Evolutionary Differential Equations, Dynamic Systems, Computation and Optimization

In this paper, stability is studied for a class of switched singular systems containing both stable and unstable modes. By introducing a time-varying piecewise Lyapunov function (TVPLF) and a mode-dependent average dwell time (ADT) switching rule, the computable sufficient conditions for system stability are derived. The time-varying piecewise Lyapunov functions are piecewise continuously differentiable on every mode (but may not be differentiable at the interpolating points of the dwell time). This Lyapunov function method is particularly advantageous in overcoming the limitations of traditional multiple Lyapunov function (MLF) methods, which may not have a feasible solution when dealing with switched systems containing only unstable modes. As such, the TVPLF offers greater flexibility in application. Compared with the conventional ADT switching rule, the mode-dependent ADT switching rule not only enables each mode to have its own ADT but also allows for its own switching strategy. Specifically, the stable mode adopts a slow switching strategy while the unstable mode adopts a fast one, thereby reducing the conservatism of the ADT switching rule. Furthermore, based on the stability analysis, the time-varying controllers are proposed to stabilize the switched singular system, which can be expressed as the sequential linear combination of a series of linear state feedback on each mode. The proposed controllers are continuous for each mode, which are different from the controllers designed through the traditional MLF and MDLF methods, where the controllers designed by traditional MLF are the time-invariant linear state feedback in each mode while the controllers designed by the MDLF are piecewise continuous for each mode. Full article

In order to improve the market competitiveness of suppliers and their resilience to emergencies, it is of great significance to discuss the investment decision making of suppliers in developing new products under uncertain and competitive environments. In this paper, with the background of knowledge spillover, absorptive capacity, initial R&D investment, and innovation efficiency asymmetry, the uncertainty of price, sales, and cost are incorporated into the evaluation system as three important risk factors. On the basis of the existing real option investment evaluation model, a real option game model of R&D investment of supplier enterprises based on multiple random variables is established. The sensitivity analysis of parameters is carried out with an example of one enterprise’s monitor R&D project. The results depict that the probability of R&D success has a great impact on the value of enterprise options, which depends on the R&D investment, innovation efficiency, and R&D performance of enterprises. Secondly, the drift rate of price, sales, and cost also has a significant impact on improvement in enterprise option value. Full article

Metaheuristic optimization algorithms (MHA) play a significant role in obtaining the best (optimal) values of the system’s parameters to improve its performance. This role is significantly apparent when dealing with systems where the classical analytical methods fail. Fractional-order (FO) systems have not yet shown an easy procedure to deal with the determination of their optimal parameters through traditional methods. In this paper, a recent, systematic. And comprehensive review is presented to highlight the role of MHA in obtaining the best set of gains and orders for FO controllers. The systematic review starts by exploring the most relevant publications related to the MHA and the FO controllers. The study is focused on the most popular controllers such as the FO-PI, FO-PID, FO Type-1 fuzzy-PID, and FO Type-2 fuzzy-PID. The time domain is restricted in the articles published through the last decade (2014:2023) in the most reputed databases such as Scopus, Web of Science, Science Direct, and Google Scholar. The identified number of papers, from the entire databases, has reached 850 articles. A Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) methodology was applied to the initial set of articles to be screened and filtered to end up with a final list that contains 82 articles. Then, a thorough and comprehensive study was applied to the final list. The results showed that Particle Swarm Optimization (PSO) is the most attractive optimizer to the researchers to be used in the optimal parameters identification of the FO controllers as it attains about 25% of the published papers. In addition, the papers that used PSO as an optimizer have gained a high citation number despite the fact that the Chaotic Atom Search Optimization (ChASO) is the highest one, but it is used only once. Furthermore, the Integral of the Time-Weighted Absolute Error (ITAE) is the best nominated cost function. Based on our comprehensive literature review, this appears to be the first review paper that systematically and comprehensively addresses the optimization of the parameters of the fractional-order PI, PID, Type-1, and Type-2 fuzzy controllers with the use of MHAs. Therefore, the work in this paper can be used as a guide for researchers who are interested in working in this field. Full article

Control engineering and state-space representation are valuable tools in the analysis and design of dynamic systems. In this research, a methodology is proposed that uses these approaches to transform a system-dynamics simulation model into a mathematical model. This is achieved by expressing input, output and state variables as input, output and state vectors, respectively, allowing the representation of the model in matrix form. The resulting model is linear and time-invariant, facilitating its analysis and design. Through the use of this methodology, the system transfer matrix is obtained, which allows the analysis and design of the optimal control of the simulation model. The Ackermann gain-control technique is used to determine the optimal control of the system, which results in a shorter settlement time. This research proposal seeks to mathematically strengthen simulation models and provide an analytical alternative through modern control engineering in SD simulation models. This would allow more informed and effective decisions in the implementation of dynamic systems. Full article

