Search

Based on phase space reconstruction theory, the root mean square error is used as a quantitative criterion for identifying the appropriate embedding dimension and time step and selecting the optimal configuration for these factors. The phase space is then reconstructed, and the convergent cross-mapping algorithm is applied to analyse the causality between time series. The causality among the variables in the Lorenz equation is first discussed, and the response of this causality to the integration step of numerical solutions to the Lorenz equation is analyzed. We conclude that changes in the integration step do not alter the causality but will affect its strength. Variables X and Y drive each other, whereas variable Z drives variables X and Y in a unidirectional manner. Second, meteorological data from 1948–2022 are used to analyse the effect of the Southern Hemisphere annular mode on the East Asian summer monsoon index and surface air temperature driving capacity. From a dynamic perspective, it is concluded that the Southern Hemisphere annular mode is the driving factor affecting the East Asian summer monsoon index and surface air temperature. Based on ideal test results and the observation data, the collaborative selection of the embedding dimension and time step is more reliable in terms of determining causality. This provides the ability to determine causality between climate indices and theoretically guarantees the selection of climate predictors. [ABSTRACT FROM AUTHOR]

The conditions for the synchronization of three interconnected generalized Lorenz systems are given. The interconnection topology contains a loop. It is shown that, under certain conditions on the strength of the coupling of the systems, the full synchronization of all three systems is guaranteed. The results are illustrated by an example. [ABSTRACT FROM AUTHOR]

This research explores the incorporation of variable order (VO) fractional calculus into the hyperchaotic Lorenz system and studies various chaotic features and attractors. Initially, we propose a variable fractional order hyperchaotic Lorenz system and numerically solve it. The solutions are obtained for multiple choices of control parameters, and these results serve as reference solutions for exploring chaos with the artificial intelligence tool radial basis function network (RBFN). We rebuild phase spaces and trajectories of system states to exhibit chaotic behavior at various levels. To further assess the sensitivity of chaotic attractors, Lyapunov exponents are calculated. The efficacy of the designed computational RBFN is validated through the RMSE and extensive error analysis. The proposed research on AI capabilities aims to introduce an innovative methodology for modeling and analyzing hyperchaotic dynamical systems with variable orders. [ABSTRACT FROM AUTHOR]

Unstable dimension variability is an extreme form of non-hyperbolic behavior that causes a severe shadowing breakdown of chaotic trajectories. This phenomenon can occur in coupled chaotic systems possessing symmetries, leading to an invariant attractor with riddled basins of attraction. We consider the coupling of two Lorenz-like systems, which exhibits chaotic synchronized and anti-synchronized states, with their respective basins of attraction. We demonstrate that these basins are riddled, in the sense that they verify both the mathematical conditions for their existence, as well as the characteristic scaling laws indicating power-law dependence of parameters. Our simulations have shown that a biased random-walk model for the log-distances to the synchronized manifold can accurately predict the scaling exponents near blowout bifurcations in this high-dimensional coupled system. The behavior of the finite-time Lyapunov exponents in directions transversal to the invariant subspace has been used as numerical evidence of unstable dimension variability. [ABSTRACT FROM AUTHOR]

Aiming to explore the subtle connection between the number of nonlinear terms in Lorenz-like systems and hidden attractors, this paper introduces a new simple sub-quadratic four-thirds-degree Lorenz-like system, where x ˙ = a (y − x) , y ˙ = c x − x 3 z , z ˙ = − b z + x 3 y , and uncovers the following property of these systems: decreasing the powers of the nonlinear terms in a quadratic Lorenz-like system where x ˙ = a (y − x) , y ˙ = c x − x z , z ˙ = − b z + x y , may narrow, or even eliminate the range of the parameter c for hidden attractors, but enlarge it for self-excited attractors. By combining numerical simulation, stability and bifurcation theory, most of the important dynamics of the Lorenz system family are revealed, including self-excited Lorenz-like attractors, Hopf bifurcation and generic pitchfork bifurcation at the origin, singularly degenerate heteroclinic cycles, degenerate pitchfork bifurcation at non-isolated equilibria, invariant algebraic surface, heteroclinic orbits and so on. The obtained results may verify the generalization of the second part of the celebrated Hilbert's sixteenth problem to some degree, showing that the number and mutual disposition of attractors and repellers may depend on the degree of chaotic multidimensional dynamical systems. [ABSTRACT FROM AUTHOR]

Misunderstandings in dyadic interactions often persist despite our best efforts, particularly between native and non-native speakers, resembling a broken duet that refuses to harmonise. This paper delves into the computational mechanisms underpinning these misunderstandings through the lens of the broken Lorenz system—a continuous dynamical model. By manipulating a specific parameter regime, we induce bistability within the Lorenz equations, thereby confining trajectories to distinct attractors based on initial conditions. This mirrors the persistence of divergent interpretations that often result in misunderstandings. Our simulations reveal that differing prior beliefs between interlocutors result in misaligned generative models, leading to stable yet divergent states of understanding when exposed to the same percept. Specifically, native speakers equipped with precise (i.e., overconfident) priors expect inputs to align closely with their internal models, thus struggling with unexpected variations. Conversely, non-native speakers with imprecise (i.e., less confident) priors exhibit a greater capacity to adjust and accommodate unforeseen inputs. Our results underscore the important role of generative models in facilitating mutual understanding (i.e., establishing a shared narrative) and highlight the necessity of accounting for multistable dynamics in dyadic interactions. [ABSTRACT FROM AUTHOR]

