Differential System Research Articles

Local stability and topological structure of completely integrable differential systems

Completely integrable systems are dynamically simple in common sense, but they may have complicated dynamics around the points where their first integrals are not functionally independent. Tudoran in 2017 provided a criterion for characterizing stability of nondegenerated regular singularities of completely integrable systems. Here we present a new criterion which is more useful than that by Tudoran, in the sense that the cases that his criterion works on are also applicable to ours, whereas there are cases that our criterion works on but his does not.

Continuous dependence of linear differential systems on polynomial matrices

Abstract Linear differential systems, as it is known, are parameterized by means of polynomial matrices. Introducing appropriate topologies, it is shown that this parametrization becomes a quotient map.

At Most Five Limit Cycles in a Class of Discontinuous Piecewise Linear Systems with Three Zones

In this paper, we study the maximum number of limit cycles where planar discontinuous piecewise differential systems can exist formed by three regions separated by a nonregular line. We show that such discontinuous piecewise linear systems can have at most five limit cycles, two of which are of the four intersection points type, the third one is of the two intersection points type and the other two are of three intersection points type. In a setting, we have solved the extended 16th Hilbert problem for these discontinuous piecewise linear differential systems.

Two-dimensional optical multiple-order differentiations based on spatial spectrum modulation

Optical differential operations have recently attracted considerable attention owing to the capabilities of ultrafast speed and low power consumption. The transfer function, which embodies the frequency-domain characteristics of differential systems, plays an important role in differentiator design. Here, we report a super-Gaussian aperture differential filter, and we reveal unique characteristics of odd- and even-order transfer functions and corresponding differential effects via spatial spectrum modulation. We show that the feature of the transfer function is well maintained, and more precise differentiation can be achieved using the designed filter. Two-dimensional first- to fifth-order full and partial differentiations are implemented both theoretically and experimentally. Our work provides an approach for engineering customized multiple-order differentiators and promotes the advancements of related areas such as optical analog computing and image processing.

Distributed iterative learning consensus tracking for singular partial differential multi-agent systems under fixed and iteration-varying topologies

This paper investigates the consensus tracking problems of singular partial differential multi-agent systems (SPD-MASs) with initial state errors under fixed and iteration-varying network topologies, where only certain follower agents have access the trajectory defined by virtual leaders. To solve this problem, a distributed iterative learning consensus protocol with initial state learning is proposed, ensuring accurate tracking of the leader's trajectory by all follower agents with initial state errors. The convergence conditions for consensus tracking error are derived through mathematical analysis between adjacent agents. Theoretical analysis shows that, under convergence conditions, the consensus error between any two agents can converge to zero along the iterative axis. Finally, numerical simulations show validate the effectiveness of the proposed distributed control protocol with initial state learning compared to traditional protocols without state learning.

Limit cycles of piecewise smooth differential systems of the type nonlinear centre and saddle

Piecewise linear differential systems separated by two parallel straight lines of the type of centre-centre-Hamiltonian saddle and the centre-Hamiltonian saddle-Hamiltonian saddle can have at most one limit cycle and there are systems in these classes having one limit cycle. In this paper, we study the limit cycles of a piecewise smooth differential system separated by two parallel straight lines formed by nonlinear centres and a Hamiltonian saddle.

On the existence of solutions for nonlocal sequential boundary fractional differential equations via ψ-Riemann–Liouville derivative

The purpose of this paper is to study a generalized Riemann–Liouville fractional differential equation and system with nonlocal boundary conditions. Firstly, some properties of the Green function are presented and then Lyapunov-type inequalities for a sequential ψ-Riemann–Liouville fractional boundary value problem are established. Also, the existence and uniqueness of solutions are proved by using Banach and Schauder fixed-point theorems. Furthermore, the existence and uniqueness of solutions to a sequential nonlinear differential system is established by means of Schauder’s and Perov’s fixed-point theorems. Examples are given to validate the theoretical results.

