Stability Of Equilibrium Point Research Articles

Study of chronic myeloid leukemia with T-cell under fractal-fractional order model

Abstract This research work is devoted to investigate myeloid leukemia mathematical model. We give some details about the existence of trivial and nontrivial equilibrium points and their stability. Also, local asymptotical stability of disease-free and endemic equilibrium points is discussed. Also, positivity of the solution has been discussed. Some sufficient results are achieved to study the local existence and uniqueness of solution to the considered model for Mittag–Leffler kernel using the Banach contraction theorem. Three numerical algorithms are derived in obtaining the numerical solution of suggested model under three different kernels using Adams–Basforth technique. Numerical results have been presented for different fractals and fractional orders to show the behavior of the proposed model.

Evolutionary Game Analysis of Governments’ and Enterprises’ Carbon-Emission Reduction

With the increasingly serious problem of global climate change, many countries are positively promoting carbon-emission-reduction actions. In order to deeply explore the interaction between enterprises’ carbon-emission reduction and governments’ regulation, this paper builds evolutionary game models between governments and enterprises under the reward-and-punishment mechanism. The peer-incentive mechanism is introduced to incentivize enterprises to reduce carbon emissions and coordinate governments and enterprises. The evolutionary-stability strategies are obtained by solving the evolutionary game models. The stability of equilibrium points under different situations is theoretically and numerically studied. The results show that the existence of peer incentives makes enterprises more inclined to positively reduce carbon emissions and governments more inclined to positively regulate. A sufficiently large peer fund can always encourage enterprises to choose positive carbon-reduction emission strategies, while governments choose positive regulation strategies. Not only the increasing rewards and fines but also lowering regulatory costs will promote carbon-emission-reduction behaviors of enterprises. Peer incentives are more effective in promoting positive emission reduction of enterprises compared with rewards and punishments. This study can provide important guidance for governments to formulate regulatory strategies and for enterprises to formulate emission-reduction strategies.

Investigating the Dynamic Behavior of Integer and Noninteger Order System of Predation with Holling’s Response

The paper’s primary objective is to examine the dynamic behavior of an integer and noninteger predator–prey system with a Holling type IV functional response in the Caputo sense. Our focus is on understanding how harvesting influences the stability, equilibria, bifurcations, and limit cycles within this system. We employ qualitative and quantitative analysis methods rooted in bifurcation theory, dynamical theory, and numerical simulation. We also delve into studying the boundedness of solutions and investigating the stability and existence of equilibrium points within the system. Leveraging Sotomayor’s theorem, we establish the presence of both the saddle-node and transcritical bifurcations. The analysis of the Hopf bifurcation is carried out using the normal form theorem. The model under consideration is extended to the fractional reaction–diffusion model which captures non-local and long-range effects more accurately than integer-order derivatives. This makes fractional reaction–diffusion systems suitable for modeling phenomena with anomalous diffusion or memory effects, improving the fidelity of simulations in turn. An adaptable numerical technique for solving this class of differential equations is also suggested. Through simulation results, we observe that one of the Lyapunov exponents has a negative value, indicating the potential for the emergence of a stable-limit cycle via bifurcation as well as chaotic and complex spatiotemporal distributions. We supplement our analytical investigations with numerical simulations to provide a comprehensive understanding of the system’s behavior. It was discovered that both the prey and predator populations will continue to coexist and be permanent, regardless of the choice of fractional parameter.