The present study introduces fractal interpolation function as an attractor of the iterated function system involving a finite number of Reich contractions on a complete metric space. Some amendments are made concerning the results of Reich contractions in the existing literature, with respect to Hausdorff metric, by imposing certain additional constraints on the Reich contractivity factors. Moreover, a non-smooth version of classical interpolant is constructed with Reich contractions as an analogue to the α -fractal function. A fractal operator is proposed to approximate any given continuous function using the newly generated Reich-type fractal function. Apart from the construction, Reich-fractal functions are employed to reconstruct the Lorenz attractor, and the results are graphically compared with fractal functions generated using other contraction mappings. [ABSTRACT FROM AUTHOR]

Estimating and controlling dynamical systems from observable time-series data are essential for understanding and manipulating nonlinear dynamics. This paper proposes a probabilistic framework for simultaneously estimating and controlling nonlinear dynamics under noisy observation conditions. Our proposed method utilizes the particle filter not only as a state estimator and a prior estimator for the dynamics but also as a controller. This approach allows us to handle the nonlinearity of the dynamics and uncertainty of the latent state. We apply two distinct dynamics to verify the effectiveness of our proposed framework: a chaotic system defined by the Lorenz equation and a nonlinear neuronal system defined by the Morris–Lecar neuron model. The results indicate that our proposed framework can simultaneously estimate and control complex nonlinear dynamical systems. [ABSTRACT FROM AUTHOR]

With its single parameter, the Lorenz-Rohde function lacks the necessary functional flexibility to adequately fit the asymmetric empirical Lorenz curve. This limitation is demonstrated by the empirical Lorenz curve constructed based on Ghana's total income employment data. Therefore, this study aimed to develop the Lorenz-Extended Rohde function that could effectively accommodate the asymmetry of the Lorenz curve. In order to achieve this goal, specific parameters were introduced to modify the Lorenz-Rohde function while ensuring its preservation. The results demonstrated that the proposed Lorenz-Extended Rohde function consists of two types: Lorenz-Extended Rohde Type 1 with three parameters and Lorenz-Extended Rohde Type 2 with two parameters. Both types exhibited notable improvements in fitting accuracy compared to the original Lorenz-Rohde function for the asymmetric Lorenz curve. [ABSTRACT FROM AUTHOR]

Previous studies have validated a performance equation (PE) and its generalized version (GPE) in describing the rotated and right-shifted Lorenz curves of organ size (e.g., leaf area and fruit volume) distributions of herbaceous plants. Nevertheless, there are still two questions that have not been adequately addressed by prior work: (i) whether the PE and GPE apply to woody plant species and (ii) how do the PE and GPE perform in comparison with other Lorenz equations when fitting data. To address these deficiencies, we measured the lamina length and width of each leaf on 60 Alangium chinense saplings to compare the performance of the PE and GPE with three other Lorenz equations in quantifying the inequality of leaf area distributions across individual trees. Leaf area is shown to be the product of a proportionality coefficient (k) and leaf length and width. To determine the numerical value of k, we scanned 540 leaves to obtain the leaf area empirically. Using the estimated k, the leaf areas of 60 A. chinense saplings were calculated. Using these data, the two performance equations and three other Lorenz equations were then compared and assessed using the root-mean-square error (RMSE) and Akaike information criterion (AIC). The PE and GPE were found to be valid in describing the rotated and right-shifted Lorenz curves of the A. chinense leaf area distributions, and GPE has the lowest RMSE and AIC values. This work validates the GPE as the best model in gauging variations in leaf area of the woody species. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new 4 - D hyperchaotic system with a nonlinear term in the form of an exponential function. The system is derived from a modified 3 - D Lü system. Firstly, we discuss the qualitative properties of the proposed system, using tools such as hyperchaotic attractors, symmetry, dissipation, equilibrium points, and Lyapunov spectrum for analysis. Next, we present an adaptive controller for stability analysis of the system. An active controller is designed to achieve complete synchronization between two identical systems. All stability results are established using Lyapunov stability theory. Finally, numerical examples and computer simulations are provided to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we explore the rich dynamics in the Chen system using Runge–Kutta method and Deep Neural Networks (DNNs). Compared with the Lorenz system, we find that the first return map P of the Chen system exhibits a more complex structure of continuous regions in the Poincaré section. Moreover, the existence of six crossing blocks with respect to the second return map implies that there is a closed invariant set Λ in the Poincaré section such that P 2 | Λ is semi-conjugate to a 6-shift map, thus demonstrating remarkably chaos in the Chen system. [ABSTRACT FROM AUTHOR]

This study thoroughly investigates the dynamical properties, analytical solutions and chaotic behavior of the high-order nonlinear Schrödinger equation with conformal fractional derivative. After obtaining the dynamic properties by a third-order polynomial complete discriminant system, the study further utilizes a fourth-order polynomial complete discrimination system to derive all traveling wave solutions, including rational solutions, solitary wave solutions, triangular function periodic solutions, and Jacobian elliptic function periodic solutions. In addition, based on different phase diagrams and largest Lyapunov exponents, the chaotic behavior is numerically observed by considering different external perturbation terms and perturbation functions. The innovation of this work lies in using the same method to qualitatively and quantitatively analyze the proposed equation. The type of equation solution is predicted using the results of qualitative analysis, and the accuracy of the prediction can be verified through specific parameter values. [ABSTRACT FROM AUTHOR]

