5,863 results
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2. Three Ways of Treating a Linear Delay Differential Equation
- Author
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Sah, Si Mohamed, Rand, Richard H., and Belhaq, Mohamed, editor
- Published
- 2018
- Full Text
- View/download PDF
3. Linear and Nonlinear Damping Effects on the Stability of the Ziegler Column
- Author
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Luongo, Angelo, D’Annibale, Francesco, and Belhaq, Mohamed, editor
- Published
- 2015
- Full Text
- View/download PDF
4. Hopf Bifurcations in Delayed Rock–Paper–Scissors Replicator Dynamics
- Author
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Wesson, Elizabeth and Rand, Richard
- Published
- 2016
- Full Text
- View/download PDF
5. On the Use of the Multiple Scale Harmonic Balance Method for Nonlinear Energy Sinks Controlled Systems
- Author
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Luongo, Angelo, Zulli, Daniele, and Belhaq, Mohamed, editor
- Published
- 2015
- Full Text
- View/download PDF
6. Evolutionary dynamics of rock-paper-scissors game in the patchy network with mutations
- Author
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Tina Verma and Arvind Kumar Gupta
- Subjects
Hopf bifurcation ,education.field_of_study ,General Mathematics ,Applied Mathematics ,Population ,Evolutionary game theory ,Biodiversity ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Metapopulation ,symbols.namesake ,Transcritical bifurcation ,Evolutionary biology ,Mutation (genetic algorithm) ,symbols ,education ,Evolutionary dynamics ,Mathematics - Abstract
Connectivity is the safety network for biodiversity conservation because connected habitats are more effective for saving the species and ecological functions. The nature of coupling for connectivity also plays an important role in the co-existence of species in cyclic-dominance. The rock-paper-scissors game is one of the paradigmatic mathematical model in evolutionary game theory to understand the mechanism of biodiversity in cyclic-dominance. In this paper, the metapopulation model for rock-paper-scissors with mutations is presented in which the total population is divided into patches and the patches form a network of complete graph. The migration among patches is allowed through simple random walk. The replicator-mutator equations are used with the migration term. When migration is allowed then the population of the patches will synchronized and attain stable state through Hopf bifurcation. Apart form this, two phases are observed when the strategies of one of the species mutate to other two species: co-existence of all the species phase and existence of one kind of species phase. The transition from one phase to another phase is taking place due to transcritical bifurcation. The dynamics of the population of species of rock, paper, scissors is studied in the environment of homogeneous and heterogeneous mutation. Numerical simulations have been performed when mutation is allowed in all the patches (homogeneous mutation) and some of the patches (heterogeneous mutation). It has been observed that when the number of patches is increased in the case of heterogeneous mutation then the population of any of the species will not extinct and all the species will co-exist.
- Published
- 2021
7. Multiple limit cycles for the continuous model of the rock–scissors–paper game between bacteriocin producing bacteria.
- Author
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Daoxiang, Zhang and Yan, Ping
- Subjects
- *
LIMIT cycles , *CONTINUOUS functions , *HOPF bifurcations , *ROCK-paper-scissors (Game) , *BACTERIOCINS - Abstract
In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rock–scissors–paper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
8. A trio of heteroclinic bifurcations arising from a model of spatially-extended Rock-Paper-Scissors
- Author
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Claire M. Postlethwaite and Alastair M. Rucklidge
- Subjects
Population ,General Physics and Astronomy ,FOS: Physical sciences ,Pattern Formation and Solitons (nlin.PS) ,01 natural sciences ,symbols.namesake ,0101 mathematics ,education ,Quantitative Biology - Populations and Evolution ,Mathematical Physics ,Saddle ,Mathematics ,Hopf bifurcation ,Equilibrium point ,education.field_of_study ,Partial differential equation ,37G15, 34C37, 37C29, 91A22 ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Ode ,Populations and Evolution (q-bio.PE) ,Heteroclinic cycle ,Statistical and Nonlinear Physics ,Nonlinear Sciences - Pattern Formation and Solitons ,010101 applied mathematics ,Ordinary differential equation ,FOS: Biological sciences ,symbols - Abstract
One of the simplest examples of a robust heteroclinic cycle involves three saddle equilibria: each one is unstable to the next in turn, and connections from one to the next occur within invariant subspaces. Such a situation can be described by a third-order ordinary differential equation (ODE), and typical trajectories approach each equilibrium point in turn, spending progressively longer to cycle around the three points but never stopping. This cycle has been invoked as a model of cyclic competition between populations adopting three strategies, characterised as Rock, Paper and Scissors. When spatial distribution and mobility of the populations is taken into account, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn. The dynamics is described by a set of partial differential equations (PDEs) that has travelling wave (in one dimension) and spiral (in two dimensions) solutions. In this paper, we explore how the robust heteroclinic cycle in the ODE manifests itself in the PDEs. Taking the wavespeed as a parameter, and moving into a travelling frame, the PDEs reduce to a sixth-order set of ODEs, in which travelling waves are created in a Hopf bifurcation and are destroyed in three different heteroclinic bifurcations, depending on parameters, as the travelling wave approaches the heteroclinic cycle. We explore the three different heteroclinic bifurcations, none of which have been observed in the context of robust heteroclinic cycles previously. These results are an important step towards a full understanding of the spiral patterns found in two dimensions, with possible application to travelling waves and spirals in other population dynamics models., Comment: 36 pages, 8 figures
- Published
- 2019
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9. Evolutionary dynamics in the rock-paper-scissors system by changing community paradigm with population flow
- Author
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Junpyo Park
- Subjects
Hopf bifurcation ,education.field_of_study ,General Mathematics ,Applied Mathematics ,Population ,General Physics and Astronomy ,Robustness (evolution) ,Statistical and Nonlinear Physics ,Fixed point ,symbols.namesake ,symbols ,Outflow ,Statistical physics ,Balanced flow ,Evolutionary dynamics ,education ,Multistability ,Mathematics - Abstract
Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups.
- Published
- 2021
10. Multiple limit cycles for the continuous model of the rock–scissors–paper game between bacteriocin producing bacteria
- Author
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Ping Yan and Zhang Dao-xiang
- Subjects
Hopf bifurcation ,Continuous modelling ,Applied Mathematics ,010102 general mathematics ,Heteroclinic cycle ,16. Peace & justice ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,symbols ,Applied mathematics ,Limit (mathematics) ,0101 mathematics ,Mathematical economics ,Mathematics - Abstract
Two limit cycles for the continuous model of the rockscissorspaper (RSP) game is constructed.The Hopf bifurcation method is used for the construction of limit cycles.The results give a partial answer to an open question posed by Neumann and Schuster. In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rockscissorspaper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game.
- Published
- 2017
11. Hopf Bifurcations in Delayed Rock–Paper–Scissors Replicator Dynamics
- Author
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Richard H. Rand and Elizabeth Wesson
- Subjects
Statistics and Probability ,Economics and Econometrics ,Population ,Interval (mathematics) ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Control theory ,Limit cycle ,0103 physical sciences ,Replicator equation ,Applied mathematics ,Limit (mathematics) ,010306 general physics ,education ,Bifurcation ,Mathematics ,Hopf bifurcation ,education.field_of_study ,Applied Mathematics ,Function (mathematics) ,Computer Graphics and Computer-Aided Design ,Computer Science Applications ,Nonlinear Sciences::Chaotic Dynamics ,Computational Mathematics ,Computational Theory and Mathematics ,symbols - Abstract
We investigate the dynamics of three-strategy (rock–paper–scissors) replicator equations in which the fitness of each strategy is a function of the population frequencies delayed by a time interval $$T$$ . Taking $$T$$ as a bifurcation parameter, we demonstrate the existence of (non-degenerate) Hopf bifurcations in these systems and present an analysis of the resulting limit cycles using Lindstedt’s method.