The integration and tests of the shattered pellet injection system on EAST

A SPI system specifically designed for EAST has been integrated and commissioned to carry out plasma rapid shutdown. This system is capable of injecting pellets with quantities ranging from 7 to 14 Pa m3 and at velocities between 150 and 400 m/s, as determined after conducting a series of bench tests. A two-stage differential pumping system designed to remove the propellant gas has had its efficiency assessed through the application of pumping equations and the fourth-order Runge-Kutta routine. For the purpose of facilitating remote control of the SPI system, a control system based on EPICS has been developed. This system enables an automated workflow for pellet formation and launch, which includes stages such as pellet formation, strengthening pellet, awaiting trigger, launching pellet, baking gun barrel, and cooling cold head, with the entire process taking 600 s. Furthermore, rapid shutdown experiments using Ne shattered pellet injection have been conducted, demonstrating the system's high reliability and reproducibility on EAST.

A constructive approach for investigating the stability of incommensurate fractional differential systems

This paper is devoted to studying the asymptotic behaviour of solutions to generalized incommensurate fractional systems. To this end, we first consider fractional systems with rational orders and introduce a criterion that is necessary and sufficient to ensure the stability of such systems. Next, from the fractional order pseudospectrum definition proposed by Šanca et al., we formulate the concept of a rational approximation for the fractional spectrum of incommensurate fractional systems with general, not necessarily rational, orders. Our first important new contribution is to show the equivalence between the fractional spectrum of a incommensurate linear system and its rational approximation. With this result in hand, we use ideas developed in our earlier work to demonstrate the stability of an equilibrium point to nonlinear systems in arbitrary finite-dimensional spaces. A second novel aspect of our work is the fact that the approach is constructive. It is effective and widely applicable in studying the asymptotic behaviour of solutions to linear incommensurate fractional differential systems with constant coefficient matrices and linearized stability theory for nonlinear incommensurate fractional differential systems. Finally, we give numerical simulations to illustrate the merit of the proposed theoretical results.

Advancements in epidemiological modeling: Bayesian regularization neural networks for smoke dynamics

Using an artificial neural network and the Bayesian Regularization Technique (NNs-BRT), the stochastic method’s strength is used to analyze the differential system, illustrating a nonlinear smoke epidemic differential model (NSED). This allows for a more precise, dependable, cost-effective, dynamic calculating approach. In addition to experiments utilizing nonlinear mathematical structures through five distinct classes of differential operators-smokers contemplating quitting, infrequent smokers, regular smokers, those who have temporarily abstained from smoking, and those who have permanently quit smoking the NSED framework has been established. By allocating 25% of the data for testing and validation and 75% for training, the proposed NNs-BRT can determine the estimated solutions of five different examples based on the numerical computation of the NSED system employing Adams techniques. To prove the accuracy of the given NNs-BRTs, a comparison of the results from the dataset obtained using the Adam method for different scenarios reflecting variance in recruitment rate, effective contact rate between C and A, natural death rate, how quickly occasional smokers become regular smokers, contact rate between smokers, and temporary quitters, number of smokers quitting, and percentage of smokers who are still leaving for good is made. The use of error histograms, mean square error, regression, correlation, and state transitions is also examined in numerical replications of the NNs-BRTs to verify their competence, dependability, accuracy, consistency, and proficiency.

Dynamics analysis of a new fractional-order SVEIR-KS model for computer virus propagation: Stability and Hopf bifurcation

This work investigates the stability and Hopf bifurcation of a fractional-order model of the computer virus known as Susceptible–Vaccinated–Exposed–Infected–Recovered–Kill Signals (SVEIR-KS) that has two delays. Utilizing the linearization technique, Laplace transform, Routh–Hurwitz criteria, and Hopf bifurcation theorem of fractional-order differential systems, the sufficient criteria for the system’s stability and Hopf bifurcation are determined. The study demonstrates that the stability and occurrence of the Hopf bifurcation of the fractional-order computer virus model are profoundly affected by fractional order q and time delays. In order to confirm the validity of the theoretical results, various simulations and examples with appropriate parameters are provided. The results show a negative correlation between the delay critical value and the fractional order q. Additionally we also dig into the effect in which kill signals prevent computer viruses from spreading over a network.

