657 results
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2. Asymptotically stable equilibrium and limit cycles in the Rock–Paper–Scissors game in a population of players with complex personalities
- Author
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Platkowski, Tadeusz and Zakrzewski, Jan
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LIMIT cycles , *GAME theory , *ROCK-paper-scissors (Game) , *BIFURCATION theory , *POLYMORPHISM (Crystallography) , *EQUILIBRIUM , *MATHEMATICAL models , *MATRICES (Mathematics) - Abstract
Abstract: We investigate a population of individuals who play the Rock–Paper–Scissors (RPS) game. The players choose strategies not only by optimizing their payoffs, but also taking into account the popularity of the strategies. For the standard RPS game, we find an asymptotically stable polymorphism with coexistence of all strategies. For the general RPS game we find the limit cycles. Their stability depends exclusively on two model parameters: the sum of the entries of the RPS payoff matrix, and a sensitivity parameter which characterizes the personality of the players. Apart from the supercritical Hopf bifurcation, we found the subcritical bifurcation numerically for some intervals of the parameters of the model. [Copyright &y& Elsevier]
- Published
- 2011
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3. Bifurcation and chaos analysis of a fractional-order delay financial risk system using dynamic system approach and persistent homology.
- Author
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He, Ke, Shi, Jianping, and Fang, Hui
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DYNAMICAL systems , *FINANCIAL risk , *HOPF bifurcations , *FINANCIAL risk management , *PHASE space , *POLYNOMIAL chaos - Abstract
A comprehensive theoretical and numerical analysis of the dynamical features of a fractional-order delay financial risk system(FDRS) is presented in this paper. Applying the linearization method and Laplace transform, the critical value of delay when Hopf bifurcation first appears near the equilibrium is firstly derived in an explicit formula. Comparison simulations clarify the reasonableness of fractional-order derivative and delay in describing the financial risk management processes. Then we employ persistent homology and six topological indicators to reveal the geometric and topological structures of FDRS in delay interval. Persistence barcodes, diagrams, and landscapes are utilized for visualizing the simplicial complex's information. The approximate values of delay when FDRS undergoes different periodic oscillations and even chaos are determined. The existence of periodic windows within the chaotic interval is correctly decided. The results of this paper contribute to capturing intricate information of underlying financial activities and detecting the critical transition of FDRS, which has promising and reliable implications for a deeper comprehension of complex behaviors in financial markets. • Determine delay τ 0 when Hopf bifurcation appears. • The effects of fractional orders and parameters on τ 0 are elucidated. • Topological features are visualized by simplicial complex in phase space. • Six indicators based on persistent homology identify varied oscillations. • A fractional-order delay system is reasonable to describe financial activities. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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4. A novel dimensionality reduction approach by integrating dynamics theory and machine learning.
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Chen, Xiyuan and Wang, Qiubao
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MACHINE learning , *MACHINE theory , *MACHINE dynamics , *HOPF bifurcations , *BIFURCATION theory , *DYNAMICAL systems , *MOTION - Abstract
This paper aims to introduce a technique that utilizes both dynamical mechanisms and machine learning to reduce dimensionality in high-dimensional complex systems. Specifically, the method employs Hopf bifurcation theory to establish a model paradigm and use machine learning to train location parameters. The effectiveness of the proposed method is evaluated by testing the Van Der Pol equation and it is found that it possesses good predictive ability. In addition, simulation experiments are conducted using a hunting motion model, which is a well-known practice in high-speed rail, demonstrating positive results. To ensure the robustness of the proposed method, we tested it on noisy data. We introduced simulated Gaussian noise into the original dataset at different signal-to-noise ratios (SNRs) of 10 db, 20 db, 30 db, and 40 db. All data and codes used in this paper have been uploaded to GitHub. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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5. Fractional order PD control of the Hopf bifurcation of HBV viral systems with multiple time delays.
- Author
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Gao, Yuequn and Li, Ning
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HOPF bifurcations ,TIME delay systems ,HEPATITIS B virus ,HEPATITIS B - Abstract
This paper will explore a fractional-order hepatitis B virus (HBV) infection model that takes into account cell-cell and virus-cell transmission and multiple treatment modalities. The desired control strategy is realized by means of a fractional order PD controller. Firstly, we calculated the basic regeneration number and equilibrium point of the HBV model. Afterwards, for the uncontrolled HBV virus system, the adequate conditions for both stability and Hopf bifurcation are systematically investigated via choosing the appropriate time delay as a parameter for bifurcation. Subsequently, under fractional order PD controller, the effect of a proposed controller on system stability and Hopf bifurcation is studied. The desired dynamic characteristics can be obtained afterwards. Finally, numerical simulations show that all three treatments significantly reduce R 0. The onset of oscillations can be delayed by decreasing the order of the fractional order. There are three control pathways for fractional order PD control, and the generation of bifurcation can also be delayed by changing the gain parameter. Using the above methods, the diffusion of HBV virus particles in the body can be effectively controlled. The conclusions drawn in this paper are extremely novel and have potential theoretical value for the future treatment of hepatitis B illness. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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6. Bifurcations of a delayed fractional-order BAM neural network via new parameter perturbations.
- Author
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Huang, Chengdai, Wang, Huanan, Liu, Heng, and Cao, Jinde
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BIDIRECTIONAL associative memories (Computer science) , *QUARTIC equations - Abstract
This paper makes a new breakthrough in deliberating the bifurcations of fractional-order bidirectional associative memory neural network (FOBAMNN). In the beginning, the corresponding bifurcation results are established according to self-regulating parameter, which is different from bifurcation outcomes available by using time delay as the bifurcation parameter, and greatly enriches the bifurcation results of continuous neural networks(NNs). The deived results manifest that a larger self-regulating parameter is more conducive to the stability of the system, which is consistent with the actual meaning of the self-regulating parameter representing the decay rate of activity. In addition to the innovation in the research object, this paper also has innovation in the procedure of calculating the bifurcation critical point. In the face of the quartic equation about the bifurcation parameters, this paper utilizes the methodology of implicit array to calculate the bifurcation critical point succinctly and effectively, which eschews the disadvantages of the conventional Ferrari approach, such as cumbersome formula and huge computational efforts. Our developed technique can be employed as a general method to solve the bifurcation point including the problem of dealing with the bifurcation critical point of delay. Ultimately, numerical experiments test the key theoretical fruits of this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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7. An extended Goodwin model with endogenous technical change and labor supply.
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Cajas Guijarro, John
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TECHNOLOGICAL innovations , *LIMIT cycles , *EMPLOYMENT statistics , *HOPF bifurcations , *DYNAMICAL systems , *LABOR productivity - Abstract
• Extended Goodwin model integrates endogenous technological change and labor supply. • Reinterpretation of induced innovation in terms of mechanization. • Application of 3D-Hopf bifurcation theorem reveals limit cycle dynamics. • Employment rate's effect on productivity as a bifurcation parameter defines cycle stability. • Numerical simulation for 10 OECD economies and sensitivity analysis. This paper extends the Goodwin model of distributive cycles by incorporating the simultaneous endogeneity of technical change and labor supply within a classical-Marxian framework. It reinterprets induced innovation, suggesting that firms optimize mechanization to maximize cost reduction, obtaining a micro-founded relationship between mechanization and the wage share. Additionally, it assumes a positive relationship between labor supply and the employment rate. The resulting three-dimensional dynamical system includes wage share, employment rate, and capital-output ratio as state variables. The Hopf bifurcation theorem reveals the emergence of limit cycles as the employment rate's effect on labor productivity (reserve-army-creation effect) approximates a critical value from below. Numerical simulations for 10 OECD countries illustrate the cyclical nature of the model and its consistency with empirical patterns. Furthermore, a sensitivity analysis explores the effect of parameters variations, emphasizing the social dimensions of productivity and labor supply as critical factors defining the stability of distributive cycles. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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8. Bifurcation analysis and chaos for a double-strains HIV coinfection model with intracellular delays, saturated incidence and Logistic growth.
