Search Results (44)

Background: Current cosmological fits typically assume a direct relation between cosmic time (t) and the scale factor ($a\left(t\right)$), yet this ansatz remains largely untested across diverse observations. Objectives: We (i) test whether a single power-law scaling ($a\left(t\right)\propto t^{\alpha }$) can reproduce late- and early-time cosmological data and (ii) explore whether a dynamically evolving ($\alpha \left(t\right)$), modeled as a scalar–tensor field, naturally induces directional asymmetry in cosmic evolution. Methods: We fit a constant-$\alpha$ model to four independent datasets: 1701 Pantheon+SH0ES supernovae, 162 gamma-ray bursts, 32 cosmic chronometers, and the Planck 2018 TT spectrum (2507 points). The CMB angular spectrum is mapped onto a logarithmic distance-like scale ($\mu ={log}_{10}{D}_{\ell }$), allowing for unified likelihood analysis. Each dataset yields slightly different preferred values for $H_0$ and $\alpha$; therefore, we also perform a global combined fit. For scalar–tensor dynamics, we integrate $\alpha \left(t\right)$ under three potentials—quadratic, cosine, and parity breaking (${\alpha }^{3}sin\alpha$)—and quantify directionality via forward/backward evolution and Lyapunov exponents. Results: (1) The constant-$\alpha$ model achieves good fits across all datasets. In combined analysis, it yields $H_0\simeq 70\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ and $\alpha \simeq 1.06$, outperforming $\Lambda$CDM globally ($\Delta \mathrm{AIC}\simeq 401254$), though $\Lambda$CDM remains favored for some low-redshift chronometer data. High-redshift GRB and CMB data drive the improved fit. Numerical likelihood evaluations are approximately three times faster than for $\Lambda$CDM. (2) Dynamical $\alpha \left(t\right)$ models exhibit time-directional behavior: under asymmetric potentials, forward evolution displays finite Lyapunov exponents (${\lambda }_L\sim {10}^{-3}$), while backward trajectories remain confined (${\lambda }_L<0$), realizing classical arrow-of-time emergence without entropy or quantum input. Limitations: This study addresses only homogeneous background evolution; perturbations and physical derivations of potentials remain open questions. Conclusions: The time-scaling approach offers a computationally efficient control scenario in cosmological model testing. Scalar–tensor extensions naturally introduce classical time asymmetry that is numerically accessible and observationally testable within current datasets. Code and full data are available. Full article

This paper introduces and analyzes a novel fractional-order Ananthakrishna model. The stability of its equilibrium points is first investigated using fractional-order stability criteria, particularly in regions where the corresponding integer-order model exhibits instability. A linear finite difference scheme is then developed, incorporating an accelerated L1 method for the fractional derivative. This enables a detailed exploration of the model’s dynamic behavior in both the time domain and phase plane. Numerical simulations, including Lyapunov exponents, bifurcation diagrams, phase and time diagrams, demonstrate that the fractional model exhibits stable and periodic behaviors across various fractional orders. Notably, as the fractional order approaches a critical threshold, the time required to reach stability increases significantly, highlighting complex stability-transition dynamics. The computational efficiency of the proposed scheme is also validated, showing linear CPU time scaling with respect to the number of time steps, compared to the nearly quadratic growth of the classical L1 and Grünwald-Letnikow schemes, making it more suitable for long-term simulations of complex fractional-order models. Finally, four types of stress-time curves are simulated based on the fractional Ananthakrishna model, corresponding to both stable and unstable domains, effectively capturing and interpreting experimentally observed repeated yielding phenomena. Full article

This paper introduces a novel model-free nonsingular fixed-time sliding mode control (MF-NFxTSMC) strategy for precise trajectory tracking in robot arm systems. Unlike conventional sliding mode control (SMC) approaches that require accurate dynamic models, the proposed method leverages the time delay estimation (TDE) approach to effectively estimate system dynamics and external disturbances in real-time, enabling a fully model-free control solution. This significantly enhances its practicality in real-world scenarios where obtaining precise models is challenging or infeasible. A significant innovation of this work lies in designing a novel fixed-time control framework that achieves faster convergence than traditional fixed-time methods. Building on this, a novel MF-NFxTSMC law is developed, featuring a novel singularity-free fixed-time sliding surface (SF-FxTSS) and a novel fixed-time reaching law (FxTRL). The proposed SF-FxTSS incorporates a dynamic proportional term and an adaptive exponent, ensuring rapid convergence and robust tracking. Notably, its smooth transition between nonlinear and linear dynamics eliminates the singularities often encountered in terminal and fixed-time sliding mode surfaces. Additionally, the designed FxTRL effectively suppresses chattering while guaranteeing fixed-time convergence, leading to smoother control actions and reduced mechanical stress on the robotic hardware. The fixed-time stability of the proposed method is rigorously proven using the Lyapunov theory. Numerical simulations on the SAMSUNG FARA AT2 robotic platform demonstrate the superior performance of the proposed method in terms of tracking accuracy, convergence speed, and control smoothness compared to existing strategies, including conventional SMC, finite-time SMC, approximate fixed-time SMC, and global fixed-time nonsingular terminal SMC (NTSMC). Overall, this approach offers compelling advantages, i.e., model-free implementation, fixed-time convergence, singularity avoidance, and reduced chattering, making it a practical and scalable solution for high-performance control in uncertain robotic systems. Full article