- Published
- 2015
12. Nonlinear dynamics with Hopf bifurcations by targeted mutation in the system of rock-paper-scissors metaphor
- Author
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Junpyo Park
- Subjects
Hopf bifurcation ,Applied Mathematics ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Biology ,01 natural sciences ,Intraspecific competition ,010305 fluids & plasmas ,symbols.namesake ,Nonlinear system ,Targeted Mutation ,Linear stability analysis ,Evolutionary biology ,0103 physical sciences ,Mutation (genetic algorithm) ,symbols ,010306 general physics ,Biological sciences ,Gene evolution ,Mathematical Physics - Abstract
The role of mutation, which is an error process in gene evolution, in systems of cyclically competing species has been studied from various perspectives, and it is regarded as one of the key factors for promoting coexistence of all species. In addition to naturally occurring mutations, many experiments in genetic engineering have involved targeted mutation techniques such as recombination between DNA and somatic cell sequences and have studied genetic modifications through loss or augmentation of cell functions. In this paper, we investigate nonlinear dynamics with targeted mutation in cyclically competing species. In different ways to classic approaches of mutation in cyclic games, we assume that mutation may occur in targeted individuals who have been removed from intraspecific competition. By investigating each scenario depending on the number of objects for targeted mutation analytically and numerically, we found that targeted mutation can lead to persistent coexistence of all species. In addition, under the specific condition of targeted mutation, we found that targeted mutation can lead to emergences of bistable states for species survival. Through the linear stability analysis of rate equations, we found that those phenomena are accompanied by Hopf bifurcation which is supercritical. Our findings may provide more global perspectives on understanding underlying mechanisms to control biodiversity in ecological/biological sciences, and evidences with mathematical foundations to resolve social dilemmas such as a turnover of group members by resigning with intragroup conflicts in social sciences.
- Published
- 2019
13. Bifurcation analysis and chaos control in Zhou's dynamical system
- Author
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Aly, E. S., El-Dessoky, M. M., Yassen, M. T., Saleh, E., Aiyashi, M. A., and Msmali, Ahmed Hussein
- Published
- 2022
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14. Limit cycles and the benefits of a short memory in rock-paper-scissors games
- Author
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James Burridge
- Subjects
Hopf bifurcation ,Computer Science::Computer Science and Game Theory ,education.field_of_study ,Recall ,media_common.quotation_subject ,Population ,Short-term memory ,Fixed point ,Asymmetry ,Computer Science::Multiagent Systems ,symbols.namesake ,Bifurcation theory ,symbols ,Limit (mathematics) ,education ,Mathematical economics ,Mathematics ,media_common - Abstract
When playing games in groups, it is an advantage for individuals to have accurate statistical information on the strategies of their opponents. Such information may be obtained by remembering previous interactions. We consider a rock-paper-scissors game in which agents are able to recall their last m interactions, used to estimate the behavior of their opponents. At critical memory length, a Hopf bifurcation leads to the formation of stable limit cycles. In a mixed population, agents with longer memories have an advantage, provided the system has a stable fixed point, and there is some asymmetry in the payoffs of the pure strategies. However, at a critical concentration of long memory agents, the appearance of limit cycles destroys their advantage. By introducing population dynamics that favors successful agents, we show that the system evolves toward the bifurcation point.
- Published
- 2015
15. Nonlinear dynamics of the rock-paper-scissors game with mutations
- Author
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Danielle F. P. Toupo and Steven H. Strogatz
- Subjects
Physics ,Hopf bifurcation ,Mutation rate ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,State (functional analysis) ,Quantitative Biology::Genomics ,Stability (probability) ,Nonlinear Sciences - Adaptation and Self-Organizing Systems ,Nonlinear system ,symbols.namesake ,Control theory ,Limit cycle ,Mutation (genetic algorithm) ,FOS: Mathematics ,symbols ,Quantitative Biology::Populations and Evolution ,Limit (mathematics) ,Statistical physics ,Mathematics - Dynamical Systems ,Adaptation and Self-Organizing Systems (nlin.AO) - Abstract
We analyze the replicator-mutator equations for the Rock-Paper-Scissors game. Various graph-theoretic patterns of mutation are considered, ranging from a single unidirectional mutation pathway between two of the species, to global bidirectional mutation among all the species. Our main result is that the coexistence state, in which all three species exist in equilibrium, can be destabilized by arbitrarily small mutation rates. After it loses stability, the coexistence state gives birth to a stable limit cycle solution created in a supercritical Hopf bifurcation. This attracting periodic solution exists for all the mutation patterns considered, and persists arbitrarily close to the limit of zero mutation rate and a zero-sum game., 6 pages, 5 figures
- Published
- 2015
16. Dynamic analysis and bifurcation control of a delayed fractional-order eco-epidemiological migratory bird model with fear effect.
- Author
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Song, Caihong and Li, Ning
- Subjects
MIGRATORY birds ,INFECTIOUS disease transmission ,COST control ,HOPF bifurcations ,PSYCHOLOGICAL feedback ,COMPUTER simulation - Abstract
In this paper, a new delayed fractional-order model including susceptible migratory birds, infected migratory birds and predators is proposed to discuss the spread of diseases among migratory birds. Fear of predators is considered in the model, as fear can reduce the reproduction rate and disease transmission rate among prey. First, some basic mathematical results of the proposed model are discussed. Then, time delay is regarded as a bifurcation parameter, and the delay-induced bifurcation conditions for such an uncontrolled system are established. A novel periodic pulse feedback controller is proposed to suppress the bifurcation phenomenon. It is found that the control scheme can successfully suppress the bifurcation behavior of the system, and the pulse width can be arbitrarily selected on the premise of ensuring the control effect. Compared with the traditional time-delay feedback controller, the control scheme proposed in this paper has more advantages in practical application, which not only embodies the advantages of low control cost and easy operation but also caters to the periodic changes of the environment. The proposed control scheme, in particular, remains effective even after the system has been disrupted by a constant. Numerical simulation verifies the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
