Unique Equilibrium Point Research Articles

Mathematical modelling and projection of Buruli ulcer transmission dynamics using classical and fractional derivatives: A case study of Cameroon

The goal of this work is to derive and study a new model which traduces the transmission dynamics of the Buruli ulcer (BU), in which we replace mass action incidences with standard incidences, consider latent period, and by using both integer and fractional derivatives. We first present reported data of Buruli in Cameroon between 01/01/2001 and 31/12/2014, and the statistical model that approximates real data as closely as possible, and thus make a forecasting in the next sixteen years. After that, we formulate a Susceptible humans-Exposed humans-Infectious humans-Recovered humans-Susceptible humans; Susceptible vectors-Infectious vectors (SEIRS-SI) type compartmental model of Buruli Ulcer using integer derivative. We compute the basic reproduction number denoted by R0 and prove the asymptotic stability of the Buruli-free equilibrium whenever R0<1. Then, for R0>1, we prove the existence of an unique endemic equilibrium point and its global stability using the general theory of Lyapunov. We perform parameter estimation to calibrate our model using real data from Cameroon, while sensitivity analysis is conducted to determine important parameters in the BU dynamics. From this model calibration, we obtain R0=2.0843 which is greater than one and implies that BU ulcer is endemic in the country. To determine whether if or not the BU admits one or multiple waves, we compute the strength number denoted by A0. We find that A0≈17>0 which means that Bu admits multiple epidemic waves. We then replace integer derivative with the Caputo fractional derivative. Asymptotic stability of equilibrium points are also performed for the fractional model. This follows by the proof of existence and the uniqueness of solutions of the fractional model. We then construct a numerical scheme based on the Adams–Bashforth–Moulton (ABM) Method. Theoretical results are validated from numerical simulations. These last also permits to evaluate the impact of varying fractional order α in the BU dynamics. This permits to conclude that for α<1, reported cases are closer to model prediction.

Dynamics, circuit design, feedback control of a new hyperchaotic system and its application in audio encryption

In this study, a 4D hyperchaotic system is constructed based on the foundation of a 3D Lü chaotic system. The newly devised hyperchaotic system possesses a sole equilibrium point, showcasing a simplified system structure that reduces complexity. This simplification offers a clearer opportunity for in-depth analysis of dynamic behaviors in the realm of scientific research. The proposed hyperchaotic system undergoes an in-depth examination of its dynamical characteristics, including chaotic attractors, equilibrium point stability, Lyapunov exponents’ spectrum, and bifurcation diagram. Numerical analysis results reveal that the attractor of this hyperchaotic system exhibits highly complex, non-periodic, and fractal structural dynamics. Its motion demonstrates extreme sensitivity and randomness, even within a wide range of variations in parameter d, affirming its hyperchaotic properties with two positive Lyapunov exponents. Hyperchaotic bifurcation diagrams typically exhibit highly intricate structures, such as fractals, branches, and period doubling characteristics, signifying that even minor parameter adjustments can lead to significant changes in system behavior, presenting diversity and unpredictability. Subsequently, to further investigate the practical utility of this hyperchaotic system, a linear feedback control strategy is implemented. Through linear feedback control, the hyperchaotic system is stabilized at its unique equilibrium point. Experimental validation is conducted using both computer software simulation Matlab, electronic circuit simulation Multisim, and embedded hardware STM32. The results of these experiments consistently align, providing theoretical support for the application of this hyperchaotic system in practical domains. Finally, leveraging the hyperchaotic keys generated by this hyperchaotic system, audio encryption is achieved using a cross-XOR algorithm, which is then realized on the embedded hardware platform STM32. The results show that the audio encryption scheme based on the hyperchaotic system is feasible, and the method is simple to implement, has nonlinear characteristics and certain algorithm complexity, which can be applied to audio encryption, image encryption, video encryption, and more.