Data-driven economic MPC with asymptotic stability and strong duality verification using Hankel matrix

We consider the problem of dynamic regulation with an economic cost function to control unknown linear systems, in which improving the economic performance and guaranteeing the stability of economical optimal equilibrium point are control objectives. A data-driven economic MPC scheme is presented using measured input-output trajectories without a prior system identification step. Our method uses Hankel matrices which include one input-output data trajectory for prediction in economic MPC, while persistently exciting of the input generating the data is needed. One of the novelties of the presented framework is directly verifying the strong duality property from input-output trajectory with the general cost function, considered as the supply rate. This is used to find a Lyapunov function for data-driven economic MPC. Under the strong duality assumption, asymptotic stability of the economical optimal equilibrium point for the closed-loop system with terminal equality constraint is guaranteed. The proposed data-driven economic MPC approach needs only persistently exciting data trajectory along with an upper bound on the system order and need no model description and no online parameter estimation. The proposed scheme applicability compared to the existing model-based economic MPC and data-driven MPC is illustrated for continuous stirred tank reactor (CSTR) and a numerical example and the robustness of the proposed scheme is evaluated in the case of measurement noise, as well as nonlinear model for CSTR system.

Energy flow investigations of Rayleigh-Plesset equation for cavitation simulations

Nonlinear Rayleigh-Plesset Equation (RPE) for cavitation simulations is investigated using energy-flow theory. Nondimensional RPE affected by Reynolds-number, surface-tension, bubble-pressure, and liquid static-dynamic pressure ratio is derived to examine its equilibrium points with stability, possible periodical/chaotic motions by the energy-flow criteria. An example is numerically analyzed to illustrate the developed method. Five cases of the example reveal: 1) It is a damped system, where initial disturbances are gradually reduced with its phase point tends to its stable equilibrium-point; 2) Reynolds-number affects the damping of system, large one corresponds small damping; 3) Bubble-pressure, surface-tension and liquid static-dynamic pressure ratio affect the position of equilibrium point; 4) Periodical orbit appears only in forced vibrations, in which free vibration is reduced with time forward and the system finally shows a stable periodical oscillation; 5) Energy flow criteria for chaotic motions is not reached, and there are no chaotic motions for the cases of example. Numerical simulations confirm the developed energy-flow means with available computer code is effective to investigate generalized RPEs to reveal their inherent characteristics affected by required parameters in engineering cavitation analysis and designs, such as considering mass transports across the boundary of bubble by evaporation or condensation of liquids.

Estimating Markov Chain Mixing Times: Convergence Rate Towards Equilibrium of a Stochastic Process Traffic Assignment Model

Network equilibrium models have been extensively used for decades. The rationale for using equilibrium as a predictor is essentially that (i) a unique and globally stable equilibrium point is guaranteed to exist and (ii) the transient period over which a system adapts to a change is sufficiently short in time that it can be neglected. However, we find transport problems without a unique and stable equilibrium in the literature. Even if it exists, it is not certain how long it takes for the system to reach an equilibrium point after an external shock onto the transport system, such as infrastructure improvement and damage by a disaster. The day-to-day adjustment process must be analysed to answer these questions. Among several models, the Markov chain approach has been claimed to be the most general and flexible. It is also advantageous as a unique stationary distribution is guaranteed in mild conditions, even when a unique and stable equilibrium does not exist. In the present paper, we first aim to develop a methodology for estimating the Markov chain mixing time (MCMT), a worst-case assessment of the convergence time of a Markov chain to its stationary distribution. The main tools are coupling and aggregation, which enable us to analyse MCMTs in large-scale transport systems. Our second aim is to conduct a preliminary examination of the relationships between MCMTs and critical properties of the system, such as travellers’ sensitivity to differences in travel cost and the frequency of travellers’ revisions of their choices. Through analytical and numerical analyses, we found key relationships in a few transport problems, including those without a unique and stable equilibrium. We also showed that the proposed method, combined with coupling and aggregation, can be applied to larger transport models. History: This paper has been accepted for the Transportation Science Special Issue on the 25th International Symposium on Transportation and Traffic Theory. Funding: This study was financially supported by the Japan Society for the Promotion of Science [Grant-in-Aid 20H00265]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/trsc.2024.0523 .