- Author
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Chen, Wei, Zhang, Long, Wang, Ning, and Teng, Zhidong
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BASIC reproduction number , *MIXED infections , *HOPF bifurcations , *VIRAL load , *HIV infection transmission , *HIV - Abstract
In this paper, a class of virus-to-cell HIV model with intracellular delays, saturated incidence and Logistic growth is proposed to characterize the interaction between two types HIV strains, i.e., wild-type and drug-resistant strains. First, a series of threshold criteria on the locally and globally asymptotic stability of (infection-free, dominant, coexistence) equilibria are discussed based on the basic reproduction number R 0. Furthermore, a detailed Hopf bifurcation analysis is performed on the coexistence equilibrium using two delays as bifurcation parameters. We find that the Hopf bifurcations induced by double-strains are evidently different and more complicated than that of single strain, the former switches from stability (periodic branches) to un-stability (chaos) more frequently and earlier than the latter since double-strains would yield more pairs of imaginary roots in the characteristic equations. Meanwhile, the total viral load of double-strains would be higher than that of single-strain as well. The emergence of drug resistance imposes either negative or positive influences on the survival of wild-type strain, which would further facilitate the transmission of HIV. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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9. Hopf bifurcation and fixed-time stability of a reaction–diffusion echinococcosis model with mixed delays.
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Chen, Weixin, Xu, Xinzhong, and Zhang, Qimin
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HOPF bifurcations , *ECHINOCOCCOSIS , *NONLINEAR systems , *COMPUTER simulation - Abstract
In this paper, a model with spatial diffusion and mixed delays is presented to describe the spread of echinococcosis between dogs and livestock. Firstly, the local stability is investigated using the Routh–Hurwitz criterion. Furthermore, when considering time delays as bifurcation parameters, the conditions for the occurrence of Hopf bifurcation are discussed based on the linear approximation of the nonlinear system. Lastly, the fixed-time stability is studied under the implementation of appropriate control measures. Numerical simulations are provided to give a better understanding of the theoretical results. [ABSTRACT FROM AUTHOR]
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- 2024
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10. Spatiotemporal patterns in a diffusive resource–consumer model with distributed memory and maturation delay.
- Author
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Shen, Hao and Song, Yongli
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HOPF bifurcations , *MEMORY , *DIFFUSION coefficients - Abstract
In this paper, we propose a diffusive consumer–resource model in which the consumer is involved with spatiotemporal memory and the resource has maturation delay. Firstly, for the case without maturation delay, the influence of the spatiotemporal distributed memory on the stability of the positive steady state is investigated. It has been shown that memory delay can induce Turing bifurcation for the negative memory-based diffusion coefficient and induce Hopf bifurcation for the positive memory-based diffusion coefficient. Then the joint effect of the memory delay and the maturation delay on the stability of the positive steady state is investigated and it has been shown that the joint effect of the memory delay and the maturation delay can induce more complicated spatiotemporal dynamics via Turing–Hopf bifurcation and double Hopf bifurcation. Finally, we apply our theoretical analysis results to a consumer–resource model with Holling-II type functional response to illustrate the existence of the different spatiotemporal patterns. • Propose a diffusive resource-consumer model with distributed memory and maturation delay. • Investigate the joint effect of the memory delay and the maturation delay on the spatiotemporal dynamics. • Find the spatially inhomogeneous spatial patterns and spatially inhomogeneous periodic patterns. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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11. Optimal control of red palm weevil model incorporating sterile insect technique, mechanical injection, and pheromone traps.
- Author
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Alnafisah, Yousef and El-Shahed, Moustafa
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PHEROMONE traps ,PALMS ,CURCULIONIDAE ,DATE palm ,INSECTS ,HOPF bifurcations - Abstract
The red palm weevil is one of the most dangerous pests affecting date palm trees in the Gulf region and the Middle East. This paper investigates the control of the red palm weevil (RPW) in date palms using several techniques, namely stem injection, sex pheromone traps, and the Sterile Insect Technique. A mathematical model describing the spread of the red palm weevil is presented, and the boundedness of the solution for the RPW system is studied. The conditions for local stability, forward bifurcation, and Hopf bifurcation are also obtained. Additionally, this research highlights the significance of optimal control in managing RPW and simulates the model using the Forward-Backward Sweep numerical method. It is observed that the parameters of the sterile insect technique, the pheromone trap, and the mechanical injection play important roles in controlling the RPW. Mechanical Injection may be more effective in controlling the palm weevil compared to the other two methods. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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12. Bifurcation analysis of a Parkinson's disease model with two time delays.
- Author
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Zeng, Qiaoyun, Zheng, Yanhong, and Yi, Dan
- Abstract
In this paper, a cortex-basal ganglia model about Parkinson's disease with two time delays is studied, and the critical conditions for Hopf bifurcation are derived. The results show that time delays can change the state of basal ganglia. The basal ganglia is stable when the delays are small. However, when the time delay is greater than the corresponding bifurcation critical point, different types of oscillations occur in the basal ganglia. The larger the time delays, the more active the neuronal clusters in the basal ganglia. Furthermore, the bidirectional Hopf bifurcation is found by studying the connection weights between different neural nuclei. Finally, the influence of connection weight and time delay which are related to the internal segment of the globus pallidus on its oscillation is discussed. Research shows that reducing the connection weight and the corresponding time delay in excitatory neuronal clusters, or increasing the connection weight and decreasing the corresponding time delay in inhibitory neuronal clusters, can improve the oscillation of Parkinson's disease. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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13. Time-delay feedback control of a cantilever beam with concentrated mass based on the homotopy analysis method.
- Author
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Li, Jia-Xuan, Yan, Yan, and Wang, Wen-Quan
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CANTILEVERS , *TIME delay systems , *HOPF bifurcations - Abstract
• Frequency response of a strongly nonlinear time delay system is obtained. • Optimal time delays of control system are given. • First statement about the feedback control relationship between displacement and velocity. • Appropriate time delay parameters can make the system softening or hardening. In this paper, the strongly nonlinear vibration of a cantilever beam system with concentrated mass at an intermediate position controlled by displacement and velocity time delay is investigated. The nonlinear governing equation is studied using the homotopy analysis method. The effects of the time delay, displacement and velocity feedback gain coefficients, as well as frequency on the amplitude of the system were studied in detail. The results indicate that the velocity feedback control is not necessarily superior to displacement feedback control. Reasonable selection of the displacement and velocity time delay parameters can avoid the Hopf bifurcation, adjust the number, amplitude and stability of periodic solutions, and exhibit the softening behavior at low frequency and hardening spring characteristic at high frequency. The theoretical research in this paper will promote the application of homotopy analysis method in the field of time delay control, and serve as a theoretical reference for the design and optimization of cantilever beam control systems with concentrated mass. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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14. Qualitative behavior in a fractional order IS-LM-AS macroeconomic model with stability analysis.
- Author
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Bazán Navarro, Ciro Eduardo and Benazic Tomé, Renato Mario
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MACROECONOMIC models , *GLOBAL asymptotic stability , *HOPF bifurcations , *STRUCTURAL stability , *ECONOMIC systems , *STRUCTURAL models - Abstract
In this article, we analyze the conditions for the structural stability of a fractional order IS-LM-AS dynamic model with adaptive expectations. It is a generalization of our previous research lately published in the literature. We also present the conditions that the structural parameters of the model must meet for the economic system to present a periodic movement when the critical value of the fractional order of the system, q * , guarantees the presence of a Hopf bifurcation of degenerate type. The theoretical analysis is complemented with numerical simulations of the phase portraits in R 3 and of the temporal trajectories of the solutions of the model in MATLAB software. Finally, it is important to highlight that unlike the results of our previous research, the qualitative results found in this paper show that all the structural parameters of the model are essential in determining its global asymptotic stability and Hopf bifurcation. • A fractional economic model generalizes an integer-order model recently reported. • The memory effect in numerical simulations of economic systems is analyzed. • The fractional order when a Hopf bifurcation occurs is determined. • All the parameters of the model are essential in determining its global stability. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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15. A delayed deterministic and stochastic [formula omitted] model: Hopf bifurcation and stochastic analysis.