AQP5 promotes epithelial-mesenchymal transition and tumor growth through activating the Wnt/β-catenin pathway in triple-negative breast cancer

BackgroundEmerging data identifies aquaporin 5 (AQP5) as a vital player in many kinds of cancers. Over expression of AQP5 was associated with increased metastasis and poor prognosis, suggesting that AQP5 may facilitate cancer cell proliferation and migration. Our previous studies also showed that AQP3 and AQP5 were highly expressed in triple-negative breast cancer (TNBC) and the expression of AQP3 and AQP5 in TNBC tissue was positive correlated with advanced clinical stage. ObjectiveWe aim to investigate the role of AQP5 in TNBC oncogenesis and development. MethodsMDA-MB-231 cells were transfected with siRNA-AQP5 and AQP5 overexpression vector to establish a differential expression system for AQP5. Cell proliferation and apoptosis of MDA-MB-231 cells were detected by CCK-8 (Cell Counting Kit-8) and FCM (flow cytometry), respectively. Cell migration and invasion abilities were evaluated by wound healing assay and transwell assay. The qRT-PCR and western blot assays were used to study the effect of AQP5 expression level on the expression of epithelial-to-mesenchymal transition (EMT) related molecules. The effects of ICG-001, a Wnt/β-catenin signaling pathway inhibitor, on the invasive and migratory capabilities of overexpressed AQP5 cells and downstream molecules were measured. Results1. The expression of AQP5 in the MDA-MB-231 cells was significantly higher than that in the MCF-10A cells. 2. Up-regulation of AQP5 significantly promoted the proliferation, migration and invasion of TNBC cells, while inhibited the cell apoptosis; in addition, up-regulation of AQP5 increased the expression of Bcl-2 and decreased the expression of Caspase-3. However, knockdown of AQP5 presented the adverse effects of AQP5 overexpression. 3. Overexpressed AQP5 induced the overexpression of EMT-related factors, which further promoted the migration and invasion of cells. 4. Overexpression of AQP5 could up-regulate the expression of β-catenin in the nucleus followed by increasing the expression levels of downstream genes in Wnt/β-catenin signaling pathway. Moreover, ICG-001, the inhibitor of Wnt/β-catenin signaling pathway, could significantly attenuate the effect of overexpression of AQP5 on cells, further confirming that AQP5 may promote the proliferation, migration and invasion of TNBC cells by activating Wnt/β-catenin signaling pathway. ConclusionsIn the TNBC cells, AQP5 modulates the expression levels of EMT-related proteins through activation of Wnt/β-catenin signaling pathway, thus enhancing the cell proliferation, migration and invasion while inhibiting the cell apoptosis.

Understanding the physics of eigenvalue-eigenfunction problems: Rotating beam problem

Eigenvalue-eigenfunction problems frequently appear in many physical areas. Some mathematical experience is needed to identify whether the differential system is an eigenvalue-eigenfunction problem or not. Apart from the mathematical nature of the problem, the eigenvalue-eigenfunction solutions have physical interpretations which have to be addressed properly for real problems. The rotating beam problem is treated to exploit the mathematical and physical nature of such problems and the conditions to divert from the eigenvalue-eigenfunction problem. The rotation of a beam about its symmetry axis along its length and about another axis parallel to its symmetry axis changes the nature of the problem. While the former is an eigenvalue-eigenfunction problem, the latter is not. The interpretations of the physical consequences of the solutions are discussed in detail. The problem can be used as supplementary material in undergraduate courses such as differential equations, mechanics and dynamics.

Precise integration boundary element method for solving nonlinear transient heat conduction problems with temperature dependent thermal conductivity and heat capacity

The precise integration boundary element method (PIBEM) is used to solve nonlinear transient heat conduction problems with temperature dependent thermal conductivity and heat capacity in this article. At first, a boundary-domain integral equation can be obtained by using the fundamental solution for steady linear heat conduction problems. Secondly, the domain integrals are converted into boundary integrals by using the radial integration method to get a pure boundary integral equation, which can retain the advantage of dimension reduction of the boundary element method. Then, after discretization, a first-order nonlinear differential equation system in time is obtained, and the precise integration method with predictor-corrector technique is used to solve the system of nonlinear equations. In the end, several numerical examples are presented to verify the accuracy and stability of the presented method. The results obtained by the presented method are less affected by the time step size and satisfactory results can be obtained even for a large time step size.