17. Hopf bifurcation in a delayed prey–predator model with prey refuge involving fear effect.
- Author
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Parwaliya, Ankit, Singh, Anuraj, and Kumar, Ajay
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PREDATION ,HOPF bifurcations ,COMPUTER simulation ,EQUILIBRIUM ,FEAR in animals - Abstract
This work investigates a prey–predator model featuring a Holling-type II functional response, in which the fear effect of predation on the prey species, as well as prey refuge, are considered. Specifically, the model assumes that the growth rate of the prey population decreases as a result of the fear of predators. Moreover, the detection of the predator by the prey species is subject to a delay known as the fear response delay, which is incorporated into the model. The paper establishes the preliminary conditions for the solution of the delayed model, including positivity, boundedness and permanence. The paper discusses the existence and stability of equilibrium points in the model. In particular, the paper considers the discrete delay as a bifurcation parameter, demonstrating that the system undergoes Hopf bifurcation at a critical value of the delay parameter. The direction and stability of periodic solutions are determined using central manifold and normal form theory. Additionally, the global stability of the model is established at axial and positive equilibrium points. An extensive numerical simulation is presented to validate the analytical findings, including the continuation of the equilibrium branch for positive equilibrium points. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
18. Fractional-order delayed Ross–Macdonald model for malaria transmission
- Author
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Cui, Xinshu, Xue, Dingyu, and Li, Tingxue
- Subjects
Original Paper ,Local stability ,Control and Systems Engineering ,Applied Mathematics ,Mechanical Engineering ,Aerospace Engineering ,Ocean Engineering ,Fractional-order ,Hopf bifurcation ,Electrical and Electronic Engineering ,Incubation periods ,Malaria - Abstract
This paper proposes a novel fractional-order delayed Ross-Macdonald model for malaria transmission. This paper aims to systematically investigate the effect of both the incubation periods of Plasmodium and the order on the dynamic behavior of diseases. Utilizing inequality techniques, contraction mapping theory, fractional linear stability theorem, and bifurcation theory, several sufficient conditions for the existence and uniqueness of solutions, the local stability of the positive equilibrium point, and the existence of fractional-order Hopf bifurcation are obtained under different time delays cases. The results show that time delay can change the stability of system. System becomes unstable and generates a Hopf bifurcation when the delay increases to a certain value. Besides, the value of order influences the stability interval size. Thus, incubation periods and the order have a major effect on the dynamic behavior of the model. The effectiveness of the theoretical results is shown through numerical simulations.
- Published
- 2022
19. A 2-D Discrete Cubic Chaotic Mapping with Symmetry
- Author
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M. Mammeri
- Subjects
Nonlinear Sciences::Chaotic Dynamics ,Physics ,Hopf bifurcation ,symbols.namesake ,Chaotic dynamical systems ,Dynamical systems theory ,Physical phenomena ,Short paper ,symbols ,Lyapunov exponent ,Symmetry breaking ,Bifurcation diagram ,Mathematical physics - Abstract
In the theoretical research of chaotic dynamical system, the different type of bifurcations is a very interesting powerful tool for analyzing the qualitative behavior of chaotic dynamical system; this short paper is devoted to analysis of a simple 2-D symmetry discrete chaotic map with quadratic and cubic nonlinearities. The dynamical behaviors of the map are investigated by mathematical analysis and simulated numerically using package of Matlab . We compute numerically the bifurcation diagram and largest Lyapunov exponent and phase portraits. The research results indicate that there are interesting nonlinear physical phenomena in this simple 2-D symmetry discrete cubic map, such as symmetry bifurcation, Hopf bifurcation, symmetry breaking bifurcation and identical symmetric attractors. The important nonlinear physical phenomena obtained in this paper would benefit the study of the cubic chaotic map and the development of the theory of chaotic discrete dynamical systems. En la investigación teórica de los sistemas dinámicos caóticos, los diferentes tipos de bifurcaciones son una herramienta poderosa muy interesante para analizar el comportamiento cualitativo de los sistemas dinámicos caóticos; este breve artículo está dedicado al análisis de un mapa caótico discreto de simetría bidimensional simple con no linealidades cuadráticas y cúbicas. Los comportamientos dinámicos del mapa se investigan mediante análisis matemático y se simulan numéricamente utilizando el paquete de Matlab . Calculamos numéricamente el diagrama de bifurcación y el mayor exponente de Lyapunov y los retratos de fase. Los resultados de la investigación indican que existen interesantes fenómenos físicos no lineales en este sencillo mapa cúbico discreto de simetría 2-D, como la bifurcación de simetría, la bifurcación de Hopf, la bifurcación de ruptura de simetría y los atractores simétricos idénticos. Los importantes fenómenos físicos no lineales obtenidos en este trabajo beneficiarían el estudio del mapa cúbico caótico y el desarrollo de la teoría de los sistemas dinámicos discretos caóticos.
- Published
- 2021
20. Mathematical derivation and mechanism analysis of beta oscillations in a cortex-pallidum model.
- Author
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Xu, Minbo, Hu, Bing, Wang, Zhizhi, Zhu, Luyao, Lin, Jiahui, and Wang, Dingjiang
- Abstract
In this paper, we develop a new cortex-pallidum model to study the origin mechanism of Parkinson's oscillations in the cortex. In contrast to many previous models, the globus pallidus internal (GPi) and externa (GPe) both exert direct inhibitory feedback to the cortex. Using Hopf bifurcation analysis, two new critical conditions for oscillations, which can include the self-feedback projection of GPe, are obtained. In this paper, we find that the average discharge rate (ADR) is an important marker of oscillations, which can divide Hopf bifurcations into two types that can uniformly be used to explain the oscillation mechanism. Interestingly, the ADR of the cortex first increases and then decreases with increasing coupling weights that are projected to the GPe. Regarding the Hopf bifurcation critical conditions, the quantitative relationship between the inhibitory projection and excitatory projection to the GPe is monotonically increasing; in contrast, the relationship between different coupling weights in the cortex is monotonically decreasing. In general, the oscillation amplitude is the lowest near the bifurcation points and reaches the maximum value with the evolution of oscillations. The GPe is an effective target for deep brain stimulation to alleviate oscillations in the cortex. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
21. Detections of bifurcation in a fractional-order Cohen-Grossberg neural network with multiple delays.
- Author
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Huang, Chengdai, Mo, Shansong, and Cao, Jinde
- Abstract
The dynamics of integer-order Cohen-Grossberg neural networks with time delays has lately drawn tremendous attention. It reveals that fractional calculus plays a crucial role on influencing the dynamical behaviors of neural networks (NNs). This paper deals with the problem of the stability and bifurcation of fractional-order Cohen-Grossberg neural networks (FOCGNNs) with two different leakage delay and communication delay. The bifurcation results with regard to leakage delay are firstly gained. Then, communication delay is viewed as a bifurcation parameter to detect the critical values of bifurcations for the addressed FOCGNN, and the communication delay induced-bifurcation conditions are procured. We further discover that fractional orders can enlarge (reduce) stability regions of the addressed FOCGNN. Furthermore, we discover that, for the same system parameters, the convergence time to the equilibrium point of FONN is shorter (longer) than that of integer-order NNs. In this paper, the present methodology to handle the characteristic equation with triple transcendental terms in delayed FOCGNNs is concise, neoteric and flexible in contrast with the prior mechanisms owing to skillfully keeping away from the intricate classified discussions. Eventually, the developed analytic results are nicely showcased by the simulation examples. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
22. Modeling Excitable Cells with Memristors.
- Author
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Sah, Maheshwar, Ascoli, Alon, Tetzlaff, Ronald, Rajamani, Vetriveeran, and Budhathoki, Ram Kaji
- Subjects
MEMRISTORS ,POTASSIUM channels ,VOLTAGE-gated ion channels ,ION channels ,BIOLOGICAL membranes ,BIOLOGICAL systems ,POTASSIUM ions - Abstract
This paper presents an in-depth analysis of an excitable membrane of a biological system by proposing a novel approach that the cells of the excitable membrane can be modeled as the networks of memristors. We provide compelling evidence from the Chay neuron model that the state-independent mixed ion channel is a nonlinear resistor, while the state-dependent voltage-sensitive potassium ion channel and calcium-sensitive potassium ion channel function as generic memristors from the perspective of electrical circuit theory. The mechanisms that give rise to periodic oscillation, aperiodic (chaotic) oscillation, spikes, and bursting in an excitable cell are also analyzed via a small-signal model, a pole-zero diagram of admittance functions, local activity, the edge of chaos, and the Hopf bifurcation theorem. It is also proved that the zeros of the admittance functions are equivalent to the eigen values of the Jacobian matrix, and the presence of the positive real parts of the eigen values between the two bifurcation points lead to the generation of complicated electrical signals in an excitable membrane. The innovative concepts outlined in this paper pave the way for a deeper understanding of the dynamic behavior of excitable cells, offering potent tools for simulating and exploring the fundamental characteristics of biological neurons. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