This investigation discusses the modified M-truncated form of the perturbed Chen–Lee–Liu (pCLL) dynamical equation. The pCLL equation is a generalization of the original CLL equation, which describes the propagation of optical solitons in optical fibers. The pCLL equation includes additional terms that account for various influences such as chromatic dispersion, nonlinear dispersion, inter-modal dispersion, and self-steepening. A new version of the generalized exponential rational function method is utilized to obtain multifarious types of soliton solutions. Moreover, the planar dynamical system of the concerned equation is created using a Hamiltonian transformation, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. Numerical simulations of the solutions are added to the analytical results to better understand the dynamic behavior of these solutions. The study's findings could be extremely useful in solving additional nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

A Modified Lorenz system is a system of three differential equations. The third differential equation in the Lorenz system is replaced with a fifth-order linear homogeneous differential equation with constant coefficients. In the paper [B. Zlatanovska and D. Dimovski, A modified Lorenz system: Definition and solution, Asian-Eur. J. Math. 13(08) (2020) 2050164], the Modified Lorenz system is defined by offering a solution, but without solving the fifth-order linear homogeneous differential equation with constant coefficients. This differential equation is solved by solving the characteristic equation of fifth degree. A formula for solving the algebraic equation of the fifth degree does not exist. Therefore, solutions for the special cases of the characteristic equation of fifth degree will be offered. The solution of the Modified Lorenz system will be completed using these forms of the characteristic equation of the fifth degree. [ABSTRACT FROM AUTHOR]

This paper introduces a numerical approach for the fractional-order Rössler chaotic systems and gives error analysis. The effectiveness of the present method is determined by comparing the numerical results of the high-precision difference scheme and predictor-correctors scheme. Some complex dynamical behaviors for the fractional-order Rössler chaotic systems are shown. [ABSTRACT FROM AUTHOR]

Based on the fractional calculus and sliding mode control (SMC) techniques, this paper presents a predefined-time synchronization scheme for fractionalorder chaotic systems (FOCSs). Firstly, a predefined-time control method is proposed for fractional-order systems. Subsequently, a novel sliding surface is presented to ensure predefined-time convergence of the synchronization error. Then, a controller is designed by combining the predefined-time stability and the SMC method to ensure that the synchronization error converges to zero within the predefined time. Finally, the feasibility and robustness of the scheme are illustrated with two numerical simulation examples. [ABSTRACT FROM AUTHOR]

Introduction: For the Navier-Stokes equation, one of the most essential tasks should be to study its completeness of the complex nonlinear systems. Also, its nature and physical practical applications would be depth explored. Moreover, as one of the routes to chaos, this equation with an external force has been investigated numerically in 1989. Recently, some information is worth noting that when the high symmetry was imposed on the velocity field, the complex nonlinear motions should occur even lead to the chaos phenomenon. However, most of the published papers are based on theoretical studies and rarely deal with the above results, which lost of the match between them and the integrity of the scientific system. Methods: This study analyzed the molecular distillation process in detail based on the basic theory of nonlinear chaotic systems. Then, the mathematical model for the process of molecular distillation with one brushless DC motor (BLDCM) is built and named the Molecular-Distillation-Navier-Stokes (MDNS) equation. Also, its complex and potentially chaotic behaviors and chaotic processes are first discovered and demonstrated, such as chaotic attractors, chaotic co-attractors, phase portraits, time-domain waveforms, Lyapunov exponent spectrums, Poincare maps, the bifurcation diagrams, and so on. Results: The good agreement among theoretical analysis, simulation and experimental results verifies the practicability and flexibility of the configured model. Discussion: The related conclusions have supplemented and improved the theoretical system for the Navier Stokes equations. Also, it reflects the significance in molecular distillation processes. Meanwhile, the novel research direction for the fields of the chaotic nonlinear and complex industrial systems have been explored and discovered. [ABSTRACT FROM AUTHOR]

We justify an analytical method for constructing periodic solutions of nonlinear systems of ordinary differential equations of polynomial type. Periodic solutions are constructed in the form of Fourier series in which the coefficients are polynomials depending on a parameter, which is not assumed to be small. Two examples are considered: the van der Pol equation and the Lorenz system. [ABSTRACT FROM AUTHOR]

The ensemble Gaussian mixture filter (EnGMF) combines the simplicity and power of Gaussian mixture models with the provable convergence and power of particle filters. The quality of the EnGMF heavily depends on the choice of covariance matrix in each Gaussian mixture. This work extends the EnGMF to an adaptive choice of covariance based on the parameterized estimates of the sample covariance matrix. Through the use of the expectation maximization algorithm, optimal choices of the covariance matrix parameters are computed in an online fashion. Numerical experiments on the Lorenz '63 equations show that the proposed methodology converges to classical results known in particle filtering. Further numerical results with more advanced choices of covariance parameterization and the medium-size Lorenz '96 equations show that the proposed approach can perform significantly better than the standard EnGMF and other classical data assimilation algorithms. [ABSTRACT FROM AUTHOR]