Dynamical behavior of fractal-fractional order monkeypox virus model

A virus that infects both humans and animals, monkeypox is common in many West African countries and has sporadically spread to other parts of the world. There are serious public health concerns around the globe as a result of the recent spike in cases of monkeypox among both endemic and non-endemic populations. This paper proposes to use the Caputo sense of fractal fractional-order derivatives to examine the dynamics of monkeypox transmission. The study uses Schauder’s fixed point theorem to evaluate the solutions qualitatively and establish their uniqueness inside the model. The next generation matrix technique is used to compute the essential reproduction number. The stability of equilibrium points is further investigated in the research, and a sensitivity analysis of model parameters is carried out. When the basic reproduction number R0 is less than 1, the equilibrium without infections is locally stable. Also, when R0 exceeds 1, this equilibrium becomes unstable. The proposed model incorporates Ulam–Hyers stability through nonlinear functional analysis. To estimate solutions for the fractal-fractional order monkeypox model, the Lagrange’s interpolation method is utilized. In addition, numerical simulations are carried out to examine the influence of some parameters on the overall dynamics of the model. Numerical simulations are performed using MATLAB software to exemplify the model behavior in the context of the Nigerian case study. The graphical representations suggest that the fractal fractional order affects the dynamics of the monkeypox. The findings indicate that isolation of infected individuals in the human population helps to reduce disease transmission.

Dynamical analysis of a stay cable with a nonlinear energy sink and time-delayed feedback control

Stay cables exist commonly in engineering and can exhibit sustainable and large vibrations under external excitations. The unexpected oscillations always lead to the fatigue and damage of structures, resulting in the economical loss. This paper studies the vibration control of a stay cable system with a nonlinear vibration absorber and time delay control. The governing equations of motion of the stay cable system under harmonic excitations are given. The stability and different bifurcations of the trivial equilibrium point of the system are analyzed. The bifurcation diagrams with respect to the control parameters are illustrated. The frequency responses of the forced system are obtained by using the harmonic balance method. Numerical simulations are performed to validate the obtained results and various dynamical phenomena are shown, such as the multi-period oscillations, quasi-periodic responses and chaotic motions. The effects of the time-delayed feedback control on the nonlinear dynamics and vibration reduction of the system are extracted. It is shown that the proper values of the time delay and feedback gain can improve the vibration reduction performance of the system, such as the suppression of the resonance peaks and elimination of the complex responses.

Comparison of Homoclinic Bifurcations Between Grid-Following and Grid-Forming Converters

A second-order model is developed to describe the essential behavior of the grid-following converter (GFLC) and the grid-forming converter (GFMC) for studying the synchronization characteristics of grid-connected converter systems. In this paper, a general set of criteria in terms of the manifolds and saddle quantity is derived for studying the homoclinic bifurcation behavior of GFLCs and GFMCs. It is shown that three unstable periodic orbits and two homoclinic bifurcations may exist for the GFLC, while only one stable periodic orbit and one homoclinic bifurcation may exist for the GFMC. It is also found that after the onset of the first homoclinic bifurcation, the stable equilibrium point (SEP) is surrounded by the stable manifolds of the unstable equilibrium point (UEP), which forms the SEP's basin of attraction, for both GFLCs and GFMCs. In this case, the converters may lose their synchronization if the trajectory crosses over the UEP. Therefore, system parameters should be designed to avoid the onset of the aforementioned bifurcation so that the SEP's basin of attraction can be significantly enlarged to contain the UEP. As a result, the GFLC or GFMC can eventually resynchronize with the grid even when its trajectory passes the UEP. Finally, experimental results are provided to verify these theoretical findings.

Nonlinear dynamics and optimal control of a fractional order cotton leaf curl virus model incorporating climate change influences

This work investigates the transmission dynamics of a proposed fractional-order cotton leaf curl virus (CLCuV) model considering the effects of climate change. High temperatures resulting from climate change and global warming significantly impact the cotton crop and CLCuV transmission, causing substantial economic losses for farmers and agricultural industries. The study aims to address these impacts and contribute to the Sustainable Development Goals (SDGs) by promoting well-being and healthy lives. The characteristics of the CLCuV model’s solution are examined, including equilibrium points and the basic reproduction number. The stability of equilibrium points is analyzed in relation to key parameters, and sensitivity analysis of controllable parameters is performed. Additionally, optimal management strategies for controlling CLCuV spread are explored, aiming for minimal cost, improved health, and reduced climate change hazards. Numerical simulations are conducted to validate the theoretical findings.

