Search Results (472)

Path planning for autonomous robots is a key problem area, particularly when faced with complicated, dynamic, or uncertain environments. Even though traditional techniques (grid-based, graph-based, sampling, and optimization-based) have already been developed to solve this problem, there are notable limitations to scalability, adaptability, and responsiveness with these methods. In this paper, we explore an alternative approach based on chaotic dynamical systems, specifically chaotic attractors like those produced by the Lorenz and Rössler systems. Chaotic systems are defined by several properties that could be leveraged: non-linearity, sensitivity to initial conditions, and dense coverage of the state space are three notable properties that could be used to generate trajectories that are organized, yet ultimately unpredictable. By applying numerical integration (Runge–Kutta) directly to robot motion through MATLAB R2025b simulations, chaotic states support more effective exploration, better obstacle avoidance, and more robust navigation in dynamic or adversarial environments. The paper also examines whether chaotic path planning can be applied in multi-robot systems through state coupled robots that emerge coordinated behavior while maintaining autonomous movement. This paper is a framework for chaos theory supporting adaptable, robust navigating behaviors for purposes such as autonomous vehicles, swarm robotics, and search and rescue and surveillance applications. Full article

This study investigates the nonlinear dynamic response and chaos evolution of a shape memory alloy hybrid composite (SMAHC) actuator beam under coupled thermal, harmonic, and aerodynamic excitations. A reduced-order nonlinear dynamic model was developed by combining Euler–Bernoulli beam theory, von Karman geometric nonlinearity, the Brinson SMA constitutive relation, and first-order piston-theory aerodynamics. The governing equations were derived from Hamilton’s principle, discretized by the weighted residual method, and solved using the Newmark-beta algorithm. Chaotic evolution was quantified using a largest Lyapunov exponent-based chaos intensity indicator rather than the exact Kolmogorov–Sinai entropy. The reduced-order model was compared with ABAQUS finite element simulations under representative coupled aerodynamic and harmonic loading. The MATLAB prediction and ABAQUS response gave a dominant frequency of approximately 9.50 Hz, close to the prescribed excitation frequency of 9.55 Hz, with peak displacement amplitudes of approximately 0.0285 mm and 0.0324 mm, respectively. A supplementary ABAQUS modal-frequency separation check supported the use of the two-mode reduced-order model for the dominant low-frequency response, while also clarifying its limitation for high-dimensional chaotic modal interactions. The parametric results showed that an increasing excitation amplitude and aerodynamic load promoted frequency broadening and chaotic transitions. The Lyapunov-based indicator rose near γ = 65 under λ* = 100 and near λ* = 328 under γ = 30. Temperature-dependent SMA recovery stress further shifted the transition threshold by modifying the effective stiffness and internal restoring action of the beam. These results provide a reduced-order framework for interpreting nonlinear response transitions in SMAHC actuator beams in thermo-aeroelastic environments. Full article