The Lorenz curve is used to describe the relationship between the cumulative proportion of household income and the number of households of an economy. The extent to which the Lorenz curve deviates from the line of equality (i.e., y = x) is quantified by the Gini coefficient. Prior models are based on the simulated and empirical data of income distributions. In biology, the Lorenz curves of cell or organ size distributions tend to have similar shapes. When the Lorenz curve is rotated by 135 degrees counterclockwise and shifted to the right by a distance of 2 , a three-parameter performance equation (PE), and its generalized version with five parameters (GPE), accurately describe this rotated and right-shifted curve. However, in prior studies, PE and GPE were not compared with the other Lorenz equations, and little is known about whether the skewness of the distribution could influence the validity of these equations. To address these two issues, simulation data from the beta distributions with different skewness values and six empirical datasets of plant (organ) size distributions were used to compare PE and GPE with three other Lorenz equations in describing the rotated and right-shifted plant (organ) size distributions. The root-mean-square error and Akaike information criterion were used to assess the validity of the two performance equations and the three other Lorenz equations. PE and GPE were both validated in describing the rotated and right-shifted simulation and empirical data of plant (organ) distributions. Nevertheless, GPE worked better than PE and the three other Lorenz equations from the perspectives of the goodness of fit, and the trade-off between the goodness of fit and the model structural complexity. Analyses indicate that GPE provides a powerful tool for quantifying size distributions across a broad spectrum of organic entities and can be used in a variety of ecological and evolutionary applications. Even for the simulation data from hypothetical extreme skewed distribution curves, GPE still worked well. [ABSTRACT FROM AUTHOR]

Let f be an expansive Lorenz map and c be the critical point. The survivor set is denoted as S f (H) : = { x ∈ [ 0 , 1 ] : f n (x) ∉ H , ∀ n ≥ 0 } , where H is an open subinterval. Here we study the hole H = (a , b) with a ≤ c ≤ b and a ≠ b . We observe that the case a = c is equivalent to the hole at 0, and the case b = c is equivalent to the hole at 1. Given any expansive Lorenz map f with a hole H = (a , b) and S f (H) ⫅̸ { 0 , 1 } , we prove that there exists a Lorenz map g such that S ~ f (H) \ Ω (g) is countable, where Ω (g) is the Lorenz-shift of g and S ~ f (H) is the symbolic representation of S f (H) . Moreover, let a be fixed, we also give a complete characterization of the maximal plateau I(b) such that for all ϵ ∈ I (b) , S f + (a , ϵ) = S f + (a , b) , and I(b) may degenerate to a single point b. As an application, when f has an ergodic acim and a is fixed, we obtain that the topological entropy function λ f (a) : b ↦ h top (f | S f (a , b)) is a devil staircase. At the special case that f being an intermediate β -transformation, using the Ledrappier-Young formula, the Hausdorff dimension function η f (a) : b ↦ dim H (S f (a , b)) is naturally a devil staircase when fixing a. All the results can be naturally extended to the case that b is fixed. As a result, we extend the devil staircases in (Kalle et al. in Ergod Th Dyn Syst 40:2482–2514, 2020; Langeveld and Samuel in Acta Math Hungar 170:269–301, 2023; Urbanski in Ergod Th Dyn Syst 6:295–309, 1986) to expansive Lorenz maps with a hole at critical point. [ABSTRACT FROM AUTHOR]

Developing chaotic systems-on-a-chip is gaining much attention due to its great potential in securing communication, encrypting data, generating random numbers, and more. The digital implementation of chaotic systems strives to achieve high performance in terms of time, speed, complexity, and precision. In this paper, the focus is on developing high-speed Field Programmable Gate Array (FPGA) cores for chaotic systems, exemplified by the Lorenz system. The developed cores correspond to numerical integration techniques that can extend to the equations of the sixth order and at high precision. The investigation comprises a thorough analysis and evaluation of the developed cores according to the algorithm complexity and the achieved precision, hardware area, throughput, power consumption, and maximum operational frequency. Validations are done through simulations and careful comparisons with outstanding closely related work from the recent literature. The results affirm the successful creation of highly efficient sixth-order Lorenz discretizations, achieving a high throughput of 3.39 Gbps with a precision of 16 bits. Additionally, an outstanding throughput of 21.17 Gbps was achieved for the first-order implementation coupled with a high precision of 64 bits. These outcomes set our work as a benchmark for high-performance characteristics, surpassing similar investigations reported in the literature. [ABSTRACT FROM AUTHOR]

The purpose of this article is to investigate the optimal control of nonlinear fractional order chaotic models of diabetes mellitus, human immunodeficiency virus, migraine and Parkinson's diseases using genetic algorithms and particle swarm optimization. Mathematical chaotic models of nonlinear fractional order type of the above diseases were presented. Then optimal control for each of the models and numerical simulation was done using genetic algorithm and particle swarm optimization algorithm. The results of the genetic algorithm method are excellent. All the results obtained for the particle swarm optimization method show that this method is also very successful and the results are very close to the genetic algorithm method. Very low values of MSE and RMSE errors indicate that the simulation is effective and efficient. Also, Lie symmetry was calculated for the proposed models and the results were presented. [ABSTRACT FROM AUTHOR]

Herringbone gear planetary gear transmission system has the advantages of high contact ratio and high bearing capacity and is widely used in various heavy load fields. However, the gear generates a large amount of heat during the meshing transmission, which affects the nonlinear dynamic characteristics of the system. In this paper, the various nonlinear factors are considered, including time-varying meshing stiffness, engagement damping, backlash, and engagement error. A nonlinear dynamic model of the herringbone planetary gear system considering temperature effects is established by using the lumped-parameter method. The nonlinear vibration differential equation of the system is solved by using the Runge–Kutta method. The impact law of temperature and engagement damping ratio changes on the bifurcation features of the herringbone planetary gear system is studied by combining the maximum Lyapunov exponent diagram, bifurcation diagram, time domain diagram, phase diagram, Poincare diagram, and spectrogram. And the chaos phenomenon of the system is analyzed and controlled by using a non-feedback control method with the external periodic signal. The result shows that with the change of temperature rise and meshing damping ratio, the system exhibits the kinematics response of chaotic, multi-period, and single-period. Higher temperature rise (∆T > 64 °C) and larger engagement damping ratio (ξ > 0.122) can make the gear system enter stable periodic motion. The chaotic motion region of the system is effectively controlled to the periodic motion orbit by introducing the periodic signal feedback controller. The relevant conclusions can provide a theoretical basis for the optimal design of the dynamic structure stability of herringbone planetary gear. [ABSTRACT FROM AUTHOR]