23. Bogdanov–Takens Bifurcation of Kermack–McKendrick Model with Nonlinear Contact Rates Caused by Multiple Exposures.
- Author
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Li, Jun and Ma, Mingju
- Subjects
HOPF bifurcations ,LIMIT cycles ,BIFURCATION diagrams ,PHASE diagrams ,DEATH rate - Abstract
In this paper, we consider the influence of a nonlinear contact rate caused by multiple contacts in classical SIR model. In this paper, we unversal unfolding a nilpotent cusp singularity in such systems through normal form theory, we reveal that the system undergoes a Bogdanov-Takens bifurcation with codimension 2. During the bifurcation process, numerous lower codimension bifurcations may emerge simultaneously, such as saddle-node and Hopf bifurcations with codimension 1. Finally, employing the Matcont and Phase Plane software, we construct bifurcation diagrams and topological phase portraits. Additionally, we emphasize the role of symmetry in our analysis. By considering the inherent symmetries in the system, we provide a more comprehensive understanding of the dynamical behavior. Our findings suggest that if this occurrence rate is applied to the SIR model, it would yield different dynamical phenomena compared to those obtained by reducing a 3-dimensional dynamical model to a planar system by neglecting the disease mortality rate, which results in a stable nilpotent cusp singularity with codimension 2. We found that in SIR models with the same occurrence rate, both stable and unstable Bogdanov-Takens bifurcations occur, meaning both stable and unstable limit cycles appear in this system. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
24. Bifurcation Analysis of a Non-Linear Vehicle Model Under Wet Surface Road Condition.
- Author
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Kumar, Abhay, Verma, Suresh Kant, and Dheer, Dharmendra Kumar
- Subjects
ACCIDENT prevention ,BIFURCATION theory ,HOPF bifurcations ,TRAFFIC accidents ,EQUILIBRIUM - Abstract
The vehicles are prone to accidents during cornering on a wet or low friction coefficient roads if the longitudinal velocity (V
x ) and steering angle (δ) are increased beyond a certain limit. Therefore, it is of major concern to analyze the behaviour and define the stability boundary of the vehicle for such scenarios. In this paper, stability analysis of a 2 degrees of freedom nonlinear bicycle model replicating a car model including lateral (sideslip angle β) and yaw (yaw rate r) dynamics only operating on a wet surface road has been performed. The stability is analysed by utilizing the phase plane method and bifurcation analysis. The obtained converging and diverging nature of the trajectories (β, r) depicts the stable and unstable equilibrium points in the phase plane. The movement of these points results in the transition of the stability known as bifurcation due to the change in the control parameters (Vx , δ). The Matcont/Matlab is utilized to obtain the bifurcation diagrams and the nature of bifurcations. The obtained results show that a saddle node (SNB) and a subcritical Hopf bifurcation (HB) exists for steering angle (±0.08 rad) and higher than (±0.08 rad) with Vx = (10 - 40) m/s respectively. The SNB and HB denotes the spinning of the vehicle and sliding of the vehicle respectively, thus generating a unstable behaviour. A stability boundary is obtained representing the stable and unstable range of parameters. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
25. Multi-modal Swarm Coordination via Hopf Bifurcations.
- Author
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Baxevani, Kleio and Tanner, Herbert G.
- Abstract
This paper outlines a methodology for the construction of vector fields that can enable a multi-robot system moving on the plane to generate multiple dynamical behaviors by adjusting a single scalar parameter. This parameter essentially triggers a Hopf bifurcation in an underlying time-varying dynamical system that steers a robotic swarm. This way, the swarm can exhibit a variety of behaviors that arise from the same set of continuous differential equations. Other approaches to bifurcation-based swarm coordination rely on agent interaction which cannot be realized if the swarm members cannot sense or communicate with one another. The contribution of this paper is to offer an alternative method for steering minimally instrumented multi-robot collectives with a control strategy that can realize a multitude of dynamical behaviors without switching their constituent equations. Through this approach, analytical solutions for the bifurcation parameter are provided, even for more complex cases that are described in the literature, along with the process to apply this theory in a multi-agent setup. The theoretical predictions are confirmed via simulation and experimental results with the latter also demonstrating real-world applicability. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
26. Complex dynamics of an epidemic model with saturated media coverage and recovery
- Author
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Yanni Xiao and Tangjuan Li
- Subjects
Original Paper ,Computer science ,Applied Mathematics ,Mechanical Engineering ,Nonlinear recovery function ,Aerospace Engineering ,Media coverage ,Ocean Engineering ,computer.software_genre ,Basic reproduction number ,Complex dynamics ,Mathematical model ,Saddle-node bifurcation ,Media impact ,Control and Systems Engineering ,Data mining ,Hopf bifurcation ,Electrical and Electronic Engineering ,Epidemic model ,computer - Abstract
During the outbreak of emerging infectious diseases, media coverage and medical resource play important roles in affecting the disease transmission. To investigate the effects of the saturation of media coverage and limited medical resources, we proposed a mathematical model with extra compartment of media coverage and two nonlinear functions. We theoretically and numerically investigate the dynamics of the proposed model. Given great difficulties caused by high nonlinearity in theoretical analysis, we separately considered subsystems with only nonlinear recovery or with only saturated media impact. For the model with only nonlinear recovery, we theoretically showed that backward bifurcation can occur and multiple equilibria may coexist under certain conditions in this case. Numerical simulations reveal the rich dynamic behaviors, including forward-backward bifurcation, Hopf bifurcation, saddle-node bifurcation, homoclinic bifurcation and unstable limit cycle. So the limitation of medical resources induces rich dynamics and causes much difficulties in eliminating the infectious diseases. We then investigated the dynamics of the system with only saturated media impact and concluded that saturated media impact hardly induces the complicated dynamics. Further, we parameterized the proposed model on the basis of the COVID-19 case data in mainland China and data related to news items, and estimated the basic reproduction number to be 2.86. Sensitivity analyses were carried out to quantify the relative importance of parameters in determining the cumulative number of infected individuals at the end of the first month of the outbreak. Combining with numerical analyses, we suggested that providing adequate medical resources and improving media response to infection or individuals' response to mass media may reduce the cumulative number of the infected individuals, which mitigates the transmission dynamics during the early stage of the COVID-19 pandemic.