- Author
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Hajri, Youssra, Allali, Amina, and Amine, Saida
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STOCHASTIC analysis , *HOPF bifurcations , *BASIC reproduction number , *WHITE noise , *STOCHASTIC models - Abstract
In this paper, we present a delayed deterministic and stochastic S I R I C V models to investigate the effects of the white noise intensities and the waning immunity of vaccinated individuals in the evolution of the disease. For the deterministic S I R I C V model, the basic reproduction number R 0 and the equilibrium points are calculated. The local stability of equilibrium points is analyzed. Particularly, when R 0 < 1 the disease-free equilibrium is locally stable for any positive value of τ. Furthermore, when R 0 > 1 , the local stability and sufficient conditions to ensure the occurrence of Hopf bifurcation for the endemic equilibrium point are established by considering the time delay τ as a bifurcation parameter. For the stochastic S I R I C V model, the conditions of the extinction and persistence of the disease are given by using the stochastic basic reproduction numbers R 0 s and R 0 s ∗. Numerical simulations are presented to enhance our analytical results and contrast the deterministic and stochastic models. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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16. Multi-scale oscillation characteristics and stability analysis of pumped-storage unit under primary frequency regulation condition with low water head grid-connected.
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Zhao, Kunjie, Xu, Yanhe, Guo, Pengcheng, Qian, Zhongdong, Zhang, Yongchuan, and Liu, Wei
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SYNCHRONOUS generators , *STABILITY of nonlinear systems , *WATER hammer , *OSCILLATIONS , *HOPF bifurcations , *NONLINEAR equations - Abstract
This paper aims to study the multi-scale oscillation characteristics and stability of pumped storage unit(PSU) under primary frequency regulation(PFR) condition with low water head grid-connected. Firstly, a novel nonlinear mathematical model of pumped storage unit governing system(PSUGS) considering the nonlinear characteristics of diversion tunnel is established, and the nonlinear state equations under power control mode is deduced. On this basis, the bifurcation characteristics and dynamic response process of PSUGS system with coupled first-order, third-order and fifth-order synchronous generator are studied respectively, and the effects of different order synchronous generator on the nonlinear PSUGS system are compared. In addition, the transient characteristics of the nonlinear PSUGS system considering the fifth order synchronous generator under different water heads are studied. Finally, the sensitivity of the regulation system parameters of PSU under PFR with low water head is analyzed, and the effects of hydraulic, mechanical and electrical factors on the multi-scale oscillation characteristics and stability of the nonlinear PSUGS system are revealed. The results show that the stability of PSU considering the first-order synchronous generator is less than that considering the third-order and fifth-order synchronous generator. The stability of PSU considering the third-order synchronous generator is close to that considering the fifth-order synchronous generator. In the operation of PSU, the lower the water head, the smaller the stability of the system. Under PFR condition, the nonlinear PSUGS system meets the supercritical Hopf bifurcation. During PFR oscillation of the unit, the high-frequency wavelet is excited by the generator excitation system and is sensitive to the q-axis parameters of the generator. The low-frequency wavelet is mainly excited by the power grid and mapped with the water hammer effect in the dynamic process. The research results of this paper provide a theoretical basis for the stability analysis and optimal regulation of PSU under PFR condition with low water head. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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17. Dynamics of a plankton community with delay and herd-taxis.
- Author
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Ding, Linglong, Zhang, Xuebing, and Lv, Guangying
- Subjects
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NEUMANN boundary conditions , *HOPF bifurcations , *PLANKTON , *JUDGMENT (Psychology) - Abstract
The movements of the plankton in the ocean are driven by random diffusion and cognitive judgement with herd-taxis. In this paper, we formulate a phytoplankton–zooplankton model with time delay in the herd-taxis effect diffusion and homogeneous Neumann boundary conditions. The conditions to guarantee the existence of the coexistence equilibrium of the model are given. By analyzing the distribution of the eigenvalues of the characteristic equation, the local asymptotic stability of the coexistence equilibrium is achieved under certain condition. When there is no time delay in the herd-taxis effect, the model can possess the Turing bifurcation when we consider the nonlinear diffusion term, which leads to instability. When taking the time delay into account, the Hopf bifurcation occurs instead as the time delay varies. Furthermore, we investigate the situation without the fact of time, that is the steady-state bifurcation and the stability of bifurcating solution. Finally, the stability of the coexistence equilibrium, the Turing bifurcation and the Hopf bifurcation of the system are modeled by numerical simulation. The simulations shown are coordinated with the theoretical results which we arrive at in the former part of the paper. The results illustrate that the time delay in the herd-taxis effect of the zooplankton influence the dynamics of the plankton system. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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18. Hopf bifurcation and global exponential stability of an epidemiological smoking model with time delay.
- Author
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Hu, Xiaomei, Pratap, A., Zhang, Zizhen, and Wan, Aying
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EXPONENTIAL stability ,HOPF bifurcations ,EPIDEMIOLOGICAL models - Abstract
There are many harmful effects of smoking. It not only engulfs the health and life of smokers, but also pollutes the air, endangers the life and health of others, and brings a heavy burden to public health. To this end, a delayed smoking model including potential smokers, occasional smokers, smokers, temporary quitters, permanent quitters and smokers with some illness, is investigated in the present paper. Firstly, local stability and existence of Hopf bifurcation of the model is conducted. Secondly, global exponential stability is explored. Lastly, we numerically simulate the correctness of the obtained theoretical results in the paper. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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19. Spatial movement with memory-induced cross-diffusion effect and toxin effect in predator.
- Author
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Ye, Luhong, Zhao, Hongyong, and Wu, Daiyong
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SPATIAL memory , *HOPF bifurcations , *PREDATOR management , *ANIMAL mechanics , *ANIMAL memory , *POISONS - Abstract
It is well known that spatial memory exists in animal movement modelling. Since spatial memory is related to the past information, then delay arises. In view of the protection of ecological environment, more and more scholars are concerned about the effects of toxic substances on it. A prey–predator system with memory-induced cross-diffusion and toxin in predator is considered in this paper. First, we discuss the fundamental dynamics in detail. Then considering the memory-induced cross-diffusion coefficient and the averaged memory period delay in predator as the controlling parameters, we get that n -mode Hopf bifurcations exist at the positive steady state. Stability switches are generated, and there are spatially nonhomogeneous periodic solutions. Namely if the cross-diffusion coefficient is small, the populations always coexist. If the cross-diffusion coefficient is moderate, two kinds of critical values of the averaged memory period of Hopf bifurcations occur and stability switches may be induced by the delay. If the cross-diffusion coefficient becomes larger, one kind of critical value of the averaged memory period of Hopf bifurcation occur. From simulations, the memory-based cross-diffusion and toxin have vital influences on stability. The moderate toxin can be good for population coexistence. When the averaged memory period is small, the toxin does not change the stability. Once the averaged memory period is bigger, the toxin can change the stability. Moreover, it shows that the averaged memory delay could switch the stability of the system. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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20. Chaos, Hopf bifurcation and control of a fractional-order delay financial system.
- Author
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Shi, Jianping, He, Ke, and Fang, Hui
- Subjects
- *
HOPF bifurcations , *LAPLACE transformation , *BUSINESS forecasting , *LONG-term memory , *FINANCIAL risk - Abstract
The evolution of financial system depends not only on the current state, but also on the previous state. Due to "long-term memory" and "non-locality" of the fractional derivative, fractional-order model can effectively characterize the dynamic features of financial process. An incommensurate fractional-order delay financial system (FDFS) is considered in this paper. Based on linearization and Laplace transformation, the characteristic equation of linearized system of FDFS is obtained. The critical value of the time delay for the occurrence of Hopf bifurcation is determined through the discussions of the eigenvalues of the characteristic equation and the transversality condition. A periodic pulse delay feedback controller is added to the FDFS to control the Hopf bifurcation and to regulate the stability domain of the system. Two illustrative examples are provided to validate our theoretical results. Moreover, numerical simulations demonstrate that the increase of the fractional-order can induce chaos in FDFS, which is detected by 0 − 1 test for chaos. This paper contributes to a better understanding of the dynamic behavior of financial market, forecasting financial risk and implementing effective financial regulation. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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21. Stability of spatial patterns in a diffusive oxygen–plankton model with time lag effect.