We consider a general metric Steiner problem, which involves finding a set $\mathcal{S}$ with the minimal length, such that $\mathcal{S}\cup A$ is connected, where A is a given compact subset of a given complete metric space X; a solution is called the Steiner tree. Paolini, Stepanov, and Teplitskaya in 2015 provided an example of a planar Steiner tree with an infinite number of branching points connecting an uncountable set of points. We prove that such a set can have a positive Hausdorff dimension, which was an open question (the corresponding tree exhibits self-similar fractal properties). Full article

This article investigates the study of Topology Optimization (TO) in 3D elasticity problems to determine the optimal topology by applying the evolutionary methods of Smoothing Evolutionary Structural Optimization (SESO), Sequential Element Rejection and Admission (SERA), and Evolutionary Structural Optimization (ESO). These procedures were implemented in MATLAB code as an extension of Top3d implemented for SIMP by using the eight-node hexahedral finite element formulation in three-dimensional elastostatic structures. The approaches conducted in the present study are demonstrated with numerical examples involving the compliance minimization criterion. Further, a brief synthesis of flexible mechanisms was studied to emphasize the performance of complaint mechanisms measured in terms of two design specifications/functionalities: mechanical and geometrical advantages, which are the highlights of this article. To show the gains of the proposed methods, numerical results obtained are compared with Solid Isotropic Material with Penalization (SIMP) models. Full article

Noise is ubiquitous in the real-world environment. At present, most scholars only include the stage of Gaussian white noise when applying noise in neural networks and regard white noise as a tool to optimize the network model, which is far from enough, because noise not only affects the optimization ability of the Hopfield neural network but can also better fit the needs of the actual use of the scene. Therefore, according to the problems in the existing research, a method is proposed to combine the neural network with colored noise according to the signal-to-noise ratio. Taking blue noise as an example, the anti-interference ability of the Hopfield neural network regarding colored noise is studied. The results show that for the Hopfield neural network driven by blue noise, by adjusting the neural network step size, excitation function and signal-to-noise ratio, it not only provides ideas for adding colored noise to the neural network but also enables the neural network model to have better optimization-seeking ability. The research results have some reference significance for improving the practical application of neural networks in noisy environments. Full article

In this article, two classes of sufficient conditions of weak solutions are given to guarantee the energy conservation of the compressible Euler equations. Our strategy is to introduce a test function $\phi \left(t\right)v^{ϵ}$ to derive the total energy. The velocity field $v$ needs to be regularized both in time and space. In contrast to the noncompressible Euler equations, the compressible flows we consider here do not have a divergence-free structure. Therefore, it is necessary to make an additional estimate of the pressure p, which takes advantage of an appropriate commutator. In addition, by using the weak convergence, we show that the energy equality is conserved in a point-wise sense. Full article

At the turn of the 21st century, the wide availability of high-frequency data aroused an increasing demand for better modeling and statistical inference. A challenging problem in statistics and econometrics is the estimation problem of the integrated volatility matrix based on high-frequency data. The existing estimators work well for diffusion processes with micro-structural noise and may get worse when jumps are considered. This paper proposes a novel estimation in the presence of jumps, micro-structural noise, and asynchronization. First, we adopt sub-sampling to synchronize the high-frequency data. Then, we use a two-time scale to realize co-volatility to handle noise. Finally, we employ the threshold parameters to remove the effect of jumps and sparsity in two steps. Both the minimax bound and the convergence rate are discussed in the paper. The estimation procedures of the heavy-tailed data will be solved in the future. Full article

Long-run bifurcation analysis aims to describe the asymptotic behavior of a dynamical system. One of the main objectives of mathematical epidemiology is to determine the acute threshold between an infection’s persistence and its elimination. In this study, we use a more comprehensive SVIR epidemic model with large jumps to tackle this and related challenging problems in epidemiology. The huge discontinuities arising from the complexity of the problem are modelled by four independent, tempered, $\alpha$-stable quadratic Lévy processes. A new analytical method is used and for the proposed stochastic model, the critical value ${\mathfrak{R}}_0^{🟉}$ is calculated. For strictly positive value of ${\mathfrak{R}}_0^{🟉}$, the stationary and ergodic properties of the perturbed model are verified (continuation scenario). However, for a strictly negative value of ${\mathfrak{R}}_0^{🟉}$, the model predicts that the infection will vanish exponentially (disappearance scenario). The current study incorporates a large number of earlier works and provides a novel analytical method that can successfully handle numerous stochastic models. This innovative approach can successfully handle a variety of stochastic models in a wide range of applications. For the tempered $\alpha$-stable processes, the Rosinski (2007) algorithm with a specific Lévy measure is implemented as a numerical application. It is concluded that both noise intensities and parameter $\alpha$ have a great influence on the dynamical transition of the model as well as on the shape of its associated probability density function. Full article