- Published
- 2022
27. Appearance of Temporal and Spatial Chaos in an Ecological System: A Mathematical Modeling Study
- Author
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S. N. Raw, B P Sarangi, P. Mishra, and B. Tiwari
- Subjects
Patter formulation ,Computer science ,General Mathematics ,General Physics and Astronomy ,Lyapunov exponent ,01 natural sciences ,Stability (probability) ,symbols.namesake ,Quantitative Biology::Populations and Evolution ,Statistical physics ,0101 mathematics ,Bifurcation ,Hopf bifurcation ,Computer simulation ,Phase portrait ,Turing instability ,010102 general mathematics ,Time evolution ,General Chemistry ,Function (mathematics) ,010101 applied mathematics ,symbols ,Chaos ,Mutual interference ,General Earth and Planetary Sciences ,General Agricultural and Biological Sciences ,Research Paper - Abstract
The ecological theory of species interactions rests largely on the competition, interference, and predator–prey models. In this paper, we propose and investigate a three-species predator–prey model to inspect the mutual interference between predators. We analyze boundedness and Kolmogorov conditions for the non-spatial model. The dynamical behavior of the system is analyzed by stability and Hopf bifurcation analysis. The Turing instability criteria for the Spatio-temporal system is estimated. In the numerical simulation, phase portrait with time evolution diagrams shows periodic and chaotic oscillations. Bifurcation diagrams show the very rich and complex dynamical behavior of the non-spatial model. We calculate the Lyapunov exponent to justify the dynamics of the non-spatial model. A variety of patterns like interference, spot, and stripe are observed with special emphasis on Beddington–DeAngelis function response. These complex patterns explore the beauty of the spatio-temporal model and it can be easily related to real-world biological systems.
- Published
- 2021
28. Study on the Strong Nonlinear Dynamics of Nonlocal Nanobeam Under Time-Delayed Feedback Using Homotopy Analysis Method
- Author
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Li, Jia-Xuan, Yan, Yan, Wang, Wen-Quan, and Wu, Feng-Xia
- Published
- 2024
- Full Text
- View/download PDF
29. Dynamical Study of an Eco-Epidemiological Delay Model for Plankton System with Toxicity
- Author
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Archana Ojha, Nilesh Kumar Thakur, and Smriti Chandra Srivastava
- Subjects
General Mathematics ,Population ,Chaotic ,General Physics and Astronomy ,01 natural sciences ,Stability (probability) ,Zooplankton ,010305 fluids & plasmas ,symbols.namesake ,0103 physical sciences ,Carrying capacity ,Quantitative Biology::Populations and Evolution ,education ,010301 acoustics ,Mathematics ,Equilibrium point ,Hopf bifurcation ,education.field_of_study ,Toxicity ,fungi ,General Chemistry ,Plankton ,System dynamics ,Local stability ,Hopf-bifurcation ,symbols ,General Earth and Planetary Sciences ,Chaos ,General Agricultural and Biological Sciences ,Biological system ,Time delay ,Research Paper - Abstract
In this paper, we analyze the complexity of an eco-epidemiological model for phytoplankton–zooplankton system in presence of toxicity and time delay. Holling type II function response is incorporated to address the predation rate as well as toxic substance distribution in zooplankton. It is also presumed that infected phytoplankton does recover from the viral infection. In the absence of time delay, stability and Hopf-bifurcation conditions are investigated to explore the system dynamics around all the possible equilibrium points. Further, in the presence of time delay, conditions for local stability are derived around the interior equilibria and the properties of the periodic solution are obtained by applying normal form theory and central manifold arguments. Computational simulation is performed to illustrate our theoretical findings. It is explored that system dynamics is very sensitive corresponding to carrying capacity and toxin liberation rate and able to generate chaos. Further, it is observed that time delay in the viral infection process can destabilize the phytoplankton density whereas zooplankton density remains in its old state. Incorporation of time delay also gives the scenario of double Hopf-bifurcation. Some control parameters are discussed to stabilize system dynamics. The effect of time delay on (i) growth rate of susceptible phytoplankton shows the extinction and double Hopf-bifurcation in the zooplankton population, (ii) a sufficiently large value of carrying capacity stabilizes the chaotic dynamics or makes the whole system chaotic with further increment.
- Published
- 2021
30. Dynamics of a diffusive model for cancer stem cells with time delay in microRNA-differentiated cancer cell interactions and radiotherapy effects.
- Author
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Essongo, Frank Eric, Mvogo, Alain, and Ben-Bolie, Germain Hubert
- Abstract
Understand the dynamics of cancer stem cells (CSCs), prevent the non-recurrence of cancers and develop therapeutic strategies to destroy both cancer cells and CSCs remain a challenge topic. In this paper, we study both analytically and numerically the dynamics of CSCs under radiotherapy effects. The dynamical model takes into account the diffusion of cells, the de-differentiation (or plasticity) mechanism of differentiated cancer cells (DCs) and the time delay on the interaction between microRNAs molecules (microRNAs) with DCs. The stability of the model system is studied by using a Hopf bifurcation analysis. We mainly investigate on the critical time delay τ c , that represents the time for DCs to transform into CSCs after the interaction of microRNAs with DCs. Using the system parameters, we calculate the value of τ c for prostate, lung and breast cancers. To confirm the analytical predictions, the numerical simulations are performed and show the formation of spatiotemporal circular patterns. Such patterns have been found as promising diagnostic and therapeutic value in management of cancer and various diseases. The radiotherapy is applied in the particular case of prostate model. We calculate the optimum dose of radiation and determine the probability of avoiding local cancer recurrence after radiotherapy treatment. We find numerically a complete eradication of patterns when the radiotherapy is applied before a time t < τ c . This scenario induces microRNAs to act as suppressors as experimentally observed in prostate cancer. The results obtained in this paper will provide a better concept for the clinicians and oncologists to understand the complex dynamics of CSCs and to design more efficacious therapeutic strategies to prevent the non-recurrence of cancers. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
31. Modeling and analysis of demand-supply dynamics with a collectability factor using delay differential equations in economic growth via the Caputo operator.
- Author
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Chen, Qiliang, Dipesh, Kumar, Pankaj, and Baskonus, Haci Mehmet
- Subjects
ECONOMIC expansion ,HOPF bifurcations ,DELAY differential equations ,LIMIT cycles ,OPERATING costs ,SUPPLY & demand - Abstract
In this paper, to investigate the dynamic interplay between supply and demand, with a focus on collectability, a novel mathematical model was introduced via conformable operator. This model considers the possibility that operating expenses or a lack of raw materials causes a manufacturing delay than the supply of goods instantly matching demand. This maturation (delay) is represented by the delay factor (τ) . Stability analysis revolves around the equilibrium point other than zero. Chaotic behavior emerges through Hopf bifurcation at the critical delay parameter value. If this delay parameter is even slightly perturbed, oscillatory limit cycles can be induced in the market dynamics, leading to equilibrium with brisk market expansion, frequent recessions, and sudden collapses. We conducted sensitivity and directional analysis on a number of factors while also examining the stability and duration of the Hopf bifurcation. Numerical findings were validated using MATLAB. Additionally, the Caputo operator was used to examine the fractional of demand and supply dynamics. Importantly, we assumed a pivotal role in advancing fair labor practices and fostering economic growth on a national scale. In this paper, to investigate the dynamic interplay between supply and demand, with a focus on collectability, a novel mathematical model was introduced via conformable operator. This model considers the possibility that operating expenses or a lack of raw materials causes a manufacturing delay than the supply of goods instantly matching demand. This maturation (delay) is represented by the delay factor . Stability analysis revolves around the equilibrium point other than zero. Chaotic behavior emerges through Hopf bifurcation at the critical delay parameter value. If this delay parameter is even slightly perturbed, oscillatory limit cycles can be induced in the market dynamics, leading to equilibrium with brisk market expansion, frequent recessions, and sudden collapses. We conducted sensitivity and directional analysis on a number of factors while also examining the stability and duration of the Hopf bifurcation. Numerical findings were validated using MATLAB. Additionally, the Caputo operator was used to examine the fractional of demand and supply dynamics. Importantly, we assumed a pivotal role in advancing fair labor practices and fostering economic growth on a national scale. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
32. Bifurcation detections of a fractional-order neural network involving three delays.
- Author
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Wang, Huanan, Huang, Chengdai, Li, Shuai, Cao, Jinde, and Liu, Heng
- Abstract
This paper lucubrates the Hopf bifurcation of fractional-order Hopfield neural network (FOHNN) with three nonidentical delays. The type of delays in the model include leakage delay, self-connection delay and communication delay. Differentiating from traditional bifurcation exploration of delayed fractional-order system, this paper presents a succinct and systematic approach as much as possible to settle the bifurcation problem when all three delays fluctuate and aren't convertible. In addition, this paper furnishes a humble opinion for solving bifurcation cases caused by arbitrary unequal delays. At length, we address three simulation examples to corroborate the correctness of key fruits. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
33. Evolutionary Game Analysis of Digital Financial Enterprises and Regulators Based on Delayed Replication Dynamic Equation.
