1,772 results
Search Results
2. 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
3. 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
- Subjects
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
4. Bifurcation analysis and control of an improved continuous traffic model considering weather effect.
- Author
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Ai, Wenhuan, Lei, Zheng Qing, Danyang, Li, Zeng, Jingming, and Liu, Dawei
- Subjects
TRAFFIC congestion ,TRAFFIC flow ,POLYNOMIAL approximation ,CHEBYSHEV approximation ,HOPF bifurcations ,SLOPE stability - Abstract
Purpose: Highway traffic systems are complex and variable, and studying the bifurcation characteristics of traffic flow systems and designing control schemes for unstable bifurcation points can alleviate traffic congestion from a new perspective. Bifurcation analysis is used to explain the changes in system stability, identify the unstable bifurcation points of the system, and design feedback controllers to realize the control of the unstable bifurcation points of the traffic system. It helps to control the sudden changes in the stable behavior of the traffic system and helps to alleviate traffic congestion, which is of great practical significance. Design/methodology/approach: In this paper, we improve the macroscopic traffic flow model by integrating severe weather factors such as rainfall, snowfall, and dust. We use traveling wave transform to convert it into a traffic flow stability model suitable for branching analysis, thus converting the traffic flow problem into a system stability analysis problem. First, this paper derives the existence conditions of the model Hopf bifurcation and saddle-node bifurcation for the improved macroscopic model, and finds the stability mutation point of the system. Secondly, the connection between the stability mutation points and bifurcation points of the traffic system is analyzed. Finally, for the unstable bifurcation point, a nonlinear system feedback controller is designed using Chebyshev polynomial approximation and stochastic feedback control method. Findings: The Hopf bifurcation is delayed and completely eliminated without changing the equilibrium point of the system, thus controlling the abrupt behavior of the traffic system. Originality/value: Currently there are fewer studies to explain the changes in the stability of the transportation system through bifurcation analysis, in this paper; we design a feedback controller for the unstable bifurcation point of the system to realize the control of the transportation system. It is a new research method that helps to control the sudden change of the stable behavior of the traffic system and helps to alleviate traffic congestion, which is of great practical significance. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
5. 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
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6. The interplay of rock-paper-scissors competition and environments mediates species coexistence and intriguing dynamics.
- Author
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Mohd, Mohd Hafiz and Park, Junpyo
- Subjects
- *
COMPETITION (Biology) , *COEXISTENCE of species , *ABIOTIC environment , *LIMIT cycles , *ECOLOGICAL models , *HOPF bifurcations , *DYNAMICAL systems - Abstract
• We introduce the effect of changing environmental carrying capacity on evolution of asymmetric rock-paper-scissors game. • According to assumptions on environmental gradients, the system can exhibit various survival states including multistability of single species survival. • Symmetry-breaking of competition rates and environmental carrying capacity can be significant factors to yield rich behavior of species survival in systems of cyclic competition. • Considering ecological factors is found to be an important issue on understanding mechanisms of evolution among cyclically competing species in the perspective of maintaining coexistence and promoting biodiversity. Asymmetrical rock-paper-scissors (RPS) competition has been perceived as a crucial factor in shaping species biodiversity, and understanding this ecological issue in a multi-species paradigm is rather difficult because community dynamics usually depend on distinct factors such as abiotic environments, biotic interactions and symmetry-breaking phenomenon. To address this problem, we employ a Lotka-Volterra competitive system consisting of both symmetrical, asymmetrical interactions and abiotic environment components. We discover that that asymmetrical RPS competition in heterogeneous environments can yield much richer dynamical behaviors, compared to the symmetrical and asymmetrical competition in homogeneous environments. While it is observed that species coexistence outcomes and/or oscillatory solutions are maintained as in the case of homogeneous environments, the nonuniformity in the environmental carrying capacities may lead to extra dynamics with regards to the appearance of survival states; for instance, coexistence of any two-species and single-species persistence states, which are not evident in the previous modelling studies. By means of bifurcation analysis, various salient features of the dynamical systems, including the emergence of certain attractors (e.g., different steady states, stable limit cycles and heteroclinic cycles) and co-dimension one bifurcations (e.g., transcritical and supercritical Hopf bifurcations) are realized in this ecological model. Overall, this modelling work provides a novel attempt to simultaneously encompass not only symmetry-breaking phenomenon through RPS competition, but also heterogeneity in the environments. This framework can provide additional insights to better understand various mechanisms underlying the effects of distinct ecological processes on multi-species communities. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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7. 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
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8. 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
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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
9. Bursting Dynamics in the General Hybrid Rayleigh-van der Pol-Duffing Oscillator with Two External Periodic Excitations.