Wiener and Lévy processes to prevent disease outbreaks: Predictable vs stochastic analysis

The study considers stochastic behavior along with modeling, mathematical analysis, theory development, and numerical simulation of the COVID-19 virus. To evaluate current trends and make future projections regarding the basic reproduction number and infection case, we have taken the modified five-compartment SEIRD mathematical model. Since the basic reproduction number R0 is not sufficient to predict the outbreak, we applied the Itô stochastic differential equations (SDEs) along with the Weiner process and Lévy jump to investigate the nature of the disease outbreak. Since COVID-19 is an infectious disease caused by the severe acute respiratory syndrome coronavirus 2 “(SARS-CoV-2)”, that has common symptoms including fever, cough, and difficulty breathing, we have illustrated the boundedness and positivity of solutions and investigated the stability of endemic and disease-free equilibrium points. The findings show that the dynamics of disease epidemics are influenced by contact patterns. For all the models, the threshold quantity, R0 is calculated which is the key reason to prove the global and local stability analysis of disease-free equilibrium and endemic equilibrium points. Using the least squares method for data fitting, we have performed a case study on COVID-19 data in Italy for this article. A sensitivity analysis is part of our investigation to find important factors. This paper investigates a stochastic epidemic model (SDE) using Lévy jumps and Weiner processes. Local and global stability in a stochastic setting are examined in the analysis. To view the analytical study, a multitude of numerical results are achieved. The study is important to comprehend the stochastic behavior of the virus in the contagious model in order to prevent similar disease outbreaks in the future.

Extreme Homogeneous and Heterogeneous Multistability in a Novel 5D Memristor-Based Chaotic System with Hidden Attractors

This paper proposes a novel five-dimensional (5D) memristor-based chaotic system by introducing a flux-controlled memristor into a 3D chaotic system with two stable equilibrium points, and increases the dimensionality utilizing the state feedback control method. The newly proposed memristor-based chaotic system has line equilibrium points, so the corresponding attractor belongs to a hidden attractor. By using typical nonlinear analysis tools, the complicated dynamical behaviors of the new system are explored, which reveals many interesting phenomena, including extreme homogeneous and heterogeneous multistabilities, hidden transient state and state transition behavior, and offset-boosting control. Meanwhile, the unstable periodic orbits embedded in the hidden chaotic attractor were calculated by the variational method, and the corresponding pruning rules were summarized. Furthermore, the analog and DSP circuit implementation illustrates the flexibility of the proposed memristic system. Finally, the active synchronization of the memristor-based chaotic system was investigated, demonstrating the important engineering application values of the new system.

Dynamic analysis and optimal control strategies of a predator–prey mathematical model for the pest eradication in oil palm plantation

Oil palm cultivation stands as a crucial industry in Indonesia, significantly contributing to the nation’s economy by generating employment opportunities and fostering social welfare for communities residing near plantations. Despite its economic importance, oil palm plantations face various challenges, with one prominent issue being the infestation of nettle caterpillar pests. These pests cause leaf skeletonization, resulting in a staggering 36% reduction in oil palm productivity over a two-year period. This paper explores diverse strategies for pest management in oil palm plantations, encompassing biological control through the stimulation of natural predators, mechanical control involving the collection and incineration of cocoons, and chemical control through pesticide application. The research introduces a predator–prey mathematical model for oil palm plantation pests, where the leaf area serves as the primary food source for caterpillars, acting as prey. Through dynamic model analysis, four equilibrium points are identified, with interconnected conditions dictating their existence and stability. These conditions are visually represented in a bifurcation plane, providing concise information. The study further includes bifurcation diagrams of equilibrium points to elucidate the influence of each parameter on pests, predators, and the leaf area of oil palm plants. Additionally, sensitivity analysis of the stable interior equilibrium point is conducted to understand the impact of individual parameters. The paper extends its investigation to optimal control strategies, evaluating six scenarios categorized into two population conditions: with predators and without predators. Within each population condition, three control strategies are considered—chemical control only, mechanical control only, and a combination of chemical and mechanical control. Simulation results from the optimal control study reveal that the presence of natural predators emerges as a pivotal strategy in effectively managing nettle caterpillars. Notably, the control of resistant pests through pupa incineration has a substantial impact on reducing the caterpillar population in subsequent life cycles.