The fluctuational transition mechanism of non-hyperbolic chaotic invariant sets

In order to reveal the general escape mechanism of non-hyperbolic chaotic invariant sets under both weak noise limit and finite noise intensity, the simplest example of Hénon map, which represents the stretching and folding of the phase space, is taken to study the escape mechanism under two kinds of global bifurcation: the fractal boundary crisis and the attractor contact crisis. In this paper, we revealed the general exit mechanism by analyzing the escape paths derived from the shooting method and Monte Carlo simulation. Finally, to further demonstrate universality, an example of high-dimensional differential dynamical system, namely the transient chaos Duffing oscillator, is examined, which underscores the main idea of this paper analyzing specific deterministic structures not only enhances the comprehensibility of the escape process, but also allows for predictions of general escape behavior under weak noise intensity.

Periodic orbits and non-existence of $$C^1$$ first integrals for analytic differential systems exhibiting a zero-Hopf bifurcation in $$\mathbb {R}^4$$

Periodic orbits and non-existence of $$C^1$$ first integrals for analytic differential systems exhibiting a zero-Hopf bifurcation in $$\mathbb {R}^4$$

Bathymetric survey and volumetric analysis of Bakolori dam reservoir North West Nigeria

Many approaches have been devised for measuring water depth in bathymetric surveys, which are used to assess sediment deposited directly in lakes and reservoirs. This technique is mostly used to calculate the reservoir's capacity and the amount of sedimentation that occurs there. The results of the volumetric analysis and bathymetric survey of the Bakolori Dam reservoir, which is situated in Northwestern Nigeria, are presented in this work. During this investigation, the differential Global Positioning System receiver (GPS), automatic level measuring tool, echo-sounder, and engine boat were utilized. ArcGis 10.0 was used to analyze the acquired data. In order to evaluate the reservoir capacity loss during the Bakolori reservoir's operating period as of 1983, the designed computed reservoir capacity was compared with the current bathymetry survey. At a spillway crest elevation of 334 meters above mean sea level (amsl), the reservoir's initial capacity is reported as 430 million cubic meters (MCM). However, the capacity has since been revised to 291 meters at the same level, with a volume change of 139,269,495 m3. As a result, the reservoir's volume changed by approximately 139 MCM during a thirty-five (35) year period of service, representing a loss of storage capacity, while the annual siltation rate was around 3,979,128 m3/a. This indicated that 33% of the reservoir's storage capacity had silted up. This is demonstrated by the aquatic weeds that have grown to a height of 334.21 meters on the lake's surface, rising from the lakebed. Therefore, to prevent total siltation, the dam reservoir needs to be adequately dredged.

Existence results of a nonlocal impulsive fractional stochastic differential systems with Atangana–Baleanu derivative

Existence results of a nonlocal impulsive fractional stochastic differential systems with Atangana–Baleanu derivative

Influence of the Effective Reproduction Number on the SIR Model with a Dynamic Transmission Rate

In this paper, we examine the epidemiological model B-SIR, focusing on the dynamic law that governs the transmission rate B. We define this dynamic law by the differential equation B′/B=F⊕−F⊖, where F⊖ represents a reaction factor reflecting the stress proportional to the active group’s percentage variation. Conversely, F⊕ is a factor proportional to the deviation of B from its intrinsic value. We introduce the notion of contagion impulse f and explore its role within the model. Specifically, for the case where F⊕=0, we derive an autonomous differential system linking the effective reproductive number with f and subsequently analyze its dynamics. This analysis provides new insights into the model’s behavior and its implications for understanding disease transmission.

A case study on effect of variable viscosity on non-newtonian nanofluid over an extendable cylindrical surface by utilizing Reynold’s model

Current artifact aims to investigate attributes of nanofluidic transport mechanism developed for dynamics of Williamson liquid over an extendable cylinder with novel physical aspects of magnetic field. Additionally, Newtonian heating is accounted at cylinder surface and Reynold viscosity model is implemented to delineate the aspects of temperature dependent viscosity. Thermophoresis, Brownian motion, and variable viscosity effects have been studied for heat and mass transfer analysis. Formulation of problems characterized by governing physical laws is conceded in the form of dimensional partial differential setup by executing boundary layer approach. Conversion of attained differential system into ODE’s is engrossed by capitalizing similarity transformations. Numerical procedures (Runge-Kutta and shooting) are implemented to resolve the problem and to attain solution. Effectiveness of sundry variables on associated distributions is revealed through graphical and tabular manner. Quantities of interest are also manipulated against the involved variables. Comparison of results to certify the numerical code is checked by making assessment with existing studies and found remarkable agreement.