- Author
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Qian, Youhua, Wang, Haolan, and Zhang, Danjin
- Subjects
HOPF bifurcations ,SELF-induced vibration ,STRUCTURAL stability ,PHASE diagrams ,OSCILLATIONS - Abstract
Purpose: The present paper investigates the bursting dynamics and its mechanism of the general hybrid Rayleigh-van der Pol-Duffing oscillator with two external periodic excitations. The study aims to avoid or reasonably utilize the vibrations induced by self-excited system in engineering. Methods: This paper mainly uses the fast–slow analysis method. First, we analyze the stability and bifurcation structure of the equilibrium point of the system with a single external excitation. Through the bifurcation set, we discover that the system has fold bifurcation and Hopf bifurcation. The direction of the Hopf bifurcation is obtained by calculating the first Lyapunov coefficient. When two external excitations exist, by employing de Moivre's equation, the original system transforms into a new form with a single slow variable. The classical machinery of fast subsystem analysis is used to investigate the bursting oscillations. Besides, the transformed phase diagrams are used to reveal the bifurcation mechanism. Results: Two bursting oscillations, namely symmetric fold/supHopf bursting and symmetric fold/fold bursting, are observed in the system with one periodic excitation. With two external excitations, the "turnover-of-fold-hystersis-induced" bursting, "symmetric periodic 2-fold" bursting, "symmetric periodic 6-fold" bursting, "symmetric periodic 10-fold" bursting and "symmetric periodic 14-fold" bursting are found and analyzed in detail. At the same time, the results show that with the increase of amplitudes of the external excitations, qualitative changes occur. Conclusions: With the addition of the second periodic excitation, the original bursting patterns induced by a single periodic excitation change completely. The phenomena will impact the vibration in the actual engineering. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
10. Nonlinear Phenomena of Fluid Flow in a Bioinspired Two-Dimensional Geometric Symmetric Channel with Sudden Expansion and Contraction.
- Author
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Yang, Liquan, Yang, Mo, and Huang, Weijia
- Subjects
FLUID flow ,REYNOLDS number ,HOPF bifurcations ,COMPUTER simulation - Abstract
Inspired by the airway for phonation, fluid flow in an idealized model within a sudden expansion and contraction channel with a geometrically symmetric structure is investigated, and the nonlinear behaviors of the flow therein are explored via numerical simulations. Numerical simulation results show that, as the Reynolds number (Re = U
0 H/ν) increases, the numerical solution undergoes a pitchfork bifurcation, an inverse pitchfork bifurcation and a Hopf bifurcation. There are symmetric solutions, asymmetric solutions and oscillatory solutions for flows. When the sudden expansion ratio (Er) = 6.00, aspect ratio (Ar) = 1.78 and Re ≤ Rec1 (≈185), the numerical solution is unique, symmetric and stable. When Rec1 < Re ≤ Rec2 (≈213), two stable asymmetric solutions and one symmetric unstable solution are reached. When Rec2 < Re ≤ Rec3 (≈355), the number of numerical solution returns one, which is stable and symmetric. When Re > Rec3, the numerical solution is oscillatory. With increasing Re, the numerical solution develops from periodic and multiple periodic solutions to chaos. The critical Reynolds numbers (Rec1, Rec2 and Rec3) and the maximum return velocity, at which reflux occurs in the channel, change significantly under conditions with different geometry. In this paper, the variation rules of Rec1, Rec2 and Rec3 are investigated, as well as the maximum return velocity with the sudden expansion ratio Er and the aspect ratio Ar. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
11. 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
12. 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
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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
13. 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
14. 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
15. Spatiotemporal patterns of a delayed diffusive prey-predator model with prey-taxis.
- Author
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Zhang, Fengrong and Chen, Ruining
- Subjects
LOTKA-Volterra equations ,SPATIOTEMPORAL processes ,BOUNDARY value problems ,HOPF bifurcations ,STABILITY theory - Abstract
This paper explored a delayed diffusive prey-predator model with prey-taxis involving the volume-filling mechanism subject to homogeneous Neumann boundary condition. To figure out the impact on the dynamic of the prey-predator model due to prey-taxis and time delay, we treated the prey-tactic coefficient χ and time delay τ as the bifurcating parameters and did stability and bifurcation analysis. It showed that the time delay will induce Hopf bifurcations in the absence of prey-taxis, and the bifurcation periodic solution at the first critical value of τ was spatially homogeneous. Hopf bifurcations occurred in the model when the prey-taxis and time delay coexisted, and at the first critical value of τ , spatially homogeneous or nonhomogeneous periodic solutions might emerge. It was also discovered that the bifurcation curves will intersect, which implied that Hopf-Hopf bifurcations can occur. Finally, we did numerical simulations to validate our outcomes. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