- Author
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Gökçe, Aytül, Yazar, Samire, and Sekerci, Yadigar
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TIME delay systems , *PLANKTON , *CLIMATE change , *HOPF bifurcations , *MARINE resources conservation , *OXYGEN , *ENVIRONMENTAL protection - Abstract
Although marine ecosystem is a highly complex phenomenon with many non-linearly interacting species, dissolved oxygen and plankton among these have perhaps the most fundamental relationship not only for the protection of marine environment but also for continuation of life on Earth. This paper deals with a generic diffusive model of dissolved oxygen, phytoplankton and zooplankton species, for which constant time delays are incorporated in growth response of phytoplankton and in the gestation time of zooplankton. We mainly focus on the stability analysis of the coexisting states and the existence of Hopf bifurcation through the characteristic equation, where time delay and oxygen production rate are considered as control parameters for all cases. Studying the effect of both time delays on a stable system, we show destabilisation of the system and irregularity in the spatio-temporal dynamical regimes, leading to chaotic oscillations. Although both delay terms have a destabilising effect, our findings indicate that time delay in zooplankton gestation may induce sharp strongly irregular pattern, whereas time delay in phytoplankton growth gives rise to more regular but higher frequency oscillations for oxygen–plankton interactions. The findings of this paper may provide new insights into main environmental issues including global warming and climate change. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
22. The green-MKS system: A baseline environmental macro-dynamic model.
- Author
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Sordi, Serena and Dávila-Fernández, Marwil J.
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ATTITUDES toward the environment , *ENDOGENOUS growth (Economics) , *EMPLOYMENT statistics , *GREENHOUSE gases , *ENVIRONMENTAL impact charges , *CAPITAL levy , *CARBON taxes , *SAVINGS - Abstract
This paper extends the Marx-Keynes-Schumpeter model in Flaschel (2015) to study the social dimension of climate change. Agents are divided between those supporting and those opposing taxing Green House Gas (GHG) emissions. The composition of the population varies according to a continuous-time version of the discrete-choice approach. Conditional to the level of interaction between players, society chooses the respective carbon tax rate. Higher taxes reduce capital accumulation but support the development of energy-saving production techniques. Output growth and employment rates will be higher or lower depending on which effect prevails. Economic activity generates GHG emissions and determines the employment rate, which, in turn, endogenously feedback on environmental sentiments. Lower emissions reinforce sustainable attitudes, while falling employment increases households' concerns with more "urgent" needs, decreasing support for taxation. Hence, the model is compatible with a positive relationship between environmental attitudes and energy efficiency but not a clear association with output. A sufficiently strong response of attitudes to emissions combined with a "sentiments-autonomous" carbon tax may lead to the disappearance of the equilibrium in which most agents oppose regulation, controlling for multi-stability. Our 3-dimensional system admits endogenous persistent and bounded fluctuations, where the interaction between green attitudes and the macro-economy appears as a novel source of growth-cycle dynamics. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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23. Bifurcations in a fractional-order BAM neural network with four different delays.
- Author
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Huang, Chengdai, Wang, Juan, Chen, Xiaoping, and Cao, Jinde
- Subjects
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BIDIRECTIONAL associative memories (Computer science) , *HOPF bifurcations - Abstract
This paper illuminates the issue of bifurcations for a fractional-order bidirectional associative memory neural network(FOBAMNN) with four different delays. On account of the affirmatory presumption, the developed FOBAMNN is firstly transformed into the one with two nonidentical delays. Then the critical values of Hopf bifurcations with respect to disparate delays are calculated quantitatively by establishing one delay and selecting remaining delay as a bifurcation parameter in the transformed model. It detects that the stability of the developed FOBAMNN with multiple delays can be fairly preserved if selecting lesser control delays, and Hopf bifurcation emerges once the control delays outnumber their critical values. The derived bifurcation results are numerically testified via the bifurcation graphs. The feasibility of theoretical analysis is ultimately corroborated in the light of simulation experiments. The analytic results available in this paper are beneficial to give impetus to resolve the issues of bifurcations of high-order FONNs with multiple delays. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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24. Bifurcations in a fractional-order neural network with multiple leakage delays.
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Huang, Chengdai, Liu, Heng, Shi, Xiangyun, Chen, Xiaoping, Xiao, Min, Wang, Zhengxin, and Cao, Jinde
- Subjects
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LEAKAGE , *HOPF bifurcations - Abstract
This paper expatiates the stability and bifurcation for a fractional-order neural network (FONN) with double leakage delays. Firstly, the characteristic equation of the developed FONN is circumspectly researched by employing inequable delays as bifurcation parameters. Simultaneously the bifurcation criteria are correspondingly extrapolated. Then, unequal delays-spurred-bifurcation diagrams are primarily delineated to confirm the precision and correctness for the values of bifurcation points. Furthermore, it lavishly illustrates from the evidence that the stability performance of the proposed FONN can be demolished with the presence of leakage delays in accordance with comparative studies. Eventually, two numerical examples are exploited to underpin the feasibility of the developed theory. The results derived in this paper have perfected the retrievable outcomes on bifurcations of FONNs embodying unique leakage delay, which can nicely serve a benchmark deliberation and provide a comparatively credible guidance for the influence of multiple leakage delays on bifurcations of FONNs. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
25. Dynamical aspects of a tuberculosis transmission model incorporating vaccination and time delay.
- Author
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Zhang, Zizhen, Zhang, Weishi, Nisar, Kottakkaran Sooppy, Gul, Nadia, Zeb, Anwar, and Vijayakumar, V.
- Subjects
LATENT tuberculosis ,TUBERCULOSIS ,HOPF bifurcations ,VACCINATION ,INFECTIOUS disease transmission - Abstract
To explore transmission dynamics of tuberculosis, a tuberculosis transmission model with vaccination and time delay is developed in current paper. Positivity and boundedness are analyzed. Local stability of tuberculosis-free equilibrium in respect of the time delay due to latent period of tuberculosis is analyzed and we have found threshold value of the time delay for the local stability of tuberculosis-free equilibrium. Then, local stability of tuberculosis-existence equilibrium following exhibition of Hopf bifurcation at the crucial value of the time delay due to latent period of tuberculosis is derived. It is shown that the developed model undergoes a Hopf bifurcation around the tuberculosis-existence when the time delay due to latent period of tuberculosis passes through the threshold value. Direction and stability of the Hopf bifurcation are investigated with the help of the normal form method and center manifold theory. Finally, numerical simulations are carried out in the justification of obtained analytical findings. The results obtained provide significant information for tuberculosis disease controlling. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
26. Dynamic behaviors and optimal control of a new delayed epidemic model.
- Author
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Liu, Qixuan, Xiang, Huili, and Zhou, Min
- Subjects
- *
PONTRYAGIN'S minimum principle , *OPTIMAL control theory , *GLOBAL analysis (Mathematics) , *HOPF bifurcations , *EPIDEMICS , *INFECTIOUS disease transmission - Abstract
We are concerned in this paper with dynamic behaviors and an optimal control problem of a new delayed epidemic model. There are three major ingredients. The first one is the dynamic behaviors of the state system. The locally asymptotic stability of the disease-free equilibrium and the endemic equilibrium are investigated and the effect of time delay on stability is also discussed. It is also found that the Hopf bifurcation appears at a specific time delay. The second, which is the main new ingredient of this paper, is an optimal control problem. Applying vaccine strategy in the system, an optimal control problem is proposed to minimize the total number of infected individuals as much as possible, maximize the total number of the uninfected individuals, and reduce the total control cost. In view of Pontryagin's maximum principle, the specific characteristics of the optimal control policy are given. The third ingredient is the numerical simulations of the theoretical results. • The model considers time delay, competition and spread of disease. • It studies dynamics and an optimal control problem of a delayed model. • Examples and numerical simulations are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
27. Environmental regulation and economic cycles.
- Author
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George, Halkos E., George, Papageorgiou J., Emmanuel, Halkos G., and John, Papageorgiou G.