Finite-time error bounds for distributed linear stochastic approximation

This paper considers a novel multi-agent linear stochastic approximation algorithm driven by Markovian noise and general consensus-type interaction, in which each agent evolves according to its local stochastic approximation process which depends on the information from its neighbors. The interconnection structure among the agents is described by a time-varying directed graph. While the convergence of consensus-based stochastic approximation algorithms when the interconnection among the agents is described by doubly stochastic matrices (at least in expectation) has been studied, less is known about the case when the interconnection matrix is simply stochastic. For any uniformly strongly connected graph sequences whose associated interaction matrices are stochastic, the paper derives finite-time bounds on the mean-square error, defined as the deviation of the output of the algorithm from the unique equilibrium point of the associated ordinary differential equation. For the case of interconnection matrices being stochastic, the equilibrium point can be any unspecified convex combination of the local equilibria of all the agents in the absence of communication. Both the cases with constant and time-varying step-sizes are considered. In the case when the convex combination is required to be a straight average and interaction between any pair of neighboring agents may be uni-directional, so that doubly stochastic matrices cannot be implemented in a distributed manner, the paper proposes a push-sum-type distributed stochastic approximation algorithm and provides its finite-time bound for the time-varying step-size case by leveraging the analysis for the consensus-type algorithm with stochastic matrices and developing novel properties of the push-sum algorithm. Distributed temporal difference learning is discussed as an illustrative application.

Stability Assessment of Microalgal Photobioreactors for Carbon Dioxide Capture under Dilution Rate Constraints

Algal cultivation is a sustainable approach which can be used not only for carbon dioxide sequestration but also for making useful products in many industries. To facilitate the widespread adoption of this approach, the current research studies the stable control of closed photobioreactors (PBRs) cultivating microalgae. More specifically, a proportional–integral (PI) controller is employed for the tracking of the microalgal concentration to a desired reference corresponding to a required amount of sequestrated carbon dioxide. In the presence of the practically positive and bounded constraints of the dilution rate, the stability and reference tracking of the closed-loop PBR system needs to be assessed. This work then derives conditions under which a unique equilibrium point exists and the closed-loop PBR system is asymptotically stable around such an equilibrium point. The derived theoretical results are validated and illustrated through numerical simulations for PBRs of the microalgae Chlorella vulgaris.

Global Attractor in the Positive Quadrant of the Lotka–Volterra System in ℝ2

We characterize when the unique equilibrium point of the positive quadrant of the twodimensional Lotka–Volterra system is a global attractor in that quadrant. Additionally, we classify the phase portraits of this class of Lokta–Volterra system.

Globally asymptotic stability analysis for memristor‐based competitive systems of reaction–diffusion delayed neural networks

The main aim of this investigation is to present a mathematical criterion that guarantees the global asymptotic stability of special class of nonlinear neural systems. Indeed, a memristor‐based competitive nonlinear reaction–diffusion Cohen–Grossberg neural network system is chosen to be investigated. Chasing the accuracy and comprehensiveness, our model has to be improved with the following elements: neutrality of the nonlinear part and being effected under time delays in the both of straight and neutral parts of the nonlinearity. Since the proposed neural network is novel, so we need to make use of a novel Lyapunov functional correspondingly to globally asymptotic stabilization of the neural network under investigation. Prior to this process, it is necessary to present a uniqueness criterion for solutions of the proposed neural network. To this aim, the well‐known M‐matrix technique of the linear algebra is applied that guarantees existence of a unique equilibrium point of the considered Cohen–Grossberg neural network. Numerical perspective of this investigation leads us to some numerical prototypes that justify our stability criterion is applicable. This investigation will be finalized with interesting discussion on the nature of time delays that are characterized as fundamental elements in modeling of the neural networks.