16. 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
17. 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
18. 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
19. 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
20. 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
21. Oscillatory solutions in a 3D predation system with time delay and age structure.
- Author
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Yang, Peng
- Subjects
TIME delay systems ,TOP predators ,PREDATION ,HOPF bifurcations ,AGE distribution ,SMOOTHNESS of functions ,BIFURCATION diagrams ,NONLINEAR oscillators - Abstract
This paper is concerned with the sustained periodic oscillation phenomenon in a three-species delayed predation system with Holling type II functional response and age structure in top predator. The top predator fertility function is regarded as a piecewise continuous smooth function with regard to their maturation period τ 2 . The complicated dynamic behavior is proved by integrated semigroup theory and Hopf bifurcation theorem for semilinear equations with non-dense domain. Through qualitative analysis and bifurcation study of the system, we yield that this system has a nontrivial periodic solution that bifurcates from the positive equilibrium age distribution when bifurcation parameter τ passes through some critical values. Numerical simulations are also provided to illustrate theoretically analytical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
22. 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
23. 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
24. Hopf bifurcation and limit cycle of the two‐variable Oregonator model for Belousov–Zhabotinsky reaction.
- Author
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He, Zecen and Zhao, Yulin
- Subjects
- *
HOPF bifurcations , *LIMIT cycles , *POSITIVE systems - Abstract
This paper is concerned with a two‐variable Oregonator model for Belousov–Zhabotinsky Reaction. Llibre and Oliveira (2022) proved that Oregonator model with an unstable node or focus has at least one limit cycle in the positive quadrant, and the system has a unique stable limit cycle for some values of the parameters. It is shown in the present paper that the positive equilibrium is not a center, and that if the system has a positive equilibrium which is a weak focus, then its order is at most 2. There exist some parameter values such that system has i(i=1,2)$$ i\left(i=1,2\right) $$ small limit cycle around the positive equilibrium. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
25. Local and Global Dynamics of a Ratio-Dependent Holling–Tanner Predator–Prey Model with Strong Allee Effect.
- Author
-
Lou, Weiping, Yu, Pei, Zhang, Jia-Fang, and Arancibia-Ibarra, Claudio
- Subjects
- *
ALLEE effect , *HOPF bifurcations , *PREDATION , *LYAPUNOV stability , *SYSTEM dynamics , *GLOBAL asymptotic stability - Abstract
In this paper, the impact of the strong Allee effect and ratio-dependent Holling–Tanner functional response on the dynamical behaviors of a predator–prey system is investigated. First, the positivity and boundedness of solutions of the system are proved. Then, stability and bifurcation analysis on equilibria is provided, with explicit conditions obtained for Hopf bifurcation. Moreover, global dynamics of the system is discussed. In particular, the degenerate singular point at the origin is proved to be globally asymptotically stable under various conditions. Further, a detailed bifurcation analysis is presented to show that the system undergoes a codimension- 1 Hopf bifurcation and a codimension- 2 cusp Bogdanov–Takens bifurcation. Simulations are given to illustrate the theoretical predictions. The results obtained in this paper indicate that the strong Allee effect and proportional dependence coefficient have significant impact on the fundamental change of predator–prey dynamics and the species persistence. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
26. Strong delayed negative feedback.
- Author
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Erneux, Thomas
- Subjects
HOPF bifurcations ,NONLINEAR theories ,DELAY differential equations ,NUMERICAL analysis ,MATHEMATICAL models - Abstract
In this paper, we analyze the strong feedback limit of two negative feedback schemes which have proven to be efficient for many biological processes (protein synthesis, immune responses, breathing disorders). In this limit, the nonlinear delayed feedback function can be reduced to a function with a threshold nonlinearity. This will considerably help analytical and numerical studies of networks exhibiting different topologies. Mathematically, we compare the bifurcation diagrams for both the delayed and non-delayed feedback functions and show that Hopf classical theory needs to be revisited in the strong feedback limit. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
27. Stability and Bifurcation Analysis in a Discrete Predator–Prey System of Leslie Type with Radio-Dependent Simplified Holling Type IV Functional Response.