- Subjects
BUSINESS cycles ,ENVIRONMENTAL regulations ,HOPF bifurcations ,FISCAL policy ,BUDGET surpluses - Abstract
This paper examines economic cycles that do not depend on exogenous economic actions. More precisely, the paper develops a positive model of government behavior in order to define the intertemporal fiscal policies that are optimal for a country and which, determine both the optimal budget level and the optimal level of environmental quality. For this purpose, we establish an optimal control model involving intertemporal subsidy strategies characteristically used by an authoritarian government similar to those found in central Europe. It will be shown that by applying the Hopf bifurcation theorem,a cyclical strategy – that is, waves of regulation, environmental subsidies alternating with deregulation and cuts in social programs – may represent an optimal policy. In this paper, we propose an extremely simple optimal control model which is applied to budget surpluses and environmental subsidies. We investigate the cyclical environmental policies as applied through the bifurcation theorem. A number of propositions are stated during the solution process. The first proposition sets the rules in the model parameters in order to establish the cyclical policies. A second proposition sets the relationship between the discount rate and the opportunity cost of capital, which is used to determine a taxing strategy which becomes the future optimal subsidy policy. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
28. Effect of wall thermal inertia upon transient thermoacoustic dynamics of a swirl-stabilized flame.
- Author
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Bonciolini, Giacomo, Ebi, Dominik, Doll, Ulrich, Weilenmann, Markus, and Noiray, Nicolas
- Abstract
Abstract This paper shows the importance of considering the thermal state of a combustor to investigate or predict its thermoacoustic stability. This aspect is often neglected or regarded as less important than the effect of the operating parameters, such as thermal power or equivalence ratio, but under certain circumstances it can have a dramatic influence on the development of the instabilities. The paper presents experimental results collected from a combustor featuring a lean swirl-stabilized flame exhibiting thermoacoustic instability at some operating conditions. It is shown that this instability is caused by a change of the flame topology that is induced by the progressive increase of the wall temperature with the thermal power. This dependence of the instability on wall temperature leads to inertial effects and hysteresis when the operating condition is changed dynamically. A low-order model of the system reproducing this remarkable dynamics is proposed and validated against the experimental data. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
29. Finite time SF bifurcation and stability analysis for stochastic time-varying delay system.
- Author
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Hu, Zhouyu, Yang, Yanling, Zhang, Congqing, and Wang, Qiubao
- Subjects
- *
TIME-varying systems , *STOCHASTIC analysis , *STOCHASTIC systems , *STOCHASTIC orders , *SYSTEM dynamics , *RANDOM noise theory , *DELAY differential equations , *LIMIT cycles - Abstract
In this paper, it is found that the systems alternate between oscillatory and steady states in studying the stability of a class of stochastic dynamical systems with fast time-varying periodic delays. We propose a new definition of bifurcation, State Flip (SF) bifurcation, by using the abrupt change in the dwell time ratio of two states. Taking the amplitude of the time-varying delay function δ as the bifurcation parameter, we investigate the dynamics of the evolution of a time-varying delay system driven by Gaussian white noise. Firstly, we convert systems with fast time-varying periodic delays into comparative systems with distributed delays, which in turn are converted into stochastic systems with constant delays for discussion. In order to reduce the stochastic delay equation to the average Itô equation, we adopted the stochastic averaging method and the generalized central manifold theory. It is used to capture the one-dimensional slow variables of a system near the critical state of the transition between two steady states during the evolution of the system. We then find that as the amplitude increases from weak to strong, the stochastic time-varying delay system changes from a two-period oscillation to alternating between oscillation and steady, and finally to a steady state. Lastly, we perform numerical simulations of the original stochastic time-varying delay system and the transformed two-dimensional stochastic constant time-delay system respectively to verify that the ideas and methods in this paper are reasonable and effective. • We have conducted an analysis of a class of stochastic system dynamics characterized by fast time-varying pointwise delay. • The method of transforming distributed delay systems into equivalent time-varying delay systems has been thoroughly investigated. • We have identified a novel bifurcation concept: state flip bifurcation. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
30. Bifurcation control of a novel fractional-order gene regulatory network with incommensurate order and time delay.
- Author
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Gao, Yuequn and Li, Ning
- Abstract
Gene regulatory networks play an extremely important role in human life activities. This paper presents a novel fractional-order gene regulatory network model (NFGRNM). The model employs incommensurate fractional orders and introduces time delay arising from the binding of miRNAs to RNA. Time delay often triggers instability such as oscillation and bifurcation, so in this paper we introduce a generalized hybrid control to control the unstable dynamical behavior induced by time delay. First we derive the NFGRNM using fractional order theory, after which the equilibrium point of the network model is calculated. Next, a systematic dynamics analysis of the NFGRNM is performed to derive sufficient conditions to generate the Hopf bifurcation. Most importantly, we introduce the extended hybrid control for the periodic oscillations generated by the time delay in the NFGRNM. Finally, numerical simulation results show that the order of the fractional order can have a great impact on the dynamical behavior of the network model and that the hybrid control has a significant role in delaying (or advancing) the Hopf bifurcation. Since infectious diseases such as SARS-Cov-2, influenza A, and HIV are still prevalent today, the study of gene regulatory networks not only promotes the study of the dynamics of complex networks but also has profound implications for the study of virus transmission in the host and various life activities in humans. • The time lag generated by miRNA-RNA binding is used as a bifurcation parameter. • The Hopf bifurcation behavior of gene regulatory network models is investigated and generalized hybrid control is added. • The effect of fractional order and control parameters on the robustness of the system is investigated. • The effect of disproportionate fractional orders and control parameters on the dynamical behavior of system bifurcations. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
31. Stability analysis and Hopf bifurcation for two-species reaction-diffusion-advection competition systems with two time delays.
- Author
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Alfifi, H.Y.
- Subjects
- *
HOPF bifurcations , *TIME delay systems , *BIFURCATION diagrams , *GALERKIN methods - Abstract
This paper examines a class of two-species reaction-diffusion-advection competition models with two time delays. A system of DDE equations was derived, both theoretically and numerically, using the Galerkin technique method. A condition is defined that helps to find the existence of Hopf bifurcation points. Full diagrams of the Hopf bifurcation points and areas of stability are investigated in detail. Furthermore, we discuss three different sources of delay on bifurcation maps, and what impacts of all these cases of delays on others free rates on the regions of the Hopf bifurcation in this model. We find two different stability regions when the delay time is positive (τ > 0), while the no-delay case (τ = 0) has only one stable region. Moreover, the effect of delays and diffusion rates on all free others parameters in this model have been considered, which can significantly impact upon the stability regions in both population concentrations. It is also found that, as diffusion increases, the time delay increases. However, as the delay maturation is increased, the Hopf points for both proliferation of the population and advection rates are decreased and it causes raises to the region of instability. In addition, bifurcation diagrams are drawn to display chosen instances of the periodic oscillation and two dimensional phase portraits for both concentrations have been plotted to corroborate all analytical outputs that investigated in the theoretical part. • To study the effect of diffusion with two different delay terms on the two-species advection competition system. • To drive theoretical outcomes of the DDE obtainable utilizing the Galerkin technique tool. • To find, theoretically, a condition that assists to obtain the existence of Hopf bifurcation points. • To construct the bifurcation diagram for the DDE and DPDE systems via numerical simulation to confirm theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
32. The role of directed cycles in a directed neural network.
- Author
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Dai, Qinrui, Zhou, Jin, and Kong, Zhengmin
- Subjects
- *
HOPF bifurcations - Abstract
This paper investigates the dynamics of a directed acyclic neural network by edge adding control. We find that the local stability and Hopf bifurcation of the controlled network only depend on the size and intersection of directed cycles, instead of the number and position of the added edges. More specifically, if there is no cycle in the controlled network, the local dynamics of the network will remain unchanged and Hopf bifurcation will not occur even if the number of added edges is sufficient. However, if there exist cycles, then the network may undergo Hopf bifurcation. Our results show that the cycle structure is a necessary condition for the generation of Hopf bifurcation, and the bifurcation threshold is determined by the number, size, and intersection of cycles. Numerical experiments are provided to support the validity of the theory. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
33. Bifurcation and stability of a reaction–diffusion–advection model with nonlocal delay effect and nonlinear boundary condition.