On a Family of Hamilton–Poisson Jerk Systems

In this paper, we construct a family of Hamilton–Poisson jerk systems. We show that such a system has infinitely many Hamilton–Poisson realizations. In addition, we discuss the stability and we prove the existence of periodic orbits around nonlinearly stable equilibrium points. Particularly, we deduce conditions for the existence of homoclinic and heteroclinic orbits. We apply the obtained results to a family of anharmonic oscillators.

Coexistence and locally exponential stability of multiple equilibrium points for fractional-order impulsive control Cohen–Grossberg neural networks

Different from the existing multiple Mittag-Leffler stability or multiple asymptotic stability, the multiple exponential stability, which has explicit and faster convergence rate, is investigated in this paper for fractional-order impulsive control Cohen–Grossberg neural networks. First, by using the definition of Dirac delta function, the fractional-order control Cohen–Grossberg neural networks are translated into fractional-order impulsive neural networks, in which pulse effect relies on the fractional order of the addressed system. Then, based on maximum norm, 1−norm and general q−norm (q=2n), a series of novel criteria are obtained respectively to ensure that such n−neuron neural networks can have ∏i=1n(2Li+1) total equilibrium points and ∏i=1n(Li+1) locally exponentially stable equilibrium points, by utilizing the known fixed point theorem, the method of average impulsive interval, the theory of fractional-order differential equations, and the method of Lyapunov function. This paper’s investigation reveals the effects of impulsive function, impulsive interval, and fractional order on the dynamic behaviors. Finally, theoretical results are shown to be effective by four illustrative examples.

Players' strategy selection in co-governance and supervision of internet platforms' monopolistic behaviors: A study on new media participation.

Incidents of monopolies among Internet platforms have seriously endangered the development of the market economy, public interest, and social fairness, making it a highly discussed topic of broad public concern. Preventing such incidents requires not only a comprehensive supervision system by governments, but also contributions from other relevant parties. The new media environment has provided a new platform to support such joint supervision from multiple parties. As such, this study constructed an evolutionary game model involving the government, Internet platforms, new media, and the public to explore the stable equilibrium point of players' strategy selections. The stability of the strategy combinations was tested using Lyapunov's first stability method, and MATLAB 2021b was used to conduct simulation analysis of the impact of each decision variable on players' strategy selection. The results showed that (1) new media participation in co-governance and public complaints/reports facilitated government supervision; (2) government's application of co-governance and supervision and public complaints/reports promotes compliance by Internet platforms; (3) new media plays a supplementary role when government supervision is lacking; the greater the impact of new media, the greater its supervisory effect on Internet platforms; and (4) effective reduction of costs stimulates the enthusiasm of the government and new media, and increases the success of the anti-monopoly co-governance and supervision system. Measures and suggestions to improve supervision of monopolistic behaviors among Internet platforms are proposed.

A proximal neurodynamic model for a system of non-linear inverse mixed variational inequalities

In this article, we introduce a system of non-linear inverse mixed variational inequalities (SNIMVIs). We propose a proximal neurodynamic model (PNDM) for solving SNIMVIs, leveraging proximal mappings. The uniqueness of the continuous solution for the PNDM is proved by assuming Lipschitz continuity. Moreover, we establish the global asymptotic stability of equilibrium points of the PNDM, contingent upon Lipschitz continuity and strong monotonicity. Additionally, an iterative algorithm involving proximal mappings for solving the SNIMVIs is presented. Finally, we provide illustrative examples to support our main findings. Furthermore, we provide an example where the SNIMVIs violate the strong monotonicity condition and exhibit the divergence nature of the trajectories of the corresponding PNDM.