This research paper investigates the Kadomtsev-Petviashvii-Benjamin-Bona-Mahony equation. The new Kudryashov and generalized Arnous methods are employed to obtain the generalized solitary wave solution. The phase plane theory examines the bifurcation analysis and illustrates phase portraits. Finally, the external perturbation terms are considered to reveal its chaotic behavior. These findings contribute to a deeper understanding of the dynamics of the Kadomtsev-Petviashvii-Benjamin-Bona-Mahony wave equation and its applications in real-world phenomena. [ABSTRACT FROM AUTHOR]

A novel simple 5D hyperchaotic system with two non-hyperbolic equilibria points is presented. The proposed system is designed by coupling between a 3D Sprott C system and a 2D linear system via a coupling strategy. Compared to the traditional systems, a new system is considered simply because it consists of five first-order ordinary differential equations with nine terms: seven linear terms and two quadratic nonlinearities. Due to the nature of equilibria points, this system belongs to the group of self-excited attractors. The attractors have been described as hyperchaotic. Finally, the projective synchronization problem of the new system has been realized through both Lyapunov stability theory and numerical simulation. The numerical simulation confirmed the validity of our analytical results. This work throws light on the great significance of the new system via designing controllers with the minimum terms possible are helpful in some practical applications. [ABSTRACT FROM AUTHOR]

The effect of time-periodic two-frequency rotation modulation on Rayleigh–Bénard convection in water with either AA7072 or AA7075 nanoparticles is investigated. The single-phase description of the Khanafer–Vafai–Lightstone model is used for modeling the nanoliquids. An asymptotic expansion procedure is adopted in the case of the linear stability to obtain the correction (due to modulation) to the Rayleigh number at marginal stability of unmodulated convection. A nonlinear regime of convection is considered with a nonautonomous generalized Lorenz model as the governing equation. The method of multiscales is then employed to obtain the coupled nonautonomous Ginzburg–Landau equations with cubic nonlinearity from the Lorenz model. These equations are presented in the phase-amplitude form and the amplitude is used to quantify the heat transport. The modulation amplitude is considered to be small (of order less than unity) and moderate frequencies of modulation are considered. We found that there is a threshold frequency beyond which the system behavior reverses. At frequencies below the threshold, the mean Nusselt number increases with an increase in the amplitude of modulation while an opposite influence is seen for values above the threshold. Such a behavior is a consequence of what is analogously seen in the case of the critical Rayleigh number. The influence of two-frequency modulation is more pronounced on the results of the linear and nonlinear regimes compared to that of the single-frequency one. The heat transport is enhanced due to the presence of dilute concentration of suspended nanoparticles (either AA7072 or AA7075 nanoalloys) in water. The influence of nanoparticles is to modify the threshold values generating chaos but it does not qualitatively alter the dynamical behavior of the system. The plots of Lyapunov exponents reveal that there is no possibility of hyper-chaos in the generalized Lorenz model when there is a rotational modulation. [ABSTRACT FROM AUTHOR]

This article explores the evolution of Lorenz trajectories within attractors. Specifically, based on the characteristics of the tangents to trajectories, we derive quantitative standards for determining the spatial position of trajectory lines. The Lorenz trajectory is decomposed into four parts. This standard is objective and quantitative and is independent of the initial field of the Lorenz equation and the calculation scheme; importantly, it is designed based on the inherent dynamic characteristics of the Lorenz equation. Linear fitting of the trajectories in the left and right equilibrium point regions shows that the trajectories lie on planes, indicating the existence of linear features in the nonlinear system. This study identifies the fundamental causes of chaos in the Lorenz equation using the microscopic evolution and local characteristics of the trajectories, and indicating that the spatial position of the initial field is important for their predictability. We theoretically demonstrate that mutation is essentially self-regulation within chaotic systems. This scheme is designed according to the evolution characteristics of Lorenz trajectories, and thus has certain methodological limitations that mean it may not be applicable to other chaotic systems. However, it does depict the causes of chaos and elucidates the sensitivity of differential equations to initial values in terms of trajectory evolution. [ABSTRACT FROM AUTHOR]

We present a data-driven and interpretable approach for reducing the dimensionality of chaotic systems using spectral submanifolds (SSMs). Emanating from fixed points or periodic orbits, these SSMs are low-dimensional inertial manifolds containing the chaotic attractor of the underlying high-dimensional system. The reduced dynamics on the SSMs turn out to predict chaotic dynamics accurately over a few Lyapunov times and also reproduce long-term statistical features, such as the largest Lyapunov exponents and probability distributions, of the chaotic attractor. We illustrate this methodology on numerical data sets including delay-embedded Lorenz and Rössler attractors, a nine-dimensional Lorenz model, a periodically forced Duffing oscillator chain, and the Kuramoto–Sivashinsky equation. We also demonstrate the predictive power of our approach by constructing an SSM-reduced model from unforced trajectories of a buckling beam and then predicting its periodically forced chaotic response without using data from the forced beam. [ABSTRACT FROM AUTHOR]