- Author
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Li, Chaochao and Guo, Shangjiang
- Subjects
- *
HOPF bifurcations , *LYAPUNOV-Schmidt equation , *ADVECTION - Abstract
In this paper, a reaction–diffusion–advection model with nonlocal delay effect and nonlinear boundary condition is investigated. By employing the Lyapunov–Schmidt reduction method, we not only establish the existence, multiplicity and stability of spatially nonhomogeneous steady-state solutions, but also obtain some sufficient conditions ensuring the occurrence of a Hopf bifurcation at the steady-state solutions. It is observed that time delay determines the existence of Hopf bifurcation when the interior reaction term is stronger than the boundary reaction term. Finally, the general theoretical results are applied to a diffusive Logistic model with advection term under monostable nonlinear boundary condition and the effect of advection on Hopf bifurcation values is also considered. The results show that Hopf bifurcation is more likely to occur in the case of small advection. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
34. Bifurcations of a single species model with spatial memory environment.
- Author
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Chen, Mengxin, Liu, Yong, and Tian, Canrong
- Subjects
- *
SPATIAL memory , *HOPF bifurcations , *SPECIES - Abstract
This paper reports on the bifurcations of a single-species model with spatial memory environment. The occurrence conditions of the Hopf bifurcation and Turing bifurcation are given with/without the spatial memory effect. Especially, we show that the positive equilibrium enjoys the stability switches with spatial memory effect. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
35. Fractional-order PD control at Hopf bifurcation in a delayed predator–prey system with trans-species infectious diseases.
- Author
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Du, Wentong, Xiao, Min, Ding, Jie, Yao, Yi, Wang, Zhengxin, and Yang, Xinsong
- Subjects
- *
PREDATION , *HOPF bifurcations , *COMMUNICABLE diseases , *FAMILY stability - Abstract
In this paper, a delayed fractional-order predator–prey system with trans-species infectious diseases is proposed and the corresponding control strategy is implemented via fractional-order proportional-derivative (PD) control. Firstly, for the uncontrolled fractional-order predator–prey system, explicit conditions of stability and Hopf bifurcation are established by selecting time delay as the bifurcation parameter. The predator–prey system will lose its stability and a family of oscillations will emerge when the time delay passes through the critical value. Secondly, under the fractional-order PD control, the influences of the controller on the system stability and bifurcation are investigated. It is demonstrated that the Hopf bifurcation can be postponed or advanced, and the desired dynamic can be achieved by choosing appropriate control gain parameters. In addition, the impacts of fractional order and control parameters on dynamics are explored. Finally, some numerical simulations are depicted to validate the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
36. Twenty Hopf-like bifurcations in piecewise-smooth dynamical systems.
- Author
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Simpson, D.J.W.
- Subjects
- *
DYNAMICAL systems , *POINCARE maps (Mathematics) , *LIMIT cycles , *HOPF bifurcations , *PREDATION , *LOTKA-Volterra equations , *BIFURCATION theory - Abstract
For many physical systems the transition from a stationary solution to sustained small amplitude oscillations corresponds to a Hopf bifurcation. For systems involving impacts, thresholds, switches, or other abrupt events, however, this transition can be achieved in fundamentally different ways. This paper reviews 20 such 'Hopf-like' bifurcations for two-dimensional ODE systems with state-dependent switching rules. The bifurcations include boundary equilibrium bifurcations, the collision or change of stability of equilibria or folds on switching manifolds, and limit cycle creation via hysteresis or time delay. In each case a stationary solution changes stability and possibly form, and emits one limit cycle. Each bifurcation is analysed quantitatively in a general setting: we identify quantities that govern the onset, criticality, and genericity of the bifurcation, and determine scaling laws for the period and amplitude of the resulting limit cycle. Complete derivations based on asymptotic expansions of Poincaré maps are provided. Many of these are new, done previously only for piecewise-linear systems. The bifurcations are collated and compared so that dynamical observations can be matched to geometric mechanisms responsible for the creation of a limit cycle. The results are illustrated with impact oscillators, relay control, automated balancing control, predator–prey systems, ocean circulation, and the McKean and Wilson–Cowan neuron models. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
37. An improved reduced order model for nonlinear stability analysis of spatial xenon oscillations.
- Author
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Chakraborty, Abhishek, Singh, Suneet, and Fernando, M.P.S.
- Subjects
- *
REDUCED-order models , *NONLINEAR analysis , *NONLINEAR difference equations , *GEOGRAPHIC spatial analysis , *XENON , *ORDINARY differential equations , *OSCILLATIONS - Abstract
In this paper, an improved reduced order model for studying spatial xenon oscillations has been developed which can be utilized for stability analysis (linear as well as nonlinear). The model presented in this paper has been developed to overcome situations of unphysical results which were obtained by earlier models which used quasi static assumption when the system moves away from the equilibrium point. The earlier model consisted of multipoint neutron kinetics equations including xenon and iodine feedback. In order to eliminate unphysical behaviour in the earlier model, the quasi-static approach is discarded and the total power control is represented by an ordinary differential equation. It represents the action of a reactivity device governed by the deviation of the total power from the steady state power level. In the current approach, a PHWR is modelled by dividing it into two regions. The linear and nonlinear stability analysis was carried out using the new model and compared with the older results. Linear stability maps were constructed in different parameter planes and were found to be identical. However, there was a change in the nonlinear characteristics of the system and the results are presented in this paper. Moreover, the effect of control device on the system behaviour especially on total power variation is studied through numerical simulations. • Improved reduced order model for stability analysis of Xenon oscillations. • Quasi steady state approximation is not adequate for nonlinear analysis. • Linear stability boundary were found to be identical with model reduction. • Effect of model reduction on bifurcation characteristics is analyzed. • Interchange of type of Hopf bifurcation is observed. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
38. Dynamical analysis of antigen-driven T-cell infection model with multiple delays.
- Author
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Prakash, M., Rakkiyappan, R., Manivannan, A., and Cao, Jinde
- Subjects
- *
CYTOTOXIC T cells , *INFECTION , *EXAMPLE - Abstract
Abstract This paper is mainly concerned with an investigation of antigen-driven T-cell infection process through a mathematical model. Several mathematical models have been introduced in the literature in order to gain insights into the dynamics of the disease progression, however, the results considering the effect of multiple factors which include antigen-driven CD4 T-cell progress, latent infection stage, activation of CTLs response, role of antiretroviral therapies (ARTs) and possibilities of multiple time-delays during the infection process are not involved. Hence, the paper introduces a six-dimensional virus infection model by involving the above factors. Particularly, (i) the effect of activation of antigen-specific T-cells; (ii) the effect of the maturation of infected cells; (iii) effect of multiple time delays, that is, during the interaction between susceptible and infectious and during the activation of immune responses; which play a significant role in preventing and modulating the Human Immunodeficiency Virus (HIV). The main aim of the paper to analyze the local and global stability of the class of mathematical models regarding the effect of time delays which provides a better pathway to the infection progress. Finally, the overall contribution of the present work is listed as follows: (1) by constructing the suitable Lyapunov functional, the global stability of the intracellular delayed model is derived; (2) detailed Hopf-bifurcation analysis is discussed with respect to immune response delay. The numerical simulation is performed to validate the effectiveness and applicability of the theoretical predictions. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
