation of mechanical systems. To address this issue, a 12-degree-of-freedom (DOF) dynamic model of a full-ceramic bearing-rotor system (BRS) is established, considering ball crack and spalling defects. The model incorporates variations in equivalent stiffness and contact forces induced by these two defect types. Subsequently, the proposed model is solved by the Newmark–β method. Bifurcation diagrams, time-domain waveforms, and frequency spectra are employed to investigate the system’s dynamic responses. In the frequency-domain analysis, particular attention is paid to characteristic frequency components, including the rotational frequency f_s, the ball spin frequency f_BSF, their harmonics, and combination frequencies. Finally, an experimental test platform is constructed to validate the accuracy of the developed model. The results indicate that crack and spalling defects exert distinctly different effects on the dynamic behavior of the system. Defect width has a significant quantitative influence on the vibration response. Under various defect conditions, the prediction errors of the developed model remain within an acceptable range, with the maximum relative error of 9.05%. The developed model offers a theoretical foundation for analyzing bearing dynamics and supporting fault diagnosis applications. Full article

In this study, we develop a novel variable-order fractional extension of an eight-dimensional (8D) hyperchaotic differential equation system modeled via the Liouville–Caputo operator. Moving beyond constant fractional-order models, our system implements time-variable orders, which enable its historical memory structure to evolve dynamically over time. To numerically approximate the trajectories of this complex 8D system, a second-order Lagrange numerical integration approach is formulated. An extensive dynamic analysis explores the behavior of this variable-order framework under two distinct configurations: a slowly periodic memory function and a smooth, monotonic hyperbolic tangent function. Topological complexity and multidimensional chaos are characterized using parameter-dependent bifurcation diagrams, phase portraits, Kaplan–Yorke fractal dimensions, and Kolmogorov–Sinai metric entropy. Numerical results show that both variable-order configurations display robust hyperchaotic dynamics characterized by four positive Lyapunov exponents. Crucially, the proposed variable-order extension enhances the phase space footprint of the baseline system, achieving a maximum Kaplan–Yorke dimension of 7.100, thereby offering excellent topological density for secure cryptographic applications. Full article

Sleep regulation is generated by interacting nonlinear feedback loops involving homeostatic pressure, circadian forcing, mutual inhibition, and neuromodulatory mechanisms. This work presents a nonlinear-dynamical interpretation of sleep with emphasis on bifurcation structure, Lyapunov stability, deterministic intermittency, and hyperchaotic fragmentation. After summarizing representative sleep models and the literature on sleep-stage dynamics, we combine bifurcation and Lyapunov exponent analyses of representative sleep-population models with a four-dimensional model in which cortical activation, sleep-promoting activity, homeostatic sleep drive, and the circadian pacemaker form a slow–fast feedback system. The model maps coordinate-dependent feedback, separated homeostatic accumulation and clearance, external sleep-debt forcing, refractory dynamics, and on–off intermittency onto physiological sleep–wake processes. The baseline diagrams identify routes from regular oscillations to period-doubling, chaotic bands, periodic windows, and positive Lyapunov regions, while the new Lyapunov maps of the four-dimensional model identify parameter regions where the largest and second-largest Lyapunov exponents are positive, supporting the hyperchaotic classification. The formulation also shows how chronic sleep debt and disease-associated or stress-related insomnia can be represented as shifts in homeostatic drive, inhibitory gain, and circadian coupling. Bifurcation theory therefore provides a useful framework for interpreting irregular sleep transitions and for inspiring future sleep-technology and digital twin applications. Full article

This study develops a nonlinear dynamic model of a human-inspired electromechanical arm system subjected to high loads. The proposed simplified representation preserves essential nonlinear dynamics using a reduced number of generalized coordinates. The model is represented by an electromechanical analog comprising a DC motor, a transmission system, and a multi-degree-of-freedom mechanical structure. The formulation is based on Lagrangian mechanics and accounts for inertia, damping, stiffness, and nonlinear kinematic coupling induced by joint misalignment. The numerical results were assessed using a consistency-based verification approach with several independent nonlinear analysis tools. The Lyapunov exponent was used in conjunction with bifurcation diagrams, Poincaré maps, and FFT spectra to identify the transition from stable operation to chaotic behavior as the external load increased. The results reveal a progressive transition from periodic motion to quasi-periodic oscillations and chaotic regimes, with fully developed chaotic behavior emerging for loads exceeding approximately 35 kg. Analysis of the Lyapunov exponent supports this interpretation, indicating stable, quasi-critical, or chaotic regimes depending on the sign of λ_max. The concordance among these independent indicators provides numerical verification of the observed stability transitions. The control gain significantly influences energy dissipation and system stability. The proposed model provides a reduced-order framework for studying nonlinear stability phenomena in human-inspired electromechanical systems. Potential applications involve rehabilitation devices and safety studies of human–robot interactions. Full article