Accurate performance prediction for large offshore wind turbines requires a principled treatment of uncertainty in both the wind resource and the rotor design parameters. In the present work, we develop a surrogate-based, multi-level uncertainty quantification (UQ) framework coupling a physics-based Blade Element Momentum (BEM) solver with a spectral Polynomial Chaos Expansion (PCE) surrogate that replaces the expensive Monte Carlo loop and apply it to the IEA 15 MW offshore reference wind turbine. The framework is completed by Sobol variance-based global sensitivity analysis. The contribution is methodological rather than algorithmic: although each individual ingredient (PCE, Sobol, BEM, and Jensen) is well established, their joint deployment in a single, internally consistent, end-to-end probabilistic workflow that simultaneously delivers (i) aerodynamic–structural UQ with analytical Sobol ranking, (ii) a like-for-like cross-comparison of three reference turbines, (iii) a quantitative leading-edge icing degradation study, and (iv) a farm-level wake-steering optimization on the same IEA 15 MW reference rotor yields a unified probabilistic envelope from which manufacturing tolerances, cold-climate investment thresholds, and farm-layout/control trade-offs can be read off consistently. Five input parameters are treated as random variables: hub-height wind speed (Weibull, k = 2.2, c = 9.8 m/s), air density, blade chord length, twist angle, and rotor speed. A degree-4 sparse PCE is built by non-intrusive spectral projection using N = 5000 Sobol quasi-random realizations, which allows the Sobol indices to be recovered analytically from the expansion coefficients at essentially no extra cost. Three parallel engineering studies complement the core UQ analysis: (A) a head-to-head comparison of the NREL 5 MW, DTU 10 MW, and IEA 15 MW reference turbines; (B) a quantitative assessment of leading-edge ice accretion at four severity levels; and (C) a Jensen-based wake optimization for a 25-turbine offshore array with static wake steering. The main results are as follows: the turbine reaches C_p,max = 0.480 at λ_opt = 8.51, and an annual energy production (AEP) of 71,261 MWh/year (PCE: 70,840 ± 2,140 MWh/year, 95% CI). Wind speed emerges as the dominant driver of C_p variance (S₁ = 0.412), followed by blade twist (0.198) and chord (0.143). Severe icing (30 kg/m) reduces C_p by 18.2% and increases the blade-root Damage Equivalent Load (DEL) by 18.5%. For the array, the optimal spacing (s_x = 8D, s_y = 6D) gives a farm efficiency of 89.6% and 1296 GWh/year, and a 15° wake-steering offset adds a further +3.2% to farm AEP. Compared with plain Monte Carlo, the sparse PCE delivers the same statistics with about 36% fewer model evaluations and a relative error below 0.8%. Full article

This work investigates, for the first time, nonlinear wave dynamics and chaos in nanocomposite micropipes conveying a viscous fluid, reinforced with graphene origami (GOr), and subjected to thermal loading. It extends the previous study by considering the influence of a transverse load and fluid viscosity, both of which were ignored previously. The Painlevé integrability of the governing equation is tested using the Ablowitz–Ramani–Segur (ARS) algorithm. Our findings prove the non-integrability of the governing equation, motivating a qualitative dynamical approach. Bifurcation theory is applied to multiple possible forms of the transverse load. In the absence of a transverse load, neither periodic nor solitary axial wave displacements exist. This is guaranteed by applying Bendixson’s criterion and confirmed through phase portraits. However, with a specific form of the transverse load, bifurcation analysis analytically provides the existence conditions for periodic, super-periodic, and solitary axial displacement waves. Furthermore, it is shown that kink and anti-kink solutions are absent. Explicit exact solutions are constructed in terms of elliptic functions, and their consistency and validity are verified through orbital degeneracy. The key material parameters’ impacts—GOr weight fraction, temperature change, hydrogen coverage, and shear layer stiffness—on the wave profiles are inspected numerically and eludicated physically. When an additional periodic transverse load is inserted, the system manifests quasi-periodic behavior at frequencies with small loads, transitioning to chaotic motion as the frequency grows. Both Lyapunov exponents and a Poincaré section are utilized to confirm this chaotic behavior. Our findings show the impact of fluid viscosity and the transverse load structure are significant in GOr-reinforced microtubes and highlight their relevance for advanced fluid-conveying systems. Full article

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. Full article

This study investigates the ($3+1$)-dimensional extended Kadomtsev–Petviashvili equation via traveling-wave phase-space geometry. The equation is reduced to a planar Hamiltonian system with cubic nonlinearity, whose conserved energy partitions the phase space into periodic orbits, separatrices, and unbounded trajectories. Closed-form profiles for the gradient variable $\phi =U_{\xi }$ are obtained through separation of variables; the corresponding field U is recovered by quadrature and must satisfy a zero-mean condition for periodic reconstruction. In particular, for $h_1>0$, the reconstructed field exhibits kink/antikink-type rather than localized-pulse behavior. Under weak periodic forcing, an explicit Melnikov amplitude factor is derived. Its exponential decay with the forcing frequency implies that the leading-order separatrix splitting distance $\mu A\left(\omega \right)$ becomes exponentially small at high frequency, while the simple-zero condition still predicts transverse intersections of stable and unstable manifolds and the onset of horseshoe chaos. Applying the complete discriminant method yields eight distinct solution families—hyperbolic, trigonometric, rational, and Jacobi elliptic—each associated with a unique orbital topology. These results enrich both the dynamical theory and the exact solution framework of higher-dimensional nonlinear evolution equations. Full article