39. The existence of a limit cycle in a pollinator–plant–herbivore mathematical model.
- Author
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Castellanos, Víctor and Sánchez-Garduño, Faustino
- Subjects
- *
CHEMICAL plants , *LIMIT cycles , *MATHEMATICAL models - Abstract
Abstract This paper deals with the existence of a limit cycle in a nonlinear ODE mathematical model which describes the interaction between three homogeneous populations. These take the form of pollinators, plants and herbivores. The interaction between the pollinators and plants is of mutualistic type given by a functional response of type II; meanwhile that for the plants–herbivores, from the demographic point of view, it can be seen as a predator–prey interaction. The resulting coupled nonlinear ODE system contains several parameters which have an important ecological interpretation. In the study we present here we give necessary conditions on the parameter values for the emergence of an attracting limit cycle and other which is unstable. These come from a supercritical and a subcritical Hopf bifurcation, respectively. Because of the amount of the involved parameters, almost any calculation becomes really massive. Thus, part of the analysis we present here intensively uses the Mathematica symbolic software. With the same software, by using a theorem authored by Kuznetsov, we calculated the first Lyapunov coefficient whose sign gives us the type of stability the limit cycle has. A number of numerical simulations also are included with the aim of showing the limit cycle in the three dimensional phase space. This is done in two main cases: when the ODE system has one positive equilibrium and when such system has two positive equilibria. The main part of the paper ends by exploring other set of parameters values which allows us to recover the previous dynamics but also gives us other interesting behaviors. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
40. The dynamics analysis of a rumor propagation model in online social networks.
- Author
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Zhu, Linhe, Liu, Mengxue, and Li, Yimin
- Subjects
- *
ONLINE social networks , *ONLINE business networks (Social networks) - Abstract
Abstract With rapid emergence of online social networks, the rumor transition has a growing significant impact on social stability and human lives. This paper investigates the local stability and optimal control of a rumor propagation model in online social network with changing number of total users, which is proposed in Ref. Dong et al. (2018). In addition, we set up a model with a delay based on the reality, and analyze the sufficient conditions for Hopf bifurcation. Finally, the rumor spreading model is simulated numerically. The results of simulation are shown the process of rumor propagation and prove the correctness of the theoretical analysis. Highlights • This paper presents an important improvement in theory and numerical analysis for Ref. Dong et al. (2018). • Bifurcation and stability switches are proved in a strict mathematical way. • Optimal control strategy of rumor propagation is investigated and compared in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
41. Stability and Hopf bifurcation analysis of a simplified six-neuron tridiagonal two-layer neural network model with delays.
- Author
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Wang, Tianshun, Cheng, Zunshui, Bu, Rui, and Ma, Runsheng
- Subjects
- *
ARTIFICIAL neural networks , *HOPF bifurcations , *NEURAL computer network stability , *TIME delay systems , *LINEAR systems - Abstract
Highlights • A general tridiagonal two-layer neural network model with delay is proposed. • A new method of Hopf bifurcation analysis is introduced by matrix decomposition. • The conditions obtained are simpler than traditional Hurwitz discriminant method. Abstract Firstly, a general tridiagonal two-layer neural network model with 2 n -neuron is proposed, where every layer has time delay. A new method of Hopf bifurcation analysis is introduced by matrix decomposition in this paper. Through factoring the tridiagonal matrix, the characteristic equation of the neural network model is simplified. Secondly, by studying the eigenvalue equations of the related linear system for the special six-neuron (three neurons per layer) two-layer neural network model, the sufficient conditions for experiencing the Hopf bifurcation are obtained. The conditions obtained by the new method proposed in this paper are simpler and more practical than those obtained by the traditional Hurwitz discriminant method. Next, based on the normal form method and the center manifold theorem, the explicit formulae about the stability of the bifurcating periodic solution and the direction of the Hopf bifurcation are established. Finally, the main results obtained in this paper are illustrated by three numerical simulation examples. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
42. Dynamical analysis for the impact of asymptomatic infective and infection delay on disease transmission.
- Author
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Wang, Ning, Qi, Longxing, and Cheng, Guangyi
- Subjects
- *
INFECTIOUS disease transmission , *ASYMPTOMATIC patients , *HOPF bifurcations , *COMMUNICABLE diseases , *INFECTION - Abstract
The influence of asymptomatic patients on disease transmission has attracted more and more attention, but the mechanism of some factors affecting disease transmission needs to be studied urgently. Considering the self-healing rate of asymptomatic patients, the cure rate of symptomatic patients, the transformation rate from asymptomatic to symptomatic and the infection delay, a type of infectious disease dynamics model S I s I a S with asymptomatic infection and infection delay is established in this paper. It is found that both the infection delay and the difference size between the cure rate and the self-healing rate not only affect the minimum value of the total number of patients in the persistent state of the disease, but also lead to disease extinction to be controlled by the proportion of symptomatic patients in patients. Moreover, the infection delay can lead to local Hopf bifurcation of periodic solutions. By using the normal form and center manifold theory the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are discussed. At last, sensitivity analysis shows that the infection delay can change the correlation of the proportion of symptomatic patients in patients and the transformation rate to the total number of patients. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
43. Backward bifurcation and stability analysis in a within-host HIV model with both virus-to-cell infection and cell-to-cell transmission, and anti-retroviral therapy.
- Author
-
Bai, Ning and Xu, Rui
- Subjects
- *
ANTIRETROVIRAL agents , *INFECTIOUS disease transmission , *REVERSE transcriptase inhibitors , *HIV infection transmission , *VIRAL load , *HIV infections , *HIV , *RETROVIRUSES - Abstract
Researches have shown that in addition to direct virus-to-cell infection, viral particles can also be transferred from a productively-infected cell to an uninfected cell through the formation of virological synapses. In order to reduce the viral load in infected individuals, different classes of antiretroviral drugs have been developed, including reverse transcriptase inhibitor (RTI), integrase inhibitor (II), protease inhibitor (PI) and so on. In this paper, we incorporate the mitotic proliferation of target cells which is described by the logistic term, both virus-to-cell infection and cell-to-cell transmission, the intracellular delay and RTI-based therapy into an in-host HIV infection model. Through mathematical analysis, we find that the model undergoes a backward bifurcation when the turn-over rate coefficient of productively-infected cells is smaller than its mitotic proliferation rate coefficient. When the turn-over rate coefficient of productively-infected cells is greater than its mitotic proliferation rate coefficient, the existence of Hopf bifurcation at the chronic-infection equilibrium with and without the intracellular delay is established, respectively. Numerical simulations suggest that the dynamics of the model be sensitive to parameter values and initial conditions, which may be of great significance to control HIV infection. We also show numerical evidence to support the fact that the smaller the therapy efficacy, the higher the viral load in infected individuals. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
44. A Goodwin type cyclical growth model with two-time delays.
- Author
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Sportelli, Mario and De Cesare, Luigi
- Subjects
- *
REAL wages , *HOPF bifurcations , *UNEMPLOYMENT statistics , *BUSINESS cycles , *ECONOMIC equilibrium , *DIFFERENTIAL equations - Abstract
• Goodwin's model with delayed investment function. • Cyclical growth model with time delays and delay dependent coefficients. • Existence of Hopf bifurcations analyzed by choosing time delays as bifurcation parameters. • Chaotic dynamics. In this paper, we reconsider the Goodwin 1967 growth-cycle model, where the antagonistic relationship between wages and profits is assimilated to the prey-predator conflict modeled by Volterra in 1931. Here we propose an extension of Goodwin's basic model by adding two important elements of the business cycle theory: (i) a finite time delay between investment orders and deliveries of finished capital goods, as theorized by Kalecki (1935); (ii) a delayed reaction of real wages to the unemployment levels, as suggested by Chiarella (1990). Both these delays preserve the two-dimensionality of the original model, but it becomes a delayed differential equation system, with two discrete time delays and one-delay dependent parameters. The qualitative study of the system shows that without lags the economic meaningful equilibrium is structurally stable. Nevertheless, as soon the time delays become positive, that equilibrium loses its stability and, according to the combinations of parameters and length of the lags, either periodic or non-periodic (chaotic) fluctuations arise. Numerical simulations supporting the economic analysis show that, in the very long run, a "strange attractor" depicts the dynamic behavior of the system. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