In this article, we introduce a novel approach to numerical integration based on a modified composite diagonal (CD) method, which is a variation of the semi-implicit Euler–Cromer method. This approach enables the finite-difference scheme to maintain the dynamic regime of the solution while adjusting the integration time step. This makes it possible to implement variable-step integration. We present a variable-step MCD (VS-MCD) version with a simple and stable Hairer step size controller. We show that the VS-MCD method is capable of changing the dynamics of the solution by changing the symmetry coefficient (reflecting the ratio between two internal steps within the composition step), which is useful for tuning the dynamics of the obtained discrete model, with no influence of the appropriate step size. We illustrate the practical application of the developed method by constructing a direct chaotic communication system based on the Sprott Case S chaotic oscillator, demonstrating high values in the largest Lyapunov exponent (LLE). The tolerance parameter of the step size controller is used as the modulation parameter to insert a message into the chaotic time series. Through numerical experiments, we show that the proposed modulation scheme has competitive robustness to noise and return map attacks in comparison with those of modulation methods based on fixed-step solvers. It can also be combined with them to achieve an extended key space. Full article

In this paper, an incommensurate fractional-order chaotic system is established based on Chua’s system. Combining fractional-order calculus theory and the Adomian algorithm, the dynamic phenomena of the incommensurate system caused by different fractional orders are studied. Meanwhile, the incommensurate system parameters and initial values are used as variables to study the dynamic characteristics of the incommensurate system. It is found that there are rich coexistence bifurcation diagrams and coexistence Lyapunov exponent spectra which are further verified with the phase diagrams. Moreover, a special dynamic phenomenon, such as chaotic degenerate dynamic behavior, is found in the incommensurate system. Secondly, for the feasibility of practical application, the equivalent analog circuit of incommensurate system is realized according to fractional-order time–frequency frequency domain algorithm. Finally, in order to overcome the limitation that the convergence time of the finite-time synchronization control scheme depends on the initial value, a fixed-time synchronization control scheme is proposed in the selection of synchronization control scheme. The rationality of this scheme is proved by theoretical analysis and numerical simulation. Full article

We study the chaotic motion of a semi-classical optomechanical system coupled to a non-Markovian environment with a finite correlation time. By studying the emergence of chaos using the Lyapunov exponent with the changing non-Markovian parameter, we show that the non-Markovian environment can significantly enhance chaos. It is observed that a non-Markovian environment characterized by the Ornstein–Uhlenbeck type noise can modify the generation of chaos with different environmental memory times. As a comparison, the crossover properties from Markov to non-Markovian regimes are also discussed. Our findings indicate that the quantum memory effects on the onset of chaos may become a useful property to be investigated in quantum manipulations and control. Full article

Turbulence has associated chaotic features. In the past couple of decades, there has been growing interest in the study of these features as an alternative means of understanding turbulent systems. Our own input to this effort is in contributing to the initial studies of chaos in Eulerian flow using direct numerical simulation (DNS). In this review, we discuss the progress achieved in the turbulence community in understanding chaotic measures including our own work. A central relation between turbulence and chaos is one by Ruelle that connects the maximum Lyapunov exponent and the Reynolds number. The first DNS studies, ours amongst them, in obtaining this relation have shown the viability of chaotic simulation studies of Eulerian flow. Such chaotic measures and associated simulation methodology provides an alternative means to probe turbulent flow. Building on this, we analyze the finite-time Lyapunov exponent (FTLE) and study its fluctuations; we find that chaotic measures could be quantified accurately even at small simulation box sizes where for comparative sizes spectral measures would be inconclusive. We further highlight applications of chaotic measures in analyzing phase transition behavior in turbulent flow and two-dimensional thin-layer turbulent systems. This work shows that chaotic measures are an excellent tool that can be used alongside spectral measures in studying turbulent flow. Full article