Dynamical analysis of a diffusion plant-wrack model with delay

In this paper, in view of the senescence of plant and the decay of wrack, time delays are introduced into the plant-wrack model. The effects of wrack decay and time delay on the dynamical behaviors of the diffusive plant-wrack model are studied analytically and numerically. When the delay is zero, the wrack decay will induce the change of stability of the unique equilibrium point, further lead to the occurrence of the Hopf bifurcation and the Turing instability. When the delay is present, the conditions for the occurrence of the Hopf bifurcation are established. By comparing the results of the model without and with delay, it is found that the increases of delay may induce no stability switches, a single stability switch or multiple stability switches, when the value of wrack decay can stabilize model with zero delay. When the value of wrack decay can destabilize model with zero delay, numerical simulations show that the small delay may cause homogeneous distributions of vegetation, while the larger delay may cause the emergence of periodic oscillation of vegetation. The obtained results provide a basis for understanding the spatiotemporal evolution of such a plant-wrack model with delay.

Bifurcation analysis of an SIS epidemic model with a generalized non-monotonic and saturated incidence rate

In this paper, a new generalized non-monotonic and saturated incidence rate was introduced into a susceptible-infected-susceptible (SIS) epidemic model to account for inhibitory effect and crowding effect. The dynamic properties of the model were studied by qualitative theory and bifurcation theory. It is shown that when the influence of psychological factors is large, the model has only disease-free equilibrium point, and this disease-free equilibrium point is globally asymptotically stable; when the influence of psychological factors is small, for some parameter conditions, the model has a unique endemic equilibrium point, which is a cusp point of co-dimension two, and for other parameter conditions the model has two endemic equilibrium points, one of which could be weak focus or center. In addition, the results of the model undergoing saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation as the parameters vary were also proved. These results shed light on the impact of psychological behavior of susceptible people on the disease transmission.

A New Cross Ring Neural Network: Dynamic Investigations and Application to WBAN

Wireless body area network (WBAN) is a crucial tool in modern medical areas. This refers to the intelligent interconnection of wireless sensor nodes installed outside or inside the human body to recover some human vital signs. The security and bandwidth saturation remain crucial problems of this technology. In this work, a cross-ring-based neural network model (CRNN) is obtained from the Hopfield neural network definition and its complex dynamics are deeply analyzed. The steady-state study shows that the model has a unique equilibrium point and its analysis shows that the model’s dynamics is self-excited. Based on the two-parameter Lyapunov spectrum, the set of synaptic weights has been quickly identified to illustrate hyperchaotic behavior of the CRNN. With the help of bifurcation plots, graphs of the Lyapunov spectrum, and phase portraits, we have characterized periodic, chaotic, and hyperchaotic patterns in the model. Furthermore, an experimental setup has been built using microcontroller technology to further support the results of the numerical simulations. A parallel compressive sensing algorithm combined with a nonlinear congruent generator has been used to show the application of the newly designed CRNN to medical image compression and security. Security and compression performances indicate an efficient scheme with the capability to resist various attacks and the capability to produce low-size data image from large-size data image. For instant from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$256\times 256 $ </tex-math></inline-formula> image size, an encrypted and compressed output image is obtained at the compression rate of 0.5, processing time of 0.147 ms, encryption throughput of 1783.3 MBits/s, for a 2.9-GHz processor, the number of cycles to process the algorithm is 1.62. Consequently, the proposal can be applied to WBANs.

Neimark–Sacker Bifurcation of a Discrete-Time Predator–Prey Model with Prey Refuge Effect

In this paper, we deduce a predator–prey model with discrete time in the interior of R+2 using a new discrete method to study its local dynamics and Neimark–Sacker bifurcation. Compared with continuous models, discrete ones have many unique properties that help to understand the changing patterns of biological populations from a completely new perspective. The existence and stability of the three equilibria are analyzed, and the formation conditions of Neimark–Sacker bifurcation around the unique positive equilibrium point are established using the center manifold theorem and bifurcation theory. An attracting closed invariant curve appears, which corresponds to the periodic oscillations between predators and prey over a long period of time. Finally, some numerical simulations and their biological meanings are given to reveal the complex dynamical behavior.