- Author
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Xu, Mengzhu, Liu, Zixin, Xu, Changjin, and Wang, Nengfa
- Subjects
CORPORATE finance ,HOPF bifurcations ,REACTION-diffusion equations ,EQUATIONS ,GAME theory ,FINANCIAL risk - Abstract
With the frequent occurrence of financial risks, financial innovation supervision has become an important research issue, and excellent regulatory strategies are of great significance to maintain the stability and sustainable development of financial markets. Thus, this paper intends to analyze the financial regulation strategies through evolutionary game theory. In this paper, the delayed replication dynamic equation and the non-delayed replication dynamic equation are established, respectively, under different reward and punishment mechanisms, and their stability conditions and evolutionary stability strategies are investigated. The analysis finds that under the static mechanism, the internal equilibrium is unstable, and the delay does not affect the stability of the system, while in the dynamic mechanism, when the delay is less than a critical value, the two sides of the game have an evolutionary stable strategy, otherwise it is unstable, and Hopf bifurcation occurs at threshold. Finally, some numerical simulation examples are provided, and the numerical results show the correctness of the proposed algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
34. Bifurcation Analysis of Time-Delayed Non-Commensurate Caputo Fractional Bi-Directional Associative Memory Neural Networks Composed of Three Neurons.
- Author
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Wang, Chengqiang, Zhao, Xiangqing, Mai, Qiuyue, and Lv, Zhiwei
- Subjects
BIDIRECTIONAL associative memories (Computer science) ,CAPUTO fractional derivatives ,HOPF bifurcations - Abstract
We are concerned in this paper with the stability and bifurcation problems for three-neuron-based bi-directional associative memory neural networks that are involved with time delays in transmission terms and possess Caputo fractional derivatives of non-commensurate orders. For the fractional bi-directional associative memory neural networks that are dealt with in this paper, we view the time delays as the bifurcation parameters. Via a standard contraction mapping argument, we establish the existence and uniqueness of the state trajectories of the investigated fractional bi-directional associative memory neural networks. By utilizing the idea and technique of linearization, we analyze the influence of time delays on the dynamical behavior of the investigated neural networks, as well as establish and prove several stability/bifurcation criteria for the neural networks dealt with in this paper. According to each of our established criteria, the equilibrium states of the investigated fractional bi-directional associative memory neural networks are asymptotically stable when some of the time delays are less than strictly specific positive constants, i.e., when the thresholds or the bifurcation points undergo Hopf bifurcation in the concerned networks at the aforementioned threshold constants. In the meantime, we provide several illustrative examples to numerically and visually validate our stability and bifurcation results. Our stability and bifurcation theoretical results in this paper yield some insights into the cause mechanism of the bifurcation phenomena for some other complex phenomena, and this is extremely helpful for the design of feedback control to attenuate or even to remove such complex phenomena in the dynamics of fractional bi-directional associative memory neural networks with time delays. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
35. Hopf bifurcation analytical expression and control strategy in direct‐drive permanent magnet synchronous generator.
- Author
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Li, Qiangqiang, Chen, Wei, Wei, Zhanhong, Wang, Kun, and Wang, Bo
- Abstract
Summary: The unstable operation of a direct‐drive permanent magnet synchronous generator (DPMSG) is exacerbated by periodic oscillation, having a significant impact on the safe and stable functioning of wind energy generation systems. This paper proposes an approximate solution method for analyzing the periodic oscillation of the Hopf bifurcation and a method for suppressing its bifurcation through H∞ robust control. Firstly, a three‐dimensional nonlinear dynamic model of the DPMSG is constructed. Secondly, the Hopf bifurcation of the system with changes in internal parameters is solved using the time domain and frequency domain methods, and the harmonic balancing method is used to approximate the periodic solution at the point of Hopf bifurcation. The order of the Hopf bifurcation neighborhood is then decreased using the central manifold theorem. Finally, this study suggests a H∞ output feedback control employing bifurcation parameters. The Nyquist criteria are used for evaluating small signal stability and convergence speed of the response system in DPMSG before and after dimensionality reduction. The simulation results suggest that the proposed strategy helps to tackle the periodic solution expression and instability difficulties caused by Hopf bifurcation. This paper provides theoretical suggestions for the future reliable operation of new energy generation systems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
36. Effect of Leakage Delays on Bifurcation in Fractional-Order Bidirectional Associative Memory Neural Networks with Five Neurons and Discrete Delays.
- Author
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Wang, Yangling, Cao, Jinde, and Huang, Chengdai
- Abstract
As is well known that time delays are inevitable in practice due to the finite switching speed of amplifiers and information transmission between neurons. So the study on the Hopf bifurcation of delayed neural networks has aroused extensive attention in recent years. However, it's worth mentioning that only the communication delays between neurons were generally considered in most existing relevant literatures. Actually, it has been proven that a kind of so-called leakage delays cannot be ignored because the self-decay process of a neuron's action potential is not instantaneous in hardware implementation of neural networks. Though leakage delays have been taken into account in a few more recent works concerning the Hopf bifurcation of fractional-order bidirectional associative memory neural networks, the addressed neural networks were low-dimension or the involved time delays were single. In this paper, we propose a five-neuron fractional-order bidirectional associative memory neural network model, which includes leakage delays and discrete communication delays to meet the characteristics of real neural networks better. Then we use the stability theory of fractional differential equations and Hopf bifurcation theory to investigate its dynamic behavior of Hopf bifurcation. The Hopf bifurcation of the proposed model are studied by taking the involved two different leakage delays as the bifurcation parameter respectively, and two kinds of sufficient conditions for Hopf bifurcation are obtained. A numerical example as well as its simulation plots and phase portraits are given at last. Our results indicate that a Hopf bifurcation rises near the zero equilibrium point when the leakage delay reaches its critical value which is given by an explicit formula. Particularly, the results of numerical simulations show that the leakage delay would narrow the stability region of the proposed system and make the Hopf bifurcation occur earlier. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
37. Hybrid control of Turing instability and bifurcation for spatial-temporal propagation of computer virus.
- Author
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Ju, Yawen, Xiao, Min, Huang, Chengdai, Rutkowski, Leszek, and Cao, Jinde
- Subjects
COMPUTER viruses ,INFORMATION technology ,HOPF bifurcations ,BACK propagation - Abstract
In this era of information technology, information leakage and file corruption due to computer virus intrusion have been serious issues. How to detect and prevent the spread of the computer virus is the major challenge we are facing now. To target this problem, a class of virus propagation models with hybrid control scheme are formulated to investigate the dynamic evolution and prevention from a spatial-temporal perspective in this paper. Diffusion-induced Turing instability is detected in response to the computer virus propagation. The introduction of hybrid control scheme can effective suppress Turing instability and turn the propagation system back to a stable state. And then, the time delay is selected as the bifurcation parameter. If the time delay exceeds the bifurcation threshold, the propagation will be destabilised and a Hopf bifurcation will occur. The hybrid control tactic can not only regulate the occurrence of Hopf bifurcation well, but also optimise the properties of bifurcating period solutions. In the end, the correctness and validity of the theoretical results are verified via numerical simulations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