This research explores the complex dynamics of a Novel Four-Dimensional Fractional Supply Chain System (NFDFSCS) that integrates a quadratic interaction term involving the actual demand of customers and the inventory level of distributors. The introduction of the quadratic term results in significantly larger maximal Lyapunov exponents (MLE) compared to the original model, indicating increased system complexity. The existence, uniqueness, and Ulam–Hyers stability of the proposed system are verified. Additionally, we establish the global Mittag-Leffler attractive set (MLAS) and Mittag-Leffler positive invariant set (MLPIS) for the system. Numerical simulations and MATLAB phase portraits demonstrate the chaotic nature of the proposed system. Furthermore, a dynamical analysis achieves verification via the Lyapunov exponents, a bifurcation diagram, a 0–1 test, and a complexity analysis. A new numerical approximation method is proposed to solve non-linear fractional differential equations, utilizing fractional differentiation with a non-singular and non-local kernel. These numerical simulations illustrate the primary findings, showing that both external and internal factors can accelerate the process. Furthermore, a robust control scheme is designed to stabilize the system in finite time, effectively suppressing chaotic behaviors. The theoretical findings are supported by the numerical results, highlighting the effectiveness of the control strategy and its potential application in real-world supply chain management (SCM). Full article

A novel control technique is presented in this paper, which is based on a first-order adaptive sliding mode that ensures convergence in a finite time without any prior information on the upper limits of the parametric uncertainties and/or external disturbances. Based on an exponent reaching law, this controller uses two dynamically adaptive control gains. Once the sliding mode is reached, the dynamic gains decrease in order to loosen the system’s constraints, which guarantees minimal control effort. The proof of convergence was demonstrated according to Lyapunov’s criterion. The proposed algorithm was applied to a drill string system to evaluate its performance because such systems present variable operating conditions caused by, for example, the type of rock. The effectiveness of the proposed controller was evaluated by conducting a comparative study that involved comparing it against a commonly used sliding mode controller, as well as other recent adaptive sliding mode control techniques. The different mathematical performance measures included energy consumption. The proposed algorithm had the best performance measures with the lowest energy consumption and it was able to significantly improve the functioning of the drill string system. The results indicated that the proposed controller had 20% less chattering than the classic SM controller. Finally, the proposed controller was the most robust to uncertainties in system parameters and external disturbances, thus demonstrating the auto-adjustable features of the controller. Full article

A Lagrangian method is introduced to analyze the tip leakage vortex (TLV) behavior in a low-speed axial compressor rotor. The finite-time Lyapunov exponent (FTLE) fields are calculated based on the delayed detached-eddy simulation (DDES) results and identifying the FTLE ridges as Lagrangian coherent structures (LCSs). The computational method of the FTLE field in three-dimensional unsteady flow fields is discussed and then applied to the instantaneous flow fields at both the design and near-stall conditions. Results show that the accuracy of the particle trajectory and the density of the initial grid of the particle trajectory greatly affect the results of the FTLE field and, thus, the LCSs. Compared to the Eulerian Q method, which is calculated based on the symmetric and anti-symmetric components of the local velocity gradient tensor, the Lagrangian method has great potential in unraveling the mechanism of complex vortex structures. The LCSs show a transport barrier between the TLV and the secondary TLV, indicating two separate vortices. The aLCSs show the bubble-like and bar-like structure in the isosurfaces corresponding to the bubble and spiral breakdown patterns. Full article

The direct ink writing (DIW) process, used for creating components with functionally graded materials, holds significant promise for advancement in various advanced fields. However, challenges persist in achieving complex gradient variations in small-sized parts. In this study, we have developed a customized pin shape for an active screw mixer using a combination of quadratic B-Spline, the response surface method, and global optimization. This tailored pin design was implemented in a two-material extrusion-based printing system. The primary objective is to facilitate the transformation of material components with shorter transition distances, overcoming size constraints and enhancing both printing flexibility and resolution. Moreover, we characterized the transition delay time for material component changes and the mixing uniformity of the extruded material by constructing a finite element simulation model based on computational fluid dynamics. Additionally, we employed a particle tracking method to obtain the Lyapunov exponent and Poincaré map of the mixing process. We employed these metrics to represent and compare the degree of chaotic mixing and dispersive mixing ability with two other structurally similar mixers. It was found that the optimized pin-type mixer can reduce the transition delay distance by approximately 30% compared to similar structures. Finally, comparative experiments were carried out to verify the printing performance of the optimized pin-type active mixer and the accuracy of the finite element model. Full article