- Author
-
Lv, Luyao and Li, Xianyi
- Subjects
PREDATION ,DISCRETE systems ,BIFURCATION theory ,LIMIT cycles ,HOPF bifurcations ,DYNAMICAL systems - Abstract
In this paper, we use a semi-discretization method to consider the predator–prey model of Leslie type with ratio-dependent simplified Holling type IV functional response. First, we discuss the existence and stability of the positive fixed point in total parameter space. Subsequently, through using the central manifold theorem and bifurcation theory, we obtain sufficient conditions for the flip bifurcation and Neimark–Sacker bifurcation of this system to occur. Finally, the numerical simulations illustrate the existence of Neimark–Sacker bifurcation and obtain some new dynamical phenomena of the system—the existence of a limit cycle. Corresponding biological meanings are also formulated. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
28. A comprehensive study of spatial spread and multiple time delay in an eco-epidemiological model with infected prey.
- Author
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Thakur, Nilesh Kumar, Srivastava, Smriti Chandra, and Ojha, Archana
- Subjects
HOPF bifurcations ,BIRD populations ,STABILITY criterion ,INFECTIOUS disease transmission ,TILAPIA - Abstract
This paper studies the dynamics of interacting Tilapia fish and Pelican bird population in the Salton Sea. We assume that the diseases spread in Tilapia fish follows the Holling type II response function, and the interaction between Tilapia and Pelican follows the Beddington–DeAngelis response function. The dynamics of diffusive and delayed system are discussed separately. Analytically, all the feasible equilibria and their stability are discussed. The criterion for Turing instability is derived. Based on the normal form theory and center manifold arguments, the existence of stability criterion and the direction of Hopf bifurcation are obtained. Numerical simulation shows the occurrence Hopf bifurcation, double Hopf bifurcation and transcritical bifurcation scenarios. The snap shot shows the spot, spot-strip mix patterns in the whole domain. Further, the stability switching phenomena is observed in the delayed system. Our comprehensive study highlights the effect of different parameters, multiple time delay and extinction in Pelican populations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
29. Bursting oscillations in dry friction system under external excitation.
- Author
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Sun, You and Zhang, Zhengdi
- Subjects
DRY friction ,OSCILLATIONS ,CONVEYOR belts ,BELT conveyors ,HOPF bifurcations - Abstract
In this paper, a system containing switching manifolds is constructed based on a non-linear conveyor belt model by introducing a periodic external excitation term. When the frequency of the external excitation is much smaller than the intrinsic frequency of the system, a series of bursting phenomena can be observed. Using the Filippov convex method, bifurcation mechanisms on the non-smooth boundary are discussed. Particularly, the explicit expression of the sliding region is obtained. Based on that, different bursting dynamics, especially the switching patterns dominated by sliding phenomena, are revealed by considering two cases of stiffness and excitation amplitude. It is shown that there exist different types of equilibrium points, leading to multiple modes of bifurcation, as stiffness changes. However, there always exist a pair of subcritical Hopf bifurcation points, which lie on the unstable equilibrium curve. Varying the excitation amplitude, different types of bursting oscillation modes may occur. In addition, both the transition of the type of equilibrium and the intrinsic structure of the system have effects on the local structure of the trajectory. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
30. Dynamical analysis of a three-dimensional discrete-time SIR model.
- Author
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Yu, Jiangqiong, Li, Lin, Liu, Lingling, and Yu, Zhiheng
- Subjects
HOPF bifurcations ,BIFURCATION theory ,THREE-dimensional modeling ,RESONANCE - Abstract
In this paper, we mainly study the dynamic properties of a class of three-dimensional SIR models. First, we use the complete discriminant theory of polynomials to obtain the parameter conditions for the topological types of each fixed point. Second, by employing the centre manifold theorem and bifurcation theory, we prove that the system may undergo transcritical, flip and Neimark–Sacker bifurcations at fixed points. Finally, we simulated all the codimension-1 bifurcations. We found that the system also has codimension-2 bifurcations, i.e. 1:2, 1:3 and 1:4 strong resonances. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