38. Bifurcation analysis of an algal blooms dynamical model in trophic interaction.
- Author
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Wei, Qian and Cai, Liming
- Abstract
In this paper, we revisit the algal blooms model of plankton interactions initially proposed by Das and Sarkar (DCDIS-A, 14(3):401–414, 2007), where the oscillatory mode in the interaction between phytoplankton and zooplankton is observed. We provide a detailed analysis of the dependence of the equilibria and their stability on various parameters in the model. The bifurcation behaviors around equilibrium (e.g., Hopf bifurcation, Bogdanov–Takens bifurcation) are further found. Meanwhile, numerical simulations verify and illustrate the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
39. Dynamics of a two-patch logistic model with diffusion and time delay.
- Author
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Sawada, Yukihiro, Takeuchi, Yasuhiro, and Dong, Yueping
- Subjects
HOPF bifurcations ,GAMMA distributions - Abstract
In this paper, we proposed a two-patch logistic model connected by diffusion, where one patch includes the Gamma type distribution time delay while the other patch does not include the time delay. In general, Routh–Hurwitz criterion is applied to the derivation for the conditions of Hopf bifurcation, but the more the order of the time delay increases the more the difficulty rises. Hence we adopt the polar form method for the characteristic equation to study the stability of coexistence equilibrium. Our findings show that the diffusion prevents the instabilization of the coexistence equilibrium. Besides, we found that the coexistence equilibrium is stable when time delay is small, and becomes unstable as the delay increases. But it can be restabilized for further increasing of time delay and continues to be stable afterwards. In other words, the diffusion and the time delay are beneficial to the stability of the coexistence equilibrium. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
40. Hopf bifurcation for a class of predator-prey system with small immigration.
- Author
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Lima, Maurıicio F. S. and Llibre, Jaume
- Subjects
HOPF bifurcations ,LOTKA-Volterra equations ,BIOLOGICAL systems ,LIMIT cycles ,COMBINATORIAL dynamics - Abstract
The subject of this paper concerns with the bifurcation of limit cycles for a predator-prey model with small immigration. Since, in general, the biological systems are not isolated, taking into account immigration in the model becomes more realistic. In this context, we deal with a model with a Holling type Ⅰ function response and study, using averaging theory of second order, the Hopf bifurcation that can emerge under small perturbation of the biological parameters. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. Fractional-Order Tabu Learning Neuron Models and Their Dynamics.
- Author
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Yu, Yajuan, Gu, Zhenhua, Shi, Min, and Wang, Feng
- Subjects
JACOBIAN matrices ,KERNEL functions ,HOPF bifurcations ,FREQUENCIES of oscillating systems ,TABOO - Abstract
In this paper, by replacing the exponential memory kernel function of a tabu learning single-neuron model with the power-law memory kernel function, a novel Caputo's fractional-order tabu learning single-neuron model and a network of two interacting fractional-order tabu learning neurons are constructed firstly. Different from the integer-order tabu learning model, the order of the fractional-order derivative is used to measure the neuron's memory decay rate and then the stabilities of the models are evaluated by the eigenvalues of the Jacobian matrix at the equilibrium point of the fractional-order models. By choosing the memory decay rate (or the order of the fractional-order derivative) as the bifurcation parameter, it is proved that Hopf bifurcation occurs in the fractional-order tabu learning single-neuron model where the value of bifurcation point in the fractional-order model is smaller than the integer-order model's. By numerical simulations, it is shown that the fractional-order network with a lower memory decay rate is capable of producing tangent bifurcation as the learning rate increases from 0 to 0.4. When the learning rate is fixed and the memory decay increases, the fractional-order network enters into frequency synchronization firstly and then enters into amplitude synchronization. During the synchronization process, the oscillation frequency of the fractional-order tabu learning two-neuron network increases with an increase in the memory decay rate. This implies that the higher the memory decay rate of neurons, the higher the learning frequency will be. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
42. Probing the effects of fiscal policy delays in macroeconomic IS–LM model.
- Author
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Rajpal, Akanksha, Bhatia, Sumit Kaur, and Kumar, Praveen
- Subjects
FISCAL policy ,HOPF bifurcations ,MACROECONOMIC models ,DIFFERENTIAL equations ,LINEAR statistical models ,MATHEMATICAL models ,DELAY differential equations - Abstract
In this paper, we address the effects of two fiscal policy delays on the dynamical analysis of macroeconomics. First, a time gap between the accrual of taxes and their payment is considered. Second, the time spent between the purchasing decisions and the actual expenditure is also taken into consideration. Since both these delays are significant in controlling macroeconomic conditions, this paper incorporates aforementioned delays into the IS–LM model. At first, a mathematical model is developed using delayed differential equations. Then a unique steady state solution is obtained. Around the equilibrium point, linear stability analysis is done. Also, the occurance of Hopf bifurcation is observed when delay crosses a critical point and switches in stability are also detected. Properties of Hopf bifurcation using center manifold theorem are discussed. Lastly, numerical simulations are run to verify our analysis. In this work, we considered a case study to perform simulation wherein GDP of India for last ten years is recorded for estimating some parameters. In different investment scenarios, numerical simulations corroborate the analytical findings of the model. Furthermore, rigorous analysis shows that adding the right mix of delays can help in maintaining/ regaining the stability after periods of instability, or even gaining stability in the long run. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
43. Bifurcation analysis of a Leslie-type predator-prey system with prey harvesting and group defense.
- Author
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Yongxin Zhang, Jianfeng Luo, Jun Hu, and Kaifa Wang
- Subjects
PREDATION ,LOTKA-Volterra equations ,STOCHASTIC analysis ,LIMIT cycles ,JACOBIAN matrices ,ALLEE effect ,BIFURCATION diagrams ,BIOLOGICAL extinction - Abstract
In this paper, we investigate a Leslie-type predator-prey model that incorporates prey harvesting and group defense, leading to a modified functional response. Our analysis focuses on the existence and stability of the system's equilibria, which are essential for the coexistence of predator and prey populations and the maintenance of ecological balance. We identify the maximum sustainable yield, a critical factor for achieving this balance. Through a thorough examination of positive equilibrium stability, we determine the conditions and initial values that promote the survival of both species. We delve into the system's dynamics by analyzing saddle-node and Hopf bifurcations, which are crucial for understanding the system transitions between various states. To evaluate the stability of the Hopf bifurcation, we calculate the first Lyapunov exponent and offer a quantitative assessment of the system's stability. Furthermore, we explore the Bogdanov-Takens (BT) bifurcation, a co-dimension 2 scenario, by employing a universal unfolding technique near the cusp point. This method simplifies the complex dynamics and reveals the conditions that trigger such bifurcations. To substantiate our theoretical findings, we conduct numerical simulations, which serve as a practical validation of the model predictions. These simulations not only confirm the theoretical results but also showcase the potential of the model for predicting real-world ecological scenarios. This in-depth analysis contributes to a nuanced understanding of the dynamics within predator-prey interactions and advances the field of ecological modeling. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
44. Hopf bifurcation and stability analysis of a delay differential equation model for biodegradation of a class of microcystins.