The linear and weakly nonlinear stability analyses of Darcy–Bénard convection of a Newtonian fluid experiencing a uniform vertical cross-flow is investigated in the paper for various aspect ratios. Making use of the Maclaurin series representation, an expression for axial eigenfunctions is obtained with the radial eigenfunction being a Bessel function of first kind. These eigenfunctions are influenced by the Peclet number, Pe, the non-dimensional number that signifies the rate of vertical cross-flow. The modified-Vadasz-Lorenz model obtained in this paper has newly defined non-dimensional parameters that capture the influence of vertical cross-flow. From the linear stability analysis, it is found that the effect of introducing vertical cross-flow is to stabilize the system. Using a weakly nonlinear stability analysis, the closed-form expression of the Hopf-Rayleigh number as a function of Pe is obtained. Furthermore, the behavior of the modified-Vadasz–Lorenz model is analyzed using the largest Lyapunov exponent and the bifurcation diagram. This gives information about the intensity of chaos and occurrence of the periodic motion. We observe that the influence of vertical cross-flow is to increase the value of the Hopf–Rayleigh number and thereby to delay the onset of chaos. Furthermore, the appearance of the first periodic point is preponed due to the vertical cross-flow. As the rate of vertical cross-flow increases, the intensity of chaos decreases, thereby indicating that the effect of introducing vertical cross-flow is to suppress chaos. [ABSTRACT FROM AUTHOR]

We introduce a predictor-corrector discretization scheme for the numerical integration of a class of stochastic differential equations and prove that it converges with weak order 1.0. The key feature of the new scheme is that it builds up sequentially (and recursively) in the dimension of the state space of the solution, hence making it suitable for approximations of high-dimensional state space models. We show, using the stochastic Lorenz 96 system as a test model, that the proposed method can operate with larger time steps than the standard Euler--Maruyama scheme and, therefore, generate valid approximations with a smaller computational cost. We also introduce the theoretical analysis of the error incurred by the new predictor-corrector scheme when used as a building block for discrete-time Bayesian filters for continuous-time systems. Finally, we assess the performance of several ensemble Kalman filters that incorporate the proposed sequential predictor-corrector Euler scheme and the standard Euler--Maruyama method. The numerical experiments show that the filters employing the new sequential scheme can operate with larger time steps, smaller Monte Carlo ensembles, and noisier systems. [ABSTRACT FROM AUTHOR]

This research paper focuses on investigating the dynamics and chaos control of two discrete systems corresponding to the singularly perturbed equation associated to the Hénon difference equation with continuous arguments. We investigate the local stability of the two systems. To achieve chaos control, we employ the state feedback control method for both systems. Numerical simulations are conducted to gain insights into the qualitative behavior of these systems. By conducting numerical simulations, we gain valuable insights into the role played by the perturbation parameter and the discrete delays in the bifurcation behavior of each system. Furthermore, we demonstrate the practical application of the chaos control method through the implementation of an on-off strategy. [ABSTRACT FROM AUTHOR]

A methodology for directly predicting the time evolution of the assumed parameters for the distribution densities based on the Liouville equation, as proposed earlier, is extended to multi–dimensional cases as well as when the systems are constrained by integrals over a part of the variable range. The extended methodology is tested against a convective energy cycle system as well as the Lorenz's stranger attractor. As a general tendency, the variance tends to collapse to a vanishing value over a finite time regardless of the chosen assumed distribution form. This general tendency is likely due to the common cause as collapse of the variance commonly found in ensemble–based data assimilation. [ABSTRACT FROM AUTHOR]

To contribute to the study of networks and graphs. We apply standard mathematical thinking. We show that the distance distribution in an undirected network Lorenz majorizes the one of a chain. As a consequence, the average and median distances in any such network are smaller than or equal to those of a chain. We restricted our investigations to undirected, unweighted networks. We are convinced that these results are useful in the study of small worlds and the so-called six degrees of separation property. To the best of our knowledge our research contains new network results, especially those related to frequencies of distances. [ABSTRACT FROM AUTHOR]

The study of the complex model associated with chaotic models is always the most complicated and fundamental in the current scientific environment. The primary goal of the current paper is to provide an illustration of the fundamental theory while analysing dynamical systems and validating the chaotic behaviour of the Lorenz–Haken (LH) equations, a system of fractional differential equations. The LH equations are used to describe the 4D chaotic laser. The Adams–Bashforth numerical method is used to extract the numerical solutions projected model. The classical model introduces the bifurcation of the parameter linked with the system. The system's uniqueness and existence are confirmed using the fixed-point hypothesis with the Caputo–Fabrizio fractional operator, followed by boundedness and dynamical analysis. Furthermore, the chaotic character of the numerical solutions with different orders obtained at different beginning circumstances is presented. [ABSTRACT FROM AUTHOR]

This study focuses on the nonlinear stability analysis of nanofluid thermal conductivity and the chaotic convection of feedback control variation. This system of fluid layer addresses external influences on feedback control, gravity, and temperature. The Galerkin method is used to solve a nonlinear system of three-dimensional Lorenz equations. Here we considered a time series analysis idea to combine feedback and nanofluids on the route from stable to chaotic solutions. These studies have been directed at Titanium dioxide (TiO2), Aluminum trioxide (Al2O3), Silver (Ag), and Copper (Cu) nanofluids. Feedback control is mainly emphasized in regulating heat transfer phenomena. This phenomenon is more widely used in engineering, medical sciences, microbiology, and medicine. It has been found that the chaotic behavior of nanofluid convection has a dwelling/delay that can be detected when feedback control changes. The results show that the feedback control has cooling system, which controls the chaotic phase of nanofluid conduction. [ABSTRACT FROM AUTHOR]