Chaotic maps have been widely studied in the field of cryptography for their complex dynamics. However, chaos-based cryptosystems have not been widely used in practice. One important reason is that the following requirements of practical engineering applications are not taken into account: computational complexity and difficulty of hardware implementation. In this paper, based on the demand for information security applications, we modify the local structure of the three-dimensional Intertwining Logistic chaotic map to improve the efficiency of software calculation and reduce the cost of hardware implementation while maintaining the complex dynamic behavior of the original map. To achieve the goal by reducing the number of floating point operations, we design a mechanism that can be decomposed into two processes. One process is that the input parameters value of the original system is fixed to $2^k$ by Scale index analysis. The other process is that the transcendental function of the original system is replaced by a nonlinear polynomial. We named the new map as “Simple intertwining logistic”. The basic chaotic dynamic behavior of the new system for controlling parameter is qualitatively analyzed by bifurcation diagram and Lyapunov exponent; the non-periodicity of the sequence generated by the new system is quantitatively evaluated by using Scale index technique based on continuous wavelet change. Fuzzy entropy (FuzzyEn) is used to evaluate the randomness of the new system in different finite precision digital systems. The analysis and evaluation results show that the optimized map could achieve the designed target. Then, a novel scheme for generating pseudo-random numbers is proposed based on new map. To ensure its usability in cryptographic applications, a series of analysis are carried out. They mainly include key space analysis, recurrence plots analysis, correlation analysis, information entropy, statistical complexity measure, and performance speed. The statistical properties of the proposed pseudo random number generator (PRNG) are tested with NIST SP800-22 and DIEHARD. The obtained results of analyzing and statistical software testing shows that, the proposed PRNG passed all these tests and have good randomness. In particular, the speed of generating random numbers is extremely rapid compared with existing chaotic PRNGs. Compared to the original chaotic map (using the same scheme of random number generation), the speed is increased by 1.5 times. Thus, the proposed PRNG can be used in the information security. Full article

Finite-dimensional equations constructed earlier to describe the motion of an aquatic drop-shaped robot due to given rotor oscillations are studied. To study the equations of motion, we use the Poincaré map method, estimates of the Lyapunov exponents, and the parameter continuation method to explore the evolution of asymptotically stable solutions. It is shown that, in addition to the so-called main periodic solution of the equations of motion for which the robot moves in a circle in a natural way, an additional asymptotically stable periodic solution can arise under the influence of highly asymmetric impulsive control. This solution corresponds to the robot’s sideways motion near the circle. It is shown that this additional periodic solution can lose stability according to the Neimark–Sacker scenario, and an attracting torus appears in its vicinity. Thus, a quasiperiodic mode of motion can exist in the phase space of the system. It is shown that quasiperiodic solutions of the equations of motion also correspond to the quasiperiodic motion of the robot in a bounded region along a trajectory of a rather complex shape. Also, strange attractors were found that correspond to the drifting motion of the robot. These modes of motion were found for the first time in the dynamics of the drop-shaped robot. Full article

Animal motion and flocking are ubiquitous nonequilibrium phenomena that are often studied within active matter. In examples such as insect swarms, macroscopic quantities exhibit power laws with measurable critical exponents and ideas from phase transitions and statistical mechanics have been explored to explain them. The widely used Vicsek model with periodic boundary conditions has an ordering phase transition but the corresponding homogeneous ordered or disordered phases are different from observations of natural swarms. If a harmonic potential (instead of a periodic box) is used to confine particles, then the numerical simulations of the Vicsek model display periodic, quasiperiodic, and chaotic attractors. The latter are scale-free on critical curves that produce power laws and critical exponents. Here, we investigate the scale-free chaos phase transition in two space dimensions. We show that the shape of the chaotic swarm on the critical curve reflects the split between the core and the vapor of insects observed in midge swarms and that the dynamic correlation function collapses only for a finite interval of small scaled times. We explain the algorithms used to calculate the largest Lyapunov exponents, the static and dynamic critical exponents, and compare them to those of the three-dimensional model. Full article

The main purpose of this article is to discuss the use of the Lyapunov exponents to evaluate the integrity of structures. The use of such coefficients is examined in an analysis that considers the geometric and physical nonlinearities, aiming to ensure the applicability of the method in robust simulations. The material nonlinearity is modeled using the multilinear isotropic elastoplastic model together with a recently developed damage model. The nonlinear equilibrium equations solution is obtained using the positional finite element method. The Newmark time-marching procedure is implemented to evaluate the Lyapunov coefficients and a nonlinear predictor technique that needs a single data series is employed. A numerical example of a frame structure is presented to illustrate the methodology applicability. Its results show that the Lyapunov exponents can be used as indicative parameters of structural integrity, since its analysis was able to detect the occurrence of the destabilization of the structure with the dynamic jump and the presence of material failures. The non-linear predictor proved to be an efficient technique for obtaining the Lyapunov exponents, with a low computational cost. The methodology presented to monitor structural integrity was shown to be a promising alternative. Full article