In this paper, a new adaptive Levenberg–Marquardt method is proposed to solve the nonlinear equations including supply chain optimization problems. We present a new adaptive update rule which is a segmented function on the ratio between the actual and predicted reductions of the objective function to accept a large number of unsuccessful iterations and avoid jumping in local areas. The global convergence and quadratic convergence of the proposed method are proved by using the trust region technique and local error bound condition, respectively. In addition, we use the proposed algorithm to test on the symmetric and asymmetric linear equations. Numerical results show that the proposed method has good numerical performance and development prospects. Furthermore, we apply the algorithm to solve the fresh agricultural products supply chain optimization problems. Full article

In this article, a novel and efficient approach based on Lucas polynomials is introduced for solving three-dimensional mixed Volterra–Fredholm integral equations for the two types (3D-MVFIEK2). This method transforms the 3D-MVFIEK2 into a system of linear algebraic equations. The error evaluation for the suggested scheme is discussed. This technique is implemented in four examples to illustrate the efficiency and fulfillment of the approach. Examples of numerical solutions to both linear and nonlinear integral equations were used. The Lucas polynomial method and other approaches were contrasted. A collection of tables and figures is used to present the numerical results. We observe that the exact solution differs from the numerical solution if the exact solution is an exponential or trigonometric function, while the numerical solution is the same when the exact solution is a polynomial. The Maple 18 program produced all of the results. Full article

Analysis of the vibration reduction characteristics of shock absorbers is crucial for engines. In this study, the fractal theory was applied to the contact surface of an under-platform damper (UPD), and the influence of the excitation force in the same and opposite directions on the roughness of the contact surface was studied. First, based on fractal geometry theory (FGT), the roughness characterization method of a UPD contact surface was proposed. Then, the friction mechanical model of the rough contact surface was established by combining it with a 3D contact mechanical model. Furthermore, a finite element dynamic model of a blade with a UPD structure was set up. Next, the harmonic balance method was used to calculate the nonlinear response characteristics of a blade under different levels of contact surface roughness. Finally, the influence of the contact surface roughness on the vibration reduction ability of a UPD under different excitation modes was analyzed. According to the simulation results, as the contact surface became rougher, the vibration suppression ability of the UPD on the blade became stronger and stronger. With the change in the centrifugal force of the UPD and the amplitude of the same/reverse excitation force, the influencing law of the contact surface roughness on the vibration suppression ability of the UPD remained unchanged, indicating that the rougher the contact surface roughness, the better the vibration suppression effect. Full article

The aim of this study is to determine a 3D incompressible Brinkman–Forchheimer-extended Darcy fluid flow. Based on global well-posedness and regularity of solutions with a periodic boundary condition, the determining modes for weak and regular solutions is achieved via the generalized Grashof number for a 3D non-autonomous Brinkman–Forchheimer-Darcy fluid flow in porous medium. Furthermore, the asymptotic determination of the complete trajectories inside an attractor via Fourier functionals is shown for a 3D autonomous Brinkman–Forchheimer-extended Darcy model. Full article

This paper is concerned with the asymptotic behavior of the solution of a Timoshenko system with two nonlinear variable exponent damping terms. We prove that the system is stable under some specific conditions on the variable exponent and the equal wave speeds of propagation. We obtain exponential and polynomial decay results by using the multiplier method, and we prove that one variable damping is enough to have polynomial and exponential decay. We observe that the decay is not necessarily improved if the system has two variable damping terms. Our results built on, developed and generalized some earlier results in the literature. Full article

The principal objective of the paper is the study of the three-dimensional Navier–Stokes system with non-autonomous perturbation force term and increasing damping term, which often appears in the fluid system within saturated porous media and other complex media. With some suitable assumptions on the system parameters and external force term, based on the known result on global well-posedness, the existence of pullback attractors is educed, and the system robustness is shown via the upper semicontinuity of system attractors as the perturbation parameter approaches a certain value. Full article

We hope this work allows one to calculate large prime numbers directly, not by trial-and-error, but following a physical law. We report—contrary to conventional assumptions—that differentiation of discontinuous functions (DDF) exists in the set Q, which becomes central to algorithms for quantum computation. DDF have been thought to exist not in the classical sense, but using distributions. However, DDF using distributions still is defined in terms of mathematical real-numbers (MRN), and do not address the Problem of Closure, here investigated. These facts lead to contradictions using MRN, solved by this work, providing a new unbounded class of physical solutions using physical numbers in quantum mechanics (QM), that were always there (just hidden), allowing DDF without distributions, or MRN. It is worthwhile to see this only in mathematics, to avoid the prejudices found in physics, as this reforms both general relativity and QM. This confirms the opinions of Nicolas Gisin that MRN are non-computable with probability 1, and Niels Bohr that physics is not reality, it is a fitting story about reality. Mathematics can get closer to reality, surprisingly. We just have to base mathematics on nature, not on how it defines nature. Full article