45. Delay-dependent bifurcation conditions in a fractional-order inertial BAM neural network.
- Author
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Huang, Chengdai, Wang, Huanan, Cao, Jinde, and Liu, Heng
- Subjects
- *
TIME delay systems , *HOPF bifurcations , *PHASE diagrams , *CAPUTO fractional derivatives - Abstract
This paper explores the stability and bifurcation of a Caputo fractional-order BAM neural network with time delay and inertia terms. Subsequently, the limitation in bifurcation characteristics of Caputo fractional-order delayed inertial BAM neural network (CFODIBAMNN) is surpassed. By analyzing the stability of the system without time delay, the direct method is applied that involves solving the eigenvalues to determine its stability. Ulteriorly, analyzing the system in the presence of time delays and choosing the time delay as the bifurcation parameter to identify the Hopf bifurcation properties of the system. Eventually, during the verifications, the critical values of bifurcation are accurately calculated, the fluctuation and phase diagrams for the ranges of different fractional order are simulated and the impact of fractional orders on system stability is analyzed. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
46. Nonlinear dynamics modeling of a light-powered liquid crystal elastomer-based perpetual motion machine.
- Author
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Liu, Junxiu, Qian, Guqian, Dai, Yuntong, Yuan, Zongsong, Song, Wenqiang, and Li, Kai
- Subjects
- *
LIQUID crystals , *HOPF bifurcations , *ROTATIONAL motion , *ENERGY harvesting , *RUNGE-Kutta formulas , *MACHINING , *TURNTABLES - Abstract
The benefits of self-excited motion include the capacity to draw energy directly from the external environment, as well as autonomy and portability, and the creation of additional self-oscillating systems holds the promise of finding applications in various fields, including medical devices, autonomous robotics, energy harvesting, sensors, and more. Inspired by the magnetic perpetual motion, this paper introduces a conceptual perpetual motion machine utilizing liquid crystal elastomer (LCE) and steady illumination. The system consists of a turntable and multiple motion ducts, each containing a mass block. A nonlinear dynamics model for the LCE-based perpetual motion machine operating under steady illumination is established, considering the dynamic LCE model. Theoretical analysis and numerical simulations, employing the classical fourth-order Runge-Kutta method, are employed to examine the system's dynamic behaviors. The system exhibits a supercritical Hopf bifurcation between static and self-rotation patterns. Notably, the external environment's energy input counteracts damping dissipation, preventing the system from halting due to damping and other factors. Theoretical condition for Hopf bifurcation in the perpetual motion machine's self-rotation is derived and validated via numerical computations. Extensive quantitative investigations are conducted to determine the influence of system parameters on self-rotation frequency. The results illustrate that through parameter adjustments, the motion pattern and frequency of the system can be controlled effectively. The proposed system offers continuous rotation in a zero-energy mode, presenting remarkable benefits, including energy efficiency, reduced noise, decreased wear, and robust stability. These attributes position it for significant future applications in power generation, energy harvesting, sensors, and various other fields. [Display omitted] • A light-powered liquid crystal elastomer-based perpetual motion machine was conceptualized. • The perpetual motion machine has two typical motion patterns: static and self-rotation pattern. • Theoretical Hopf bifurcation condition was derived and validated via numerical computations. • The self-rotation frequency can be designed by several parameters. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
47. Self-organized pattern dynamics of somitogenesis model in embryos.
- Author
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Guan, Linan and Shen, Jianwei
- Subjects
- *
SOMITOGENESIS , *VERTEBRATE embryology , *SELF-organizing systems , *DEVELOPMENTAL biology , *HOPF bifurcations - Abstract
Somitogenesis, the sequential formation of a periodic pattern along the anteroposterior axis of vertebrate embryos, is one of the most obvious examples of the segmental patterning processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. In this paper, we investigate the effect of diffusion on pattern formation use a modified two dimensional model which can be used to explain somitogenesis during embryonic development. This model is suitable for exploring a design space of somitogenesis and can explain many aspects of somitogenesis that previous models cannot. In the present paper, by analyzing the local linear stability of the equation, we acquired the conditions of Hopf bifurcation and Turing bifurcation. In addition, the amplitude equation near the Turing bifurcation point is obtained by using the methods of multi-scale expansion and symmetry analysis. By analyzing the stability of the amplitude equation, we know that there are various complex phenomena, including Spot pattern, mixture of spot–stripe patterns and labyrinthine. Finally, numerical simulation are given to verify the correctness of our theoretical results. Somitogenesis occupies an important position in the process of biological development, and as a pattern process can be used to investigate many aspects of embryogenesis. Therefore, our study helps greatly to cell differentiation, gene expression and embryonic development. What is more, it is of great significance for the diagnosis and treatment of human diseases to study the related knowledge of model biology. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
48. Impact of leakage delay on bifurcation in high-order fractional BAM neural networks.
- Author
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Huang, Chengdai and Cao, Jinde
- Subjects
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BIDIRECTIONAL associative memories (Computer science) , *ARTIFICIAL neural networks , *BIFURCATION theory , *SIGNAL processing , *ARTIFICIAL intelligence , *FRACTIONAL calculus - Abstract
The effects of leakage delay on the dynamics of neural networks with integer-order have lately been received considerable attention. It has been confirmed that fractional neural networks more appropriately uncover the dynamical properties of neural networks, but the results of fractional neural networks with leakage delay are relatively few. This paper primarily concentrates on the issue of bifurcation for high-order fractional bidirectional associative memory(BAM) neural networks involving leakage delay. The first attempt is made to tackle the stability and bifurcation of high-order fractional BAM neural networks with time delay in leakage terms in this paper. The conditions for the appearance of bifurcation for the proposed systems with leakage delay are firstly established by adopting time delay as a bifurcation parameter. Then, the bifurcation criteria of such system without leakage delay are successfully acquired. Comparative analysis wondrously detects that the stability performance of the proposed high-order fractional neural networks is critically weakened by leakage delay, they cannot be overlooked. Numerical examples are ultimately exhibited to attest the efficiency of the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
49. Analysis of a mathematical model arising from stage-structured predator–prey in a chemostat.
- Author
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Zhou, Hui
- Subjects
- *
CHEMOSTAT , *MATHEMATICAL analysis , *MATHEMATICAL models , *HOPF bifurcations , *OSCILLATIONS - Abstract
In this article, we consider a 4-dimensional predator–prey chemostat model of nitrogen-phytoplankton-rotifer interactions with staged structure proposed by Blasius et al. (2020). Although it is still difficult to prove the simulation observations in Blasius et al. (2020) by mathematical arguments, we explore the dynamics in order to better understand the dynamical mechanism of cyclic persistence for this model. We firstly investigate the corresponding system without staged structure, i.e., when the juvenile is absent, the asymptotical behavior of the solutions is given. When the juvenile is present, a threshold condition for the uniform persistence of the 4-dimensional system is provided. Finally, by choosing the life development time delay as a bifurcation parameter, we show that the system admits periodic solutions near one semi-equilibrium undergoing Hopf bifurcation. The rigorous theoretical analytic work in this paper provides some helpful transient information between coherent oscillation and non-coherent oscillation described by the experimental data of Blasius et al. (2020). [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
50. Analysis and simulation on dynamical behaviors of a reaction–diffusion system with time-delay.
- Author
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Suriguga, Jia, Yunfeng, Wang, Jingjing, and Li, Yanling
- Abstract
Time-delay effect and bifurcation phenomenon are important topics in the study of reaction–diffusion equations. In this paper, we consider a three-species predator–prey system with diffusion and incubation delay for predator. The stability and Hopf bifurcation are mainly discussed. We conclude that there exists a critical value of delay, such that the internal equilibrium is stable or unstable as the delay crosses the critical value. Especially, the system emerges Hopf bifurcation phenomenon at this critical value. For bifurcation solution, the conclusions of stability, period and bifurcation direction are also presented. Additionally, numerical simulations are proceeded to support the main results. In biology, the existence of bifurcation solution means that when the delay of predator reaches to a certain extent, the predator and prey will coexist within a period of time. It turns out that the related computations and analyses are much more complicated than that of two-species time-delay systems. • A predation model with time-delay and diffusion is considered. • Results on stability and Hopf-bifurcations are obtained. • Hopf-bifurcation graphs are performed to illustrate the theoretical analysis. • It is found that time-delay has significant impacts on the pattern formations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
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