- Author
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Luyao Zhao, Mou Li, and Wanbiao Ma
- Subjects
HOPF bifurcations ,DELAY differential equations ,MICROCYSTINS ,LOTKA-Volterra equations ,BIODEGRADATION ,SPHINGOMONAS - Abstract
In this paper, a delay differential equation model is investigated, which describes the biodegradation of microcystins (MCs) by Sphingomonas sp. and its degrading enzymes. First, the local stability of the positive equilibrium and the existence of the Hopf bifurcation are obtained. Second, the global attractivity of the positive equilibrium is obtained by constructing suitable Lyapunov functionals, which implies that the biodegradation of microcystins is sustainable under appropriate conditions. In addition, some numerical simulations of the model are carried out to illustrate the theoretical results. Finally, the parameters of the model are determined from the experimental data and fitted to the data. The results show that the trajectories of the model fit well with the trend of the experimental data. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
45. Research on pattern dynamics of a class of predator-prey model with interval biological coefficients for capture.
- Author
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Xiao-Long Gao, Hao-Lu Zhang, and Xiao-Yu Li
- Subjects
PREDATION ,BIOLOGICAL models ,HOPF bifurcations ,NONLINEAR analysis ,LOTKA-Volterra equations ,DIFFUSION coefficients ,MEASUREMENT errors ,NATURAL disasters - Abstract
Due to factors such as climate change, natural disasters, and deforestation, most measurement processes and initial data may have errors. Therefore, models with imprecise parameters are more realistic. This paper constructed a new predator-prey model with an interval biological coefficient by using the interval number as the model parameter. First, the stability of the solution of the fractional order model without a diffusion term and the Hopf bifurcation of the fractional order α were analyzed theoretically. Then, taking the diffusion coefficient of prey as the key parameter, the Turing stability at the equilibrium point was discussed. The amplitude equation near the threshold of the Turing instability was given by using the weak nonlinear analysis method, and different mode selections were classified by using the amplitude equation. Finally, we numerically proved that the dispersal rate of the prey population suppressed the spatiotemporal chaos of the model. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
46. SIRS epidemic modeling using fractional-ordered differential equations: Role of fear effect.
- Author
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Mangal, Shiv, Misra, O. P., and Dhar, Joydip
- Subjects
DIFFERENTIAL equations ,FRACTIONAL differential equations ,EPIDEMICS ,HOPF bifurcations ,COMMUNICABLE diseases ,BASIC reproduction number ,CLASSICAL swine fever - Abstract
In this paper, an SIRS epidemic model using Grunwald–Letnikov fractional-order derivative is formulated with the help of a nonlinear system of fractional differential equations to analyze the effects of fear in the population during the outbreak of deadly infectious diseases. The criteria for the spread or extinction of the disease are derived and discussed on the basis of the basic reproduction number. The condition for the existence of Hopf bifurcation is discussed considering fractional order as a bifurcation parameter. Additionally, using the Grunwald–Letnikov approximation, the simulation is carried out to confirm the validity of analytic results graphically. Using the real data of COVID-19 in India recorded during the second wave from 15 May 2021 to 15 December 2021, we estimate the model parameters and find that the fractional-order model gives the closer forecast of the disease than the classical one. Both the analytical results and numerical simulations presented in this study suggest different policies for controlling or eradicating many infectious diseases. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
47. The number of limit cycles of Josephson equation.
- Author
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Yu, Xiangqin, Chen, Hebai, and Liu, Changjian
- Subjects
LIMIT cycles ,DIFFERENTIAL equations ,EQUATIONS ,HOPF bifurcations - Abstract
In this paper, the existence and number of non-contractible limit cycles of the Josephson equation $ \beta \frac{d^{2}\Phi}{dt^{2}}+(1+\gamma \cos \Phi)\frac{d\Phi}{dt}+\sin \Phi = \alpha $ are studied, where $ \phi\in \mathbb S^{1} $ and $ (\alpha,\beta,\gamma)\in \mathbb R^{3} $. Concretely, by using some appropriate transformations, we prove that such type of limit cycles are changed to limit cycles of some Abel equation. By developing the methods on limit cycles of Abel equation, we prove that there are at most two non-contractible limit cycles, and the upper bound is sharp. At last, combining with the results of the paper (Chen and Tang, J. Differential Equations, 2020), we show that the sum of the number of contractible and non-contractible limit cycles of the Josephson equation is also at most two, and give the possible configurations of limit cycles when two limit cycles appear. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
48. An ingenious scheme to bifurcations in a fractional-order Cohen–Grossberg neural network with different delays
- Author
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Huang, Chengdai, Mo, Shansong, Li, Zhouhong, Liu, Heng, and Cao, Jinde
- Published
- 2024
- Full Text
- View/download PDF
49. Stability and Hopf bifurcation analysis of a fractional-order ring-hub structure neural network with delays under parameters delay feedback control.
- Author
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Ma, Yuan and Dai, Yunxian
- Subjects
HOPF bifurcations ,STABILITY theory ,ARTIFICIAL neural networks ,FEEDBACK control systems ,COMPUTER simulation - Abstract
In this paper, a fractional-order two delays neural network with ring-hub structure is investigated. Firstly, the stability and the existence of Hopf bifurcation of proposed system are obtained by taking the sum of two delays as the bifurcation parameter. Furthermore, a parameters delay feedback controller is introduced to control successfully Hopf bifurcation. The novelty of this paper is that the characteristic equation corresponding to system has two time delays and the parameters depend on one of them. Selecting two time delays as the bifurcation parameters simultaneously, stability switching curves in (τ 1 , τ 2) plane and crossing direction are obtained. Sufficient criteria for the stability and the existence of Hopf bifurcation of controlled system are given. Ultimately, numerical simulation shows that parameters delay feedback controller can effectively control Hopf bifurcation of system. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
50. The impact of fear effect on the dynamics of a delayed predator–prey model with stage structure.
- Author
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Cao, Qi, Chen, Guotai, and Yang, Wensheng
- Subjects
HOPF bifurcations ,PREDATION ,ORDINARY differential equations ,NONLINEAR differential equations ,COMPETITION (Biology) ,FUZZY neural networks ,POSITIVE systems - Abstract
In this paper, a stage structure predator–prey model consisting of three nonlinear ordinary differential equations is proposed and analyzed. The prey populations are divided into two parts: juvenile prey and adult prey. From extensive experimental data, it has been found that prey fear of predators can alter the physiological behavior of individual prey, and the fear effect reduces their reproductive rate and increases their mortality. In addition, we also consider the presence of constant ratio refuge in adult prey populations. Moreover, we consider the existence of intraspecific competition between adult prey species and predator species separately in our model and also introduce the gestation delay of predators to obtain a more realistic and natural eco-dynamic behaviors. We study the positivity and boundedness of the solution of the non-delayed system and analyze the existence of various equilibria and the stability of the system at these equilibria. Next by choosing the intra-specific competition coefficient of adult prey as bifurcation parameter, we demonstrate that Hopf bifurcation may occur near the positive equilibrium point. Then by taking the gestation delay as bifurcation parameter, the sufficient conditions for the existence of Hopf bifurcation of the delayed system at the positive equilibrium point are given. And the direction of Hopf bifurcation and the stability of the periodic solution are analyzed by using the center manifold theorem and normal form theory. What's more, numerical experiments are performed to test the theoretical results obtained in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
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