Deep reinforcement learning (DRL) algorithms are suitable for modeling and controlling complex systems. Methods for controlling chaos, a difficult task, require improvement. In this article, we present a DRL‐based control method that can control a nonlinear chaotic system without any prior knowledge of the system's equations. We use proximal policy optimization (PPO) to train an agent. The environment is a Lorenz chaotic system, and our goal is to stabilize this chaotic system as quickly as possible and minimize the error by adding extra control terms to the chaotic system. Therefore, the reward function accounts for the total triaxial error. The experimental results demonstrated that the trained agent can rapidly suppress chaos in the system, regardless of the system's random initial conditions. A comprehensive comparison of different DRL algorithms indicated that PPO is the most efficient and effective algorithm for controlling the chaotic system. Moreover, different maximum control forces were applied to determine the relationship between the control forces and controller performance. To verify the robustness of the controller, random disturbances were introduced during training and testing, and the empirical results indicated that the agent trained with random noise performed better. The chaotic system has highly nonlinear characteristics and is extremely sensitive to initial conditions, and DRL is suitable for modeling such systems. [ABSTRACT FROM AUTHOR]

Control and synchronisation of chaotic systems have received much attention due to their various potential applications. In this paper, the predefined-time master–slave synchronisation of two chaotic systems is investigated. With the help of two different predefined-time stability theorems, two controllers are designed to achieve the predefined-time synchronisation of two nonidentical chaotic systems. Then based on inequality techniques and a Lyapunov function, some new sufficient criteria are obtained to ensure that two nonidentical chaotic systems achieve predefined-time synchronisation. Finally, numerical examples are given to show the effectiveness of our new results. [ABSTRACT FROM AUTHOR]

Traditional sensitivity analysis methods fail for chaotic systems due to the unstable characteristics of the linearized equations. To overcome these issues two methods have been developed in the literature, one being the shadowing approach, which results in a minimization problem, and the other being numerical viscosity, where a damping term is added to the linearized equations to suppress the instability. The shadowing approach is computationally expensive but produces accurate sensitivities, while numerical viscosity can produce less accurate sensitivities but with significantly reduced computational cost. However, it is not fully clear how the solutions generated by these two approaches compare to each other. In this work we aim to bridge this gap by introducing a control term, found with optimal control theory techniques, to prevent the exponential growth of solution of the linearized equations. We will refer to this method as optimal control shadowing. We investigate the computational aspects and performance of this new method on the Lorenz and Kuramoto--Sivashinsky systems and compare its performance with simple numerical viscosity schemes. We show that the tangent solution generated by the proposed approach is similar to that generated by shadowing methods, suggesting that optimal control attempts to stabilize the unstable shadowing direction. Further, for the spatially extended system, we examine the energy budget of the tangent equation and show that the control term found via the solution of the optimal control problem acts only at length scales where production of tangent energy dominates dissipation, which is not necessarily the case for the numerical viscosity methods. [ABSTRACT FROM AUTHOR]

Possibility of chaos is studied in Darcy–Bénard convection using the Dirichlet and the Robin boundary condition at the lower and upper boundaries, respectively. Comparison is made with the results of Dirichlet (classical-Darcy–Bénard convection, CDBC) and Neumann boundary condition (Barletta–Darcy–Bénard convection, BDBC). It is found that the cell size at onset is bigger in the case of BDBC compared to the generalized-Darcy–Bénard convection (GDBC) and much bigger compared to CDBC. The critical-Darcy–Rayleigh number of BDBC is found to be the least and that of CDBC is the largest. Nonlinear-stability-analysis is performed leading to the scaled-generalized-Vadasz–Lorenz model (SGVLM). In deriving this model, help is sought from a local-nonlinear-stability-analysis that yields the form of the convective-mode. The SGVLM is shown to be dissipative and conservative, with its bounded solution trapped within an ellipsoid. Onset of chaos and its characteristics are studied using the Hopf–Rayleigh-number, the Lorenz-butterfly-diagram, and the plot of the amplitude of the convective-mode vs the control-parameter, R, which is the eigenvalue. Chaos sets in earlier in CDBC and much later in BDBC when compared to that in GDBC. Beyond the onset of chaos is seen a sequence of chaotic and periodic motions, with the latter sometimes being present for an extended period. [ABSTRACT FROM AUTHOR]

Koopman operators are infinite‐dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra and infinite‐dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data‐driven algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme for computing the spectra and pseudospectra of general Koopman operators from snapshot data without spectral pollution. Using the resolvent operator and ResDMD, we compute smoothed approximations of spectral measures associated with general measure‐preserving dynamical systems. We prove explicit convergence theorems for our algorithms (including for general systems that are not measure‐preserving), which can achieve high‐order convergence even for chaotic systems when computing the density of the continuous spectrum and the discrete spectrum. Since our algorithms have error control, ResDMD allows aposteri verification of spectral quantities, Koopman mode decompositions, and learned dictionaries. We demonstrate our algorithms on the tent map, circle rotations, Gauss iterated map, nonlinear pendulum, double pendulum, and Lorenz system. Finally, we provide kernelized variants of our algorithms for dynamical systems with a high‐dimensional state space. This allows us to compute the spectral measure associated with the dynamics of a protein molecule with a 20,046‐dimensional state space and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number >105 that has a 295,122‐dimensional state space. [ABSTRACT FROM AUTHOR]