31. Modeling the Nonmonotonic Immune Response in a Tumor–Immune System Interaction.
- Author
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Liu, Yu, Ma, Yuhang, Yang, Cuihong, Peng, Zhihang, Takeuchi, Yasuhiro, Banerjee, Malay, and Dong, Yueping
- Subjects
IMMUNE response ,HOPF bifurcations ,ORBITS (Astronomy) ,IMMUNE system ,CELL physiology - Abstract
Tumor–immune system interactions are very complicated, being highly nonlinear and not well understood. A large number of tumors can potentially weaken the immune system through various mechanisms such as secreting cytokines that suppress the immune response. In this paper, we propose a tumor–immune system interaction model with a nonmonotonic immune response function and adoptive cellular immunotherapy (ACI). The model has a tumor-free equilibrium and at most three tumor-presence equilibria (low, moderate and high ones). The stability of all equilibria is studied by analyzing their characteristic equations. The consideration of nonmonotonic immune response results in a series of bifurcations such as the saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. In addition, numerical simulation results show the coexistence of periodic orbits and homoclinic orbits. Interestingly, along with various bifurcations, we also found two bistable scenarios: the coexistence of a stable tumor-free as well as a high-tumor-presence equilibrium and the coexistence of a stable-low as well as a high-tumor-presence equilibrium, which can show symmetric and antisymmetric properties in a range of model parameters and initial cell concentrations. The new findings indicate that under ACI, patients can possibly reach either a stable tumor-free state or a low-tumor-presence state in the presence of nonmonotonic immune response once the immune system is activated. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
32. Steady-State Bifurcation and Hopf Bifurcation in a Reaction–Diffusion–Advection System with Delay Effect.
- Author
-
Liu, Di and Jiang, Weihua
- Subjects
HOPF bifurcations ,IMPLICIT functions ,ADVECTION - Abstract
A general time-delay reaction–diffusion–advection system with the Dirichlet boundary condition and spatial heterogeneity is investigated in this paper. By using the implicit function theorem, we obtain the existence and asymptotic expression of the spatially non-homogeneous positive steady-state solution. This is the steady-state bifurcation from zero equilibrium. Via analyzing the corresponding characteristic equation, the stability of the spatially non-homogeneous positive steady-state solution and the occurrence of Hopf bifurcation at the positive steady-state solution are obtained, and the spatially non-homogeneous periodic solution is derived from Hopf bifurcation, this is the secondary bifurcation behavior of the system. Utilizing the normal form method and center manifold theory, we prove that the direction of Hopf bifurcation is supercritical and the bifurcating spatially non-homogeneous periodic solution is stable. Furthermore, We show that there exist two sequences Hopf bifurcation values and the orders of two sequences Hopf bifurcation values are given. Moreover, theoretical and numerical results are applied to competition and cooperation systems, respectively. Finally, the effect of the advection rate and spatial heterogeneity are discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
33. Double Hopf Bifurcation Analysis in the Memory-based Diffusion System.
- Author
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Song, Yongli, Peng, Yahong, and Zhang, Tonghua
- Subjects
HOPF bifurcations ,PREDATION ,DIFFUSION coefficients ,NONLINEAR oscillators - Abstract
In this paper, we derive the algorithm for calculating the normal form of the double Hopf bifurcation that appears in a memory-based diffusion system via taking memory-based diffusion coefficient and the memory delay as the perturbation parameters. Using the obtained theoretical results, we study the dynamical classification near the double Hopf bifurcation point in a predator-prey system with Holling type II functional response. We show the existence of different kinds of stable spatially inhomogeneous periodic solutions, the transition from one kind to the other as well as the coexistence of two types of periodic solutions with different spatial profiles by varying the memory-based diffusion coefficient and the memory delay. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
34. Fractional order-induced bifurcations in a delayed neural network with three neurons.
- Author
-
Huang, Chengdai, Wang, Huanan, and Cao, Jinde
- Subjects
IMPLICIT functions ,HOPF bifurcations ,NEURONS - Abstract
This paper reports the novel results on fractional order-induced bifurcation of a tri-neuron fractional-order neural network (FONN) with delays and instantaneous self-connections by the intersection of implicit function curves to solve the bifurcation critical point. Firstly, it considers the distribution of the root of the characteristic equation in depth. Subsequently, it views fractional order as the bifurcation parameter and establishes the transversal condition and stability interval. The main novelties of this paper are to systematically analyze the order as a bifurcation parameter and concretely establish the order critical value through an implicit function array, which is a novel idea to solve the critical value. The derived results exhibit that once the value of the fractional order is greater than the bifurcation critical value, the stability of the system will be smashed and Hopf bifurcation will emerge. Ultimately, the validity of the developed key fruits is elucidated via two numerical experiments. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
35. Adaptive control to actively damp bistabilities in highly interrupted turning processes using a hardware-in-the-loop simulator.