The article examines forecasts of electromobility development across seven European countries over a ten-year horizon (until 2035). The introduction provides a characterization and statistical analysis of the electromobility market within the framework of sustainable development. The analysis includes both leading electromobility markets and lower-income countries with relatively small electromobility sectors. First, forecasts for the total number of registered passenger vehicles of all drive types will be generated for each country, followed by forecasts for the number of passenger electric vehicles (Battery Electric Vehicle (BEV) and Plug-in Hybrid Electric Vehicle (PHEV)). Based on this data, the degree of electromobility development—defined as the percentage of passenger electric vehicles among all registered passenger vehicles through 2035—will be established. The forecasts will be conducted using an artificial intelligence model, a deterministic chaos theory model and selected trend extrapolation methods. The multi-stage approach applied to the problem, together with the use of single-type models within ensembles and the model selection procedure, constitutes an original, proprietary solution. To the author’s knowledge, a similar approach has not been reported for a forecasting task in the context of electromobility. Three ensemble projections will be presented: low, middle, and high. The article concludes with findings regarding the implementation of European Union (EU) sustainable development goals, specifically the degree of passenger vehicle electrification. Full article

As an important option for energy storage projects, pumping stations can also generate electricity when the upstream has surplus water and the pump system operates as a turbine (PAT mode). When it switches from pump mode to PAT mode, the pump operation state changes significantly. This study adopts a numerical simulation to investigate the flow characteristics, time-frequency domain performance and chaotic features of pressure pulsation in a vertical mixed-flow pump device when it operates in different PAT modes. The results show that, when the pump operates in PAT mode, the flow in the straight passage remains smooth, but it deteriorates in the elbow-shaped draft tube, such as developing a spiral stream in the straight section, a disordered stream in the elbow section, and vortexes and flow separation at the beginning of the diffuser section, but it gradually becomes smooth after passing through the diffuser section. Under low-head PAT conditions, circumferential circulation cross flow occurs at the impeller inlet, reducing energy conversion efficiency. Under all PAT conditions, the flow on the blade surface near the hub is stable, but obvious vortexes happen near the shroud. As the head increases, the small-scale vortexes disappear on the mid-blade surface, and the flow becomes smoother on the blade surface near the shroud of the impeller. Except at the impeller outlet, pressure pulsation of the monitoring probes exhibits clear periodicity, with dominant frequencies corresponding to the rotational frequency, and its amplitudes decreasing from shroud to hub. Pressure pulsation under all PAT conditions is chaotic, and phase trajectories exhibit ring-shaped structures consisting of the ring circle and the ring surface. Differences in the circle spacing, size, and spatial position of the ring circle phase locus and ring surface phase locus are observed, and these variations are closely related to the PAT conditions. A correlative relationship exists between the chaotic correlation dimension and flow performance, which is of great significance for the condition monitoring and fault diagnosis of pump units. These findings not only enrich the theoretical research on the PAT mode of pumps, but also provide a reference for similar engineering applications and offer new insights into condition monitoring of hydraulic machinery. Full article

Technological progress reduces carbon emissions by promoting energy structure optimization while fostering new industries and improving efficiency, thus achieving a win–win situation for economic growth and low-carbon development. From the perspective of mechanism analysis, this paper constructs a new dynamic system model of technological progress–carbon emissions–economic growth–energy structure based on the interdependent and mutually restrictive causal relationships among technological progress, carbon emissions, economic growth and energy structure within an economic period. The dynamical behaviors of the system and its subsystems are analyzed using Lyapunov exponents, bifurcation diagrams, equilibrium point stability theory and other methods. Numerical simulations show that the system parameter $a_2$ (the driving coefficient of economic growth on carbon emissions) determines the threshold of state transition. With the increase in $a_2$, the system exhibits a clear evolutionary path from stable equilibrium to periodic state and then to chaotic state. The system enters chaos when $a_2$ falls within the interval [0.741, 0.79]. Model parameters are estimated based on real data, the evolutionary relationships of technological progress, carbon emissions, energy structure and economic growth over time are presented, and the impacts of different regulation strategies on carbon emission reduction and economic growth are analyzed. Full article