The Qualitative Analysis of Host–Parasitoid Model with Inclusion of Spatial Refuge Effect

The objective of this work was to investigate the dynamics of host–parasitoid model with spatial refuge effect. For this, two discrete host–parasitoid models were considered under spatial refuge effect. Suppose that a constant population of hosts may seek refuge and protection from an attack of parasitoids. We found the parametric factors affecting the existence of the equilibrium points and uniqueness of equilibrium points. A local stability analysis of host–parasitoid models was also carried out. Bifurcation theory was used to observe that the host–parasitoid models undergo Neimark–Sacker bifurcation. The effect of the existence of constant refuge effect on the local stability and bifurcation of models was also explored. Hybrid chaos control methodology was used to control the chaotic behavior of model. In addition, numerical simulations, bifurcation diagrams, and phase portraits of the models are also presented.

Modeling the potential influence of economic migration on Ebola virus disease transmission dynamics

IntroductionThe good health of the economy of developed countries is an attracting factor for economic migrants from poorer countries whose movements can be associated with the spread of diseases such as the Ebola virus disease.MethodsIn this paper, we formulate a model of Ebola virus disease that considers two patches with different economic statuses represented by the respective gross-national incomes of the two patches. First, we consider a one-directional movement from a poorer patch to a rich patch. Second, we consider a two-way migration model where people move between patches.ResultsThe steady states of the model are determined and analyzed. The analysis shows also that the disease free and the endemic equilibrium points are locally stable. The analysis also shows that a unique endemic equilibrium point exists in the poor patch when the reproduction number in this patch is greater than one and multiple endemic equilibria exist in the rich patch when the reproduction number in the poor patch is less than one. The model dynamics of the rich patch present a backward bifurcation which does not facilitate easy Ebola virus disease control. Sensitivity analysis is carried out to determine the most sensitive parameters that must be carefully estimated for the successful control of the Ebola virus disease.DiscussionNumerical simulations indicate a decrease in the number of infected individuals in the rich patch when movements of populations are limited through the improvement of the economy in the poor patch. So, the improvement of the economy of poorer countries may be critical in avoiding potential outbreaks of Ebola virus disease.

Dynamics and Stability of a Fractional-Order Tumor–Immune Interaction Model with B-D Functional Response and Immunotherapy

In this paper, we investigate a fractional-order tumor–immune interaction model with B-D function item and immunotherapy. First, the existence, uniqueness and nonnegativity of the solutions of the model are established. Second, the local and global asymptotic stability of some tumor-free equilibrium points and a unique positive equilibrium point are obtained. Finally, we use numerical simulation method to visualize and verify the theoretical conclusions. It is known that the fractional-order parameter β has a stabilization effect, and the tumor cells can be destroyed or controlled by using immunotherapy.

Joint Management of Compute and Radio Resources in Mobile Edge Computing: A Market Equilibrium Approach

Edge computing has been recently introduced to bring computational capabilities closer to end-users of modern network-based services, supporting existing and future delay-sensitive applications by effectively addressing the high propagation delay issue that affects cloud computing. However, the problem of efficiently and fairly managing the system resources presents particular challenges due to the limited capacity of both edge nodes and wireless access networks and the heterogeneity of resources and services’ requirements. To this end, we propose a techno-economic market where service providers act as buyers, securing both radio and computing resources to execute their associated end-users’ jobs while being constrained by a budget limit. We design an allocation mechanism that employs convex programming to find the unique market equilibrium point that maximizes fairness while ensuring that all buyers receive their preferred resource bundle. Additionally, we derive theoretical properties that confirm how the market equilibrium approach strikes a balance between fairness and efficiency. We also propose alternative allocation mechanisms and give a comparison with the market-based mechanism. Finally, we conduct simulations to numerically analyze and compare the performance of the mechanisms and confirm the market model's theoretical properties.