- Author
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Sahu, Govind N., Law, Mohit, and Wahi, Pankaj
- Subjects
ADAPTIVE control systems ,HOPF bifurcations ,CUTTING force ,METAL cutting - Abstract
Interruptions in turning make the process forces non-smooth and nonlinear. Smooth nonlinear cutting forces result in the process of being stable for small perturbations and unstable for larger ones. Re-entry after interruptions acts as perturbations making the process exhibit bistabilities. Stability for such processes is characterized by Hopf bifurcations resulting in lobes and period-doubling bifurcations resulting in narrow unstable lenses. Interrupted turning remains an important technological problem, and since experimentation to investigate and mitigate instabilities are difficult, this paper instead emulates these phenomena on a controlled hardware-in-the-loop simulator. Emulated cutting on the simulator confirms that bistabilities persist with lobes and lenses. Cutting in bistable regimes should be avoided due to conditional stability. Hence, we demonstrate the use of active damping to stabilize cutting with interruptions/perturbations. To stabilize cutting with small/large perturbations, we successfully implement an adaptive gain tuning scheme that adapts the gain to the level of interruption/perturbation. To facilitate real-time detection of instabilities and their control, we characterize the efficacy of the updating scheme for its dependence on the time required to update the gain and for its dependence on the levels of gain increments. We observe that higher gain increments with shorter updating times result in the process being stabilized quicker. Such results are instructive for active damping of real processes exhibiting conditional instabilities prone to perturbations. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
36. Evolutionary dynamics of rock-paper-scissors game in the patchy network with mutations.
- Author
-
Verma, Tina and Gupta, Arvind Kumar
- Subjects
- *
COEXISTENCE of species , *LIMIT cycles , *BIODIVERSITY conservation , *RANDOM walks , *HOPF bifurcations , *HABITATS - Abstract
• The rock-paper-scissors model is presented for two-person non-zero sum game in which the strategies of rock mutate to scissors and paper. • The evolutionary dynamics of the population of species rock, paper and scissors is studied when the patches are connected through random walk. • When the patches are coupled, the state of synchronization and stability is observed. • When mutation is allowed, the limit cycle converges to stable state and the transition from one phase to another phase is observed. • The replicator-mutator equations are solved analytically as well as numerically. 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. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
37. Stability and Hopf bifurcation analysis of a fractional-order ring-hub structure neural network with delays under parameters delay feedback control.
- Author
-
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
38. The impact of fear effect on the dynamics of a delayed predator–prey model with stage structure.
- Author
-
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
39. Complex Dynamic Analysis for a Rent-Seeking Game with Political Competition and Policymaker Costs.
- Author
-
Yang, Xiuqin, Liu, Feng, and Wang, Hua
- Subjects
POLITICAL competition ,RENT seeking ,NASH equilibrium ,ORBITS (Astronomy) ,LOTKA-Volterra equations ,GAMES ,HOPF bifurcations - Abstract
This paper investigates the dynamics of rent-seeking games that include political competition and policymaker cost model. The local asymptotic stability of multiple equilibrium points and Nash equilibrium points are studied. In the rent-seeking model, the existence and stability of Flip bifurcation and Neimark–Sacker bifurcation are examined, and the corresponding theorems and conditions are derived. The theoretical conclusions of the paper are verified by numerical simulations with different parameters. The simulation graphics show that the rent-seeking game model exhibits rich dynamic behaviors, such as multi-periodic orbits, Flip bifurcation, Neimark–Sacker bifurcation, and chaotic sets. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
40. Modeling and Nonlinear Characteristics Analysis of Fluorescent Lamp Driven by a Full-Bridge Inverter.
- Author
-
Yang, Jiahang, Wang, Faqiang, and Wang, Xian
- Subjects
FLUORESCENT lamps ,NONLINEAR analysis ,HOPF bifurcations ,POWER electronics ,FREQUENCIES of oscillating systems ,BRIDGES - Abstract
Addressing the issue of system stability is a crucial step towards the successful integration of memristive devices in power electronics applications. This paper focuses on fluorescent lamps possessing memristive characteristics and investigates the instability phenomena and mechanisms within a full-bridge inverter with a fluorescent lamp load. Based on the memristive characteristics of fluorescent lamps, this paper establishes the averaged model of the system, whose coefficient matrix is nonlinear, periodic, and time-varying. This study identifies the occurrence of low-frequency oscillations within the system and elucidates the fundamental mechanism underlying the emergence of low-frequency oscillations. Furthermore, this paper establishes the stability boundaries of the system across different parameter planes. The research findings indicate that the low-frequency oscillations within the system are attributed to the occurrence of Hopf bifurcations in a frequency range higher than line frequency but significantly lower than switching frequency. Lastly, the PSpice circuit of the system is designed, and simulation results are provided for validation. This study can offer guidance on parameters and control strategies for ensuring the stable operation of a full-bridge inverter with fluorescent lamps. Moreover, it can facilitate the comprehension of instability mechanisms in systems incorporating memristive devices, thereby offering a foundation for the expansion of memristor applications. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