Particle swarm optimization (PSO) is an optimizing method that is based on the theory of swarm intelligence. PSO is an effective algorithm that is used to search in a parallel manner compared to other methods. However, PSO has a tendency towards local optima when tackling complex multimodal optimization problems. It also has the disadvantages of slow convergence process and poor stability in the latter evolutionary period. In view of these demerits, a hybrid PSO method based on chaotic opposition-based initialization and an adaptive learning strategy is presented in this work (abbreviated as ACMPSO). First, the chaos initialization and opposition-based learning (OBL) are employed to produce high-quality initial particles in the feasible region, which is able to improve the quality of the initial solutions. Second, the logistic mapping embedded inertia weight is formulated to better trade off the global and local search process. Third, the global optimal particle is regulated by an exclusive velocity and position updating strategy whereas the rest particles are adjusted by the standard updating mechanism so as to prevent particles from premature convergence. Furthermore, an adaptive position update paradigm is developed to finely regulate the global exploration and local exploitation. Finally, conducted experiments on CEC’13 and CEC’22 reveal that the proposed ACMPSO outperforms several other advanced PSO variants regarding their convergence rate and accuracy. Alternatively, to further illustrate the effect of ACMPSO, we have applied it to two real-world problems, and simulation results ascertain its effectiveness and robustness. Full article

Thisarticle develops and analyzes a fractional-order Leslie–Gower prey–predator–parasite system incorporating two discrete delays and nonlocal spatial diffusion. The model’s central novelty lies in the simultaneous integration of three biologically realistic features that have not previously been combined: (i) fractional-order memory effects via a Caputo derivative of order $\alpha \in (0,1]$, (ii) two distinct biological delays—an infection transmission delay ${\tau }_1$ and a predator handling delay ${\tau }_2$—and (iii) nonlocal spatial dispersal modeled through fractional Laplacian operators ${(-\Delta )}^{\gamma /2}$. This triple integration enables the model to capture long-range temporal memory, delayed biological responses, and nonlocal spatial interactions simultaneously, offering insights into dynamics that are challenging to capture with classical integer-order or single-delay formulations. The fractional Laplacian generalizes classical diffusion by allowing long-range dispersal events (Lévy flights), where individuals can occasionally move over large distances with heavy-tailed step-size distributions—a phenomenon observed in many animal movement patterns but absent from standard diffusion models. We provide rigorous proofs of solution existence, uniqueness, non-negativity, and boundedness in both temporal and spatiotemporal settings. Local asymptotic stability conditions are derived for all feasible equilibrium states via characteristic equation analysis. The coexistence equilibrium undergoes a Hopf bifurcation when either delay crosses a critical threshold, with fractional order $\alpha$ modulating the bifurcation point and post-bifurcation oscillation frequency. A Lyapunov functional demonstrates global asymptotic stability of the infection-free equilibrium under biologically interpretable conditions. Turing instability analysis reveals conditions for spontaneous pattern formation, with the fractional exponent $\gamma$ controlling pattern wavelength and correlation length. Numerical simulations validate theoretical predictions, including spatial patterns, traveling waves, and chaos. To bridge theory with potential applications, we outline a statistical framework for parameter estimation and uncertainty quantification, suggesting that $\beta$, $\alpha$, and ${\tau }_1$ may be priority targets for parameter estimation. Full article

Wireless sensor networks (WSNs) have increasing demand on lightweight, efficient, and secure encryption techniques for devices with limited resources, since traditional algorithms require high computation which make them impractical. This preliminary study presents an encryption algorithm based on chaos designed for transmitting short data, using the Lorenz system and Euler’s method for computation. It is combined with a synchronization model based on data array. It inserts iteration parameters within the ciphertext to ensure consistent key reproduction while decrypting. Within the broader context of e-health data streams, encryption efficiency is critical: continuous ECG signals generate large volumes of data that challenge real-time secure transmission, whereas individual blood pressure readings are far smaller and lightweight. While this work delimits its scope to short, low-power transmissions, simulations and hardware implementation on an nRF chip using the Enhanced ShockBurst (ESB) protocol demonstrated efficiency, with the lowest encryption speed of 0.154 ms for a 1-byte payload. Security analysis using the NIST Statistical Test Suite confirmed high statistical randomness of the generated keystream, and theoretical key-space analysis supports robustness. By focusing on short-stream encryption in preliminary form, the scheme contributes toward inclusive secure communication technologies for resource-constrained IoT healthcare systems and diverse user populations. Full article