A Half-Tangent Phase-Locked Loop for Variable-Frequency Grids of the More Electric Aircraft

In this article, a half-tangent phase-locked loop (HTan-PLL) is proposed for variable-frequency grids of more electric aircrafts. First, the pros and cons of the synchronous reference frame PLL (SRF-PLL) are discussed in a nonlinear framework. To address the disadvantages, a half-tangent phase detector (PD) is developed and a novel PLL is then proposed. Subsequently, a phase portrait-based analysis and a Lyapunov function-based argument are provided. It is proven that the HTan-PLL has a good performance in terms of steady-state accuracy and fast dynamics, even under large frequency changes. Compared with the conventional SRF-PLL, the proposed HTan-PLL outperforms with the following features: faster dynamics than the SRF-PLL when the frequency varies largely; no unstable equilibrium points, while the SRF-PLL has infinite unstable equilibrium points; and when the grid frequency jumps widely, the HTan-PLL converges to the unique equilibrium point (0,0). By contrast, the SRF-PLL goes to different equilibrium points. Experimental tests are provided, which verify the superior performance of the HTan-PLL.

Multistability analysis of quaternion-valued neural networks with cosine activation functions

It is significant to design a system with high storage capacity for associative memory and pattern recognition. To address this issue, this paper first proposes a quaternion-valued neural network (QVNN) model with multiple equilibrium points in which the cosine function is used as the activation function of QVNN. Then, based on the Brouwer fixed point theorem and the geometric properties of the activation function, sufficient conditions for QVNN to have unique equilibrium points, finite equilibrium points, and countable infinite equilibrium points are obtained, respectively. Furthermore, sufficient conditions for the exponential stability of equilibrium points are derived, and the attraction basins of the stable equilibrium points are given. Finally, two numerical examples are given to confirm the validity of the proposed theoretical results.

A New Simple 6D Hyperchaotic System with Hyperbolic Equilibrium and Its Electronic Circuit

This paper represents a sequential design of controlling from a 3D chaotic system to a simple 6D hyperchaotic system by adding three linear controllers via state feedback control and coupling strategies. The new 6D system has a unique non-hyperbolic equilibrium point with four positive Lyapunov exponents (LE). Moreover, some dynamic properties are analyzed which inclined equilibrium and their stability, dissipative, conservative, and multistability. In addition, the corresponding analog electronic circuit is designed and implemented to verify the new simple 6D hyperchaotic system. Finally, the NI Multisim simulation results observed from the digital oscilloscope are consistent with the mathematical simulation based on the Matlab R2021a.

Complex dynamics and impulsive control of a chemostat model under the ratio threshold policy

In this paper, we study the periodic solution and global stability of a chemostat model under impulsive control. First, we investigate the positivity and boundedness of the solution of the controlled system. Second, we find the periodic solution of the controlled system by employing the Poincare map and Brouwer’s fixed-point theorem. Furthermore, we obtain a sufficient condition which allows the existence of orbitally stable order-k periodic solutions (k=1,2) by using the comparison method and the vector field analysis. We find that the controlled system exists a unique positive equilibrium point that is globally asymptotically stable (GAS) under some conditions. Finally, we provide two numerical examples to verify the correctness of the theoretical results.

A study of Wiener-Hopf dynamical systems for variational inequalities in the setting of fractional calculus

&lt;abstract&gt;&lt;p&gt;In this paper, we consider a new fractional dynamical system for variational inequalities using the Wiener Hopf equations technique. We show that the fractional Wiener-Hopf dynamical system is exponentially stable and converges to its unique equilibrium point under some suitable conditions. We also discuss some special cases, which can be obtained from our main results.&lt;/p&gt;&lt;/abstract&gt;

Mathematical Modeling and Analysis of Corruption Dynamics in Kenya

This study presents a mathematical model that aims to study corruption in Kenya. The model is validated both epidemiologically and mathematically, with all solutions demonstrating positivity and boundedness within a meaningful set of initial conditions. By investigating unique corruption-free and endemic equilibrium points, as well as computing the basic reproduction number, we assess the system’s behavior. Our analysis reveals that a locally asymptotically stable corruption-free equilibrium point is achieved when the reproduction number is below one, while a locally asymptotically stable endemic equilibrium point is attained when the reproduction number exceeds one. Simulation results confirm the agreement with analytical findings. This research enhances our understanding of corruption dynamics and provides valuable insights for designing effective anti-corruption strategies in Kenya.