Most of the dynamics displayed by highly complicated nonlinear systems also appear in simple nonlinear systems. The purpose of the two-dimensional map with a strange attractor was for it to be a simple mapping that possesses similar properties to the Lorenz system and its Poincare map. Henon map is investigated, periodic points are found, and chaotic attractors are produced. In this paper, we will demonstrate an orbit of the Henon map with 10000 points, which vary on the initial conditions of the orbit and the values of the two parameters of the system. Known that the chaotic attractors in the Henon map are neither area filling (dimension 2) nor a simple curve (of dimension 1), the dimensions of these complicated geometries must be non-integer values between 1 and 2, and the chaotic attractors are then called fractals or strange attractors. Thee capacity or box-counting dimension dbox is the simplest possible way to measure such pathologies. We use The OGY (Ott, Grebogi, and Yorke) Method with the idea to make small time-dependent linear perturbations to the control parameter p in order to nudge the state towards the stable manifold of the desired fixed point. Acting on 500 equally spaced initial points (x0,y0) on a circle or a square, represent an numerical experiment which may give us some hints about why Jupiter's red spot and Saturn's hexagon-shaped hurricane seem to exist forever without contracting. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to study the fuzzy stability of a fractional Lorenz system in the sense of the Caputo-Fabrizio derivative and a fractional financial crisis in the sense of ♄−Hilfer derivative. Defining a new type of fuzzy control function that has a dynamic situation helps us to investigate new stability results for these mathematical models. [ABSTRACT FROM AUTHOR]

Precipitation concentration indices have become a popular tool for analyzing the structure of daily precipitation amounts. Among the existing indices, the concentration index (CI) is widely used. In calculating the CI, an important aspect is adjusting the Lorenz curve based on the observed precipitation data. Usually, the fit has been carried out with equations of the type y = axebx. However, in some research work, it has been observed that sometimes, the fit obtained only partially describes the behavior of the data. Thus, this work evaluated an alternative functional form to fit the Lorenz curve. For this, daily precipitation data from 44 climatological stations in Mexico were used to assess two equations for adjusting the Lorenz curve. Once the fit was made, the goodness of fit was evaluated to determine which of the functional forms best described the behavior of the data. Results showed that the two functional forms produced similar results for low precipitation concentrations. However, when the concentration increased, the alternative functional form generated results following the behavior of the observations. Thus, it is recommended to use the alternative functional form to avoid overestimations of the concentration of daily precipitation in areas where it is known that a high concentration occurs. [ABSTRACT FROM AUTHOR]

This paper proposes an approach for identifying unknown switching topology in a complex dynamical network. The setup is divided into two components: a primary drive network and a specialized response network equipped with switched topology observers. Each class of observers is dedicated to tracking a specific topology structure. The updating law for these observers is dynamically adjusted based on the operational status of the corresponding topology in the drive network—active if engaged and dormant if not. The sufficient conditions for successful identification are obtained by employing adaptive synchronization control and the Lyapunov function method. In particular, this paper abandons the generally used assumption of linear independence and adopts an easily verifiable condition for accurate identification. The result shows that the proposed identification method is applicable for any finite switching periods. By employing the chaotic Lü system and the Lorenz system as the local dynamics of the networks, numerical examples demonstrate the effectiveness of the proposed topology identification method. [ABSTRACT FROM AUTHOR]

Nonlinear fractional difierential equations and chaotic systems can be modeled with variable-order difierential operators. We propose a generalized numerical scheme to simulate variableorder fractional difierential operators. Fractional calculus' fundamental theorem and Lagrange polynomial interpolation are used. Two methods, Atangana-Baleanu-Caputo and Atangana-Seda derivatives, were used to solve a chaotic Newton-Leipnik system problem with fractional operators. Our scheme examined the existence and uniqueness of the solution. We analyze the model qualitatively using its equivalent integral through an iterative convergence sequence. This novel method is illustrated with numerical examples. Simulated and analytical results agree. We contribute to real-world mathematical applications. Finally, we applied a numerical successive approximation method to solve the fractional model. [ABSTRACT FROM AUTHOR]

The random structures in the Aquila optimization algorithm are modeled with fractional chaotic oscillators, and the fractional-order chaotic oscillator-based Aquila optimization (FOCOBAO) algorithm was suggested in this study. First of all, the basic AO algorithm was examined. In particular, random variables that affect the optimization performance of the AO algorithm have been determined. Then, instead of the determined random variables, the coefficients were derived with fractional chaotic oscillators and used in the FOCOBAO. The superiority of the proposed algorithm was primarily demonstrated via twenty-three benchmark functions. The results were matched with GO, EO, GWO, MPA, WOA, SMA and basic AO optimization algorithms. Then, the design of the Lorenz chaotic oscillator, according to maximum chaotic objective function, is a topic that remains up to date in the literature. In this study, a fractional chaotic Lorenz oscillator was designed with FOCOBAO as an engineering application. Especially for maximum chaoticity, maximum positive Lyapunov exponents were determined. In this way, a different design process has been proposed in the literature. The basic AO algorithm, which includes stochastic processes, was developed with fractional chaotic oscillators, and a deterministic method was obtained. The parameters of the Lorenz system were calculated for maximum chaoticity, and the results were presented comparatively. [ABSTRACT FROM AUTHOR]