Classical homogenization theory treats critical exponents as universal quantities depending only on spatial dimension, but recent evidence shows that this assumption fails for continuum composites once the mechanism of randomness generation is taken into account. We synthesize three complementary frameworks—structural approximation, structural sums, and self-similar renormalization—to develop a unified geometric theory of criticality in random composites. Dilute-regime expansions for the effective conductivity and shear modulus are expressed in terms of structural sums whose ensemble statistics depend sensitively on the randomness protocol. To bridge the dilute and critical regimes, we employ self-similar factor approximants, iterated-root approximants, additive approximants, and renormalization schemes based on minimal-difference and minimal-sensitivity conditions, combined with Borel summation. For maximally disordered protocols $\mathcal{P}\left(\tau \right)$, the conductivity index s and the elasticity index S fall within comparable numerical ranges, indicating a shared geometric origin and spectral response to the continuous breaking of translational symmetry. A regular periodic arrangement of inclusions ($\tau =0$) possesses full discrete translational symmetry; as a stochastic protocol $\mathcal{P}\left(\tau \right)$ is applied ($\tau$ increases), this symmetry is gradually degraded until statistical chaos is reached. For instance, the parameter $\tau$ can be considered as a time of stirring. During this evolution, the system traverses a continuous spectrum of critical indices, $s=s\left[\mathcal{P}\right(\tau \left)\right],$ which encodes the geometric and topological memory of the initial ordered state. It is established that the classical “universality” of percolation corresponds to a fixed point $\tau$ within a broader manifold of protocol-dependent critical behaviors. The framework developed here provides a coherent basis for inverse design, diagnostics, and classification of random composites by their disorder history, offering a geometric alternative to the universality paradigm. Full article

While general network dynamics have been extensively modeled using stochastic methods, the emergence of dense Internet of Things (IoT) ecosystems demands a more specialized analytical framework. IoT environments are characterized by extreme non-linearity and sensitivity to initial conditions, where traditional models often fail to account for chaotic latency and packet loss. This paper introduces a specialized approach that integrates Chaos Theory with the innovative paradigm of Vibe Coding—an AI-assisted development and analysis methodology that allows for the ‘encoding’ and interpretation of the dynamic ‘vibe’ or signature of network fluctuations in real-time. By categorizing network behavior into four distinct scenarios (quiescent, perturbed, attacked, and perturbed–Attacked), the proposed framework utilizes deep learning to transform chaotic signals into actionable intelligence. Our findings demonstrate that this specialized synergy between chaos analysis and Vibe Coding provides superior classification of adversarial threats, such as DoS and injection attacks, fostering intelligent native security for next-generation IoT infrastructures. Full article

Different methodologies have been developed for the analysis and study of dynamical systems, including both theoretical models and natural systems. Examples span a wide range of applications, such as astronomy, financial and economic time series, biophysical systems, physiological phenomena, and Earth sciences, including seismicity and climatic processes. The study of these complex systems is commonly based on the analysis of the signals they generate, using mathematical tools to extract relevant information. A broad spectrum of mathematical disciplines converges in this context, including stochastic, probability and statistical theory, entropic and informational measures, fractal and multifractal analysis, natural time analysis, modeling of non-linearity and recurrence methods, generalized entropies, non-extensive systems, machine learning, and high-dimensional and multivariate complexity. Research in this area is largely focused on the characterization of complex systems, providing indicators of determinism or stochasticity, distinguishing between regularity, chaos, and noise, and identifying topological as well as disorder-regularity features. In addition, short- and long-term forecasting, together with the identification of short- and long-range correlations, play a central role in such characterization. To address these objectives, numerous mathematical tools have been developed for the analysis of time series and point processes, each designed to capture specific signal properties. In this work, many of the most important tools used in time series analysis are compiled and reviewed, highlighting their main characteristics and the different types of complex systems to which they have been applied. Full article