41. HOPF BIFURCATIONS IN DYNAMICAL SYSTEMS VIA ALGEBRAIC TOPOLOGICAL METHOD.
- Author
-
Jawarneh, Ibrahim and Altawallbeh, Zuhier
- Subjects
HOPF bifurcations ,DYNAMICAL systems ,DIFFERENTIAL equations ,LIMIT cycles - Abstract
A nonlinear phenomenon in nature is often modeled by a system of differential equations with parameters. The bifurcation occurs when a parameter varies in such systems, causing a qualitative change in its solution. In this paper, we study one of the most exciting bifurcations, which is Hopf bifurcation. We use tools from algebraic topology to analyze and reveal supercritical and subcritical Hopf bifurcations. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
42. A simple chaotic circuit based on memristor and its analyzation of bifurcation.
- Author
-
Zhao, Shaoqing, Cui, Yan, Lu, Chenhui, and Zhou, Liuyuan
- Subjects
HOPF bifurcations ,BIFURCATION theory ,PROBLEM solving ,CIRCUIT elements ,MEMORIZATION ,HOPFIELD networks - Abstract
Memristor is a kind of nonlinear resistance with memory function, which is a typical nonlinear circuit component. In this paper, the problem of a simple chaotic circuit based on the memristor was studied. We constructed a new simple circuit of multiple memorizing components, summarized the mathematical model and found seven types of chaotic attractors without reference to classical circuits that substituted memristors for elements in the original circuit. Traditional methods of dynamic analysis are used to analyze the equilibrium point and stability of the system. Furthermore, we analyzed the dynamical behavior with the varying coefficient of the system specifically. In order to reduce the deviation from the actual physical circuit system as much as possible so as to facilitate the follow-up research of scholars, we solved the problem of Hopf bifurcation in this system under the condition of time delay through the use of the canonical form and Hopf bifurcation theory. Theoretical analysis and simulation results show that there existed the state transition in seven chaotic attractors of system and the bifurcation was stable when we changed the parameter τ slightly. As a small unit circuit, this paper lays a foundation for the research and control of large-scale system with memorizing components. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
43. Crucial impact of component Allee effect in predator-prey system.
- Author
-
Sahoo, Kalyanashis and Sahoo, Banshidhar
- Subjects
ALLEE effect ,PREDATION ,POPULATION ecology ,TOP predators ,ECOLOGICAL disturbances ,HOPF bifurcations - Abstract
Allee effect in models in interacting species of predator prey system has great significance in ecological context. Allee effect plays crucial role in population dynamics in ecology, where it is the challenging fact that per capita population growth rate is positively dependent on the population density of a species. In this paper, we inspect the famous Hastings and Powell (HP) (Hastings and Powell 1991 Ecology 72 896–903) model incorporating component Allee effect on top predator's reproduction. We analyse the updated model with the help of both analytical and numerical phenomena. Considering cost of Allee effect, half-saturation constant of prey as the key parameters, the Hopf bifurcations are also analysed. The directions of Hopf bifurcations at the critical values of Allee parameter and half-saturation constant of prey are studied theoretically by using normal form theory introduced by Hassard et al (1981 Theory and Applications of Hopf Bifurcation vol 41 (CUP Archive)). The formulated model indicates that the system demonstrates chaotic, periodic and stable dynamics in the variation of key parameters. The chaos can be controlled for proper application of the large values of parameter used as the cost of Allee effect and also for small values of the parameter used as the half saturation constant of prey population. The results of this study are applicable in the field of marine and wild ecosystem dynamics. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