This paper investigates the dynamical behavior of a three-species commensalism–amensalism system with Beddington–DeAngelis functional response. The model describes a novel tripartite interaction: a neutral–commensal species (e.g., sea anemone) simultaneously engages in a commensal relationship with a commensal–amensal species (e.g., clownfish) and an indirect nutritional coupling with a neutral–amensal species (e.g., crustacean), while the commensal–amensal and neutral–amensal species interact amensalistically. This paper makes three principal contributions. First, by constructing a Volterra-type Lyapunov function, we rigorously prove the global asymptotic stability of the unique positive equilibrium of the $(x,z)$-subsystem in the positive quadrant, and further establish the global asymptotic stability of both the amensal-free equilibrium $E_3$ and the coexistence equilibrium $E_4$ in the positive octant. Second, selecting the commensal benefit coefficient c as the bifurcation parameter, we present a complete and rigorous proof of a transcritical bifurcation. Third, we provide systematic Maple-based numerical verification: the bifurcation diagram exhibits excellent agreement with the theoretical curve, and logarithmic-scale plots confirm exponential convergence rates. Ecologically, our results reveal a sharp threshold phenomenon: when the commensal benefit coefficient lies below a critical value $c^{\ast }$, the commensal species inevitably goes extinct and the system collapses to a two-species state; when c exceeds $c^{\ast }$, the commensal species can invade and achieve stable three-species coexistence. The explicit formula for this threshold provides a quantitative criterion for determining the minimum mutualistic strength required for persistence in conservation contexts. The results obtained in this paper substantially extend the existing theoretical understanding of three-species commensalism–amensalism systems. Full article

The study investigates the nonlinear dynamics of a harmonically excited Duffing oscillator coupled with an unforced two-degrees-of-freedom nonlinear energy sink. The external excitation is applied only to the primary oscillator; meanwhile, the NES response is induced through nonlinear internal coupling. The governing nonlinear ordinary differential equations are analyzed using the proposed non-perturbation approach, which does not rely on small-parameter assumptions or Taylor-series expansions. The formulation is used to obtain amplitude-dependent equivalent linear representations and analytical approximations of the coupled system. The analytical results are compared with direct numerical simulations, showing overall agreement with the full nonlinear model. The stability of the steady-state solutions is examined under variations of the main system parameters. The results indicate that the nonlinear coupling and stiffness parameters significantly affect the response amplitudes, stability characteristics, and overall dynamical behavior. Additional analyses using bifurcation diagrams, Lyapunov exponents, Poincaré maps, and basins of attraction reveal transitions between periodic, quasi-periodic, and chaotic regimes, as well as the presence of multi-stability and sensitivity to initial conditions. The proposed framework provides a useful analytical tool in studying the dynamics and stability of nonlinear oscillatory systems over a wide range of operating conditions. Full article

This study examines a discrete fractional-order predator–prey system incorporating a Holling type-III functional response. The Caputo fractional derivative is employed because it naturally incorporates memory and hereditary effects while preserving biologically meaningful initial conditions. The system is formulated from a biologically relevant continuous fractional-order framework through the application of the piecewise constant argument approach, enabling an analysis of how memory-dependent effects and discrete dynamics influence predator–prey interactions. The existence and local stability of fixed points are determined by using the Jacobian matrix and eigenvalue conditions. The bifurcation of the positive fixed point is analyzed by using the center manifold and normal form methods. Numerical simulations, including bifurcation diagrams, phase portraits, and maximum Lyapunov exponent plots, confirm our analytical results and reveal periodic, quasiperiodic, and chaotic behavior. The findings of this study reveal that the combined influence of memory-dependent dynamics, nonlinear predator–prey interactions, and discrete-time effects can generate rich and complicated behaviors in fractional-order predator-prey systems. Full article