Analysis of Prey-Predator Scheme in Conjunction with Help and Gestation Delay

This paper presents a three-dimensional continuous time dynamical system of three species, two of which are competing preys and one is a predator. We also assume that during predation, the members of both teams of preys help each other and the rate of predation of both teams is different. The interaction between prey and predator is assumed to be governed by a Holling type II functional response and discrete type gestation delay of the predator for consumption of the prey. In this work, we establish the local asymptotic stability of various equilibrium points to understand the dynamics of the model system. Different conditions for the coexistence of equilibrium solutions are discussed. Persistence, permanence of the system, and global stability of the positive interior equilibrium solution are discussed by constructing suitable Lyapunov functions when the gestation delay is zero, and there is no periodic orbit within the interior of the first quadrant of state space around the interior equilibrium. As we introduced time delay due to the gestation of the predator, we also discuss the stability of the delayed model. It is observed that the existence of stability switching occurs around the interior equilibrium point as the gestation delay increases through a certain critical threshold. Here, a phenomenon of Hopf bifurcation occurs, and a stable limit cycle corresponding to the periodic solution of the system is also observed. This study reveals that the delay is taken as a bifurcation parameter and also plays a significant role for the stability of the proposed model. Computer simulations of numerical examples are given to explain our proposed model. We have also addressed critically the biological implications of our analytical findings with proper numerical examples.

Nonlinear dynamics and optimal control strategies of a novel fractional-order lumpy skin disease model

Fractional order differential equations are often employed to better and accurately model several natural and physical processes with memory characteristics than the integer-order models. In this work, a proposed fractional-order model is presented to simulate one of the most dangerous viral cattle diseases. The present fractional-order Lumpy Skin Disease model considers the interactions between different compartments of cattle and virus vectors. The solution of the model is examined in terms of existence, uniqueness, positivity and boundedness. Then, the stability analysis of the different equilibrium points in the model is carried out. The basic reproduction number, R0, has been obtained too. The dependence of stability regions on the values of key parameters is demonstrated and the sensitivity of R0 to different parameters in the system is also investigated. Finally, a proposed optimal control scheme is adopted to successfully diminish the disease spread. It is found that the stability of disease-free equilibrium point is verified at small values of the infection rate of cattle caused by infected cattle or insects. In addition, the disease-free equilibrium point maintains its stability in cases where the recovery rate of cattle or the vaccination rate of cattle are increased. Moreover, the high-efficiency vaccination program helps in stabilizing the disease-free equilibrium point. Also, the fractional-order parameter is found to provide additional degree of freedom in the model to represent different scenarios of evolution for solution orbits. The optimal control parameters of the model are examined in different scenarios to limit or suppress the spread of the disease. It is demonstrated that two control parameters are sufficient to achieve the goals of disease elimination and stabilization of an infection-free steady state. Numerical simulations are verified to confirm our theoretical findings and estimations.

Memory effects in disease modelling through kernel estimates with oscillatory time history

We design a linear chain trick algorithm for dynamical systems for which we have oscillatory time histories in the distributed time delay. We make use of this algorithmic framework to analyse memory effects in disease evolution in a population. The modelling is based on a susceptible-infected-recovered SIR—model and on a susceptible-exposed-infected-recovered SEIR—model through a kernel that dampens the activity based on the recent history of infectious individuals. This corresponds to adaptive behavior in the population or through governmental non-pharmaceutical interventions. We use the linear chain trick to show that such a model may be written in a Markovian way, and we analyze the stability of the system. We find that the adaptive behavior gives rise to either a stable equilibrium point or a stable limit cycle for a close to constant number of susceptibles, i.e. locally in time. We also show that the attack rate for this model is lower than it would be without the dampening, although the adaptive behavior disappears as time goes to infinity and the number of infected goes to zero.