44. Investigating the Dynamic Behavior of Integer and Noninteger Order System of Predation with Holling's Response.
- Author
-
Owolabi, Kolade M., Jain, Sonal, and Pindza, Edson
- Subjects
BIFURCATION theory ,HOPF bifurcations ,LYAPUNOV exponents ,PREDATION ,QUANTITATIVE research ,REACTION-diffusion equations ,LIMIT cycles - Abstract
The paper's primary objective is to examine the dynamic behavior of an integer and noninteger predator–prey system with a Holling type IV functional response in the Caputo sense. Our focus is on understanding how harvesting influences the stability, equilibria, bifurcations, and limit cycles within this system. We employ qualitative and quantitative analysis methods rooted in bifurcation theory, dynamical theory, and numerical simulation. We also delve into studying the boundedness of solutions and investigating the stability and existence of equilibrium points within the system. Leveraging Sotomayor's theorem, we establish the presence of both the saddle-node and transcritical bifurcations. The analysis of the Hopf bifurcation is carried out using the normal form theorem. The model under consideration is extended to the fractional reaction–diffusion model which captures non-local and long-range effects more accurately than integer-order derivatives. This makes fractional reaction–diffusion systems suitable for modeling phenomena with anomalous diffusion or memory effects, improving the fidelity of simulations in turn. An adaptable numerical technique for solving this class of differential equations is also suggested. Through simulation results, we observe that one of the Lyapunov exponents has a negative value, indicating the potential for the emergence of a stable-limit cycle via bifurcation as well as chaotic and complex spatiotemporal distributions. We supplement our analytical investigations with numerical simulations to provide a comprehensive understanding of the system's behavior. It was discovered that both the prey and predator populations will continue to coexist and be permanent, regardless of the choice of fractional parameter. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
45. Bifurcation Analysis of a Class of Two-Delay Lotka–Volterra Predation Models with Coefficient-Dependent Delay.
- Author
-
Li, Xiuling and Fan, Haotian
- Subjects
HOPF bifurcations ,DELAY differential equations ,PREDATION - Abstract
In this paper, a class of two-delay differential equations with coefficient-dependent delay is studied. The distribution of the roots of the eigenequation is discussed, and conditions for the stability of the internal equilibrium and the existence of Hopf bifurcation are obtained. Additionally, using the normal form method and the central manifold theory, the bifurcation direction and the stability for the periodic solution of Hopf bifurcation are calculated. Finally, the correctness of the theory is verified by numerical simulation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
46. Dynamics analysis and optimal control of delayed SEIR model in COVID-19 epidemic.
- Author
-
Liu, Chongyang, Gao, Jie, and Kanesan, Jeevan
- Subjects
COVID-19 pandemic ,COVID-19 ,HOPF bifurcations ,COST functions ,COST control - Abstract
The coronavirus disease 2019 (COVID-19) remains serious around the world and causes huge deaths and economic losses. Understanding the transmission dynamics of diseases and providing effective control strategies play important roles in the prevention of epidemic diseases. In this paper, to investigate the effect of delays on the transmission of COVID-19, we propose a delayed SEIR model to describe COVID-19 virus transmission, where two delays indicating the incubation and recovery periods are introduced. For this system, we prove its solutions are nonnegative and ultimately bounded with the nonnegative initial conditions. Furthermore, we calculate the disease-free and endemic equilibrium points and analyze the asymptotical stability and the existence of Hopf bifurcations at these equilibrium points. Then, by taking the weighted sum of the opposite number of recovered individuals at the terminal time, the number of exposed and infected individuals during the time horizon, and the system cost of control measures as the cost function, we present a delay optimal control problem, where two controls represent the social contact and the pharmaceutical intervention. Necessary optimality conditions of this optimal control problem are exploited to characterize the optimal control strategies. Finally, numerical simulations are performed to verify the theoretical analysis of the stability and Hopf bifurcations at the equilibrium points and to illustrate the effectiveness of the obtained optimal strategies in controlling the COVID-19 epidemic. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
47. Analysis of the Dynamical Properties of Discrete Predator-Prey Systems with Fear Effects and Refuges.
- Author
-
Li, Wei, Zhang, Chunrui, and Wang, Mi
- Subjects
PREDATION ,DISCRETE systems ,BIFURCATION theory ,HOPF bifurcations ,LYAPUNOV exponents ,BIFURCATION diagrams ,COMPUTER simulation - Abstract
This paper examines the dynamic behavior of a particular category of discrete predator-prey system that feature both fear effect and refuge, using both analytical and numerical methods. The critical coefficients and properties of bifurcating periodic solutions for Flip and Hopf bifurcations are computed using the center manifold theorem and bifurcation theory. Additionally, numerical simulations are employed to illustrate the bifurcation phenomenon and chaos characteristics. The results demonstrate that period-doubling and Hopf bifurcations are two typical routes to generate chaos, as evidenced by the calculation of the maximum Lyapunov exponents near the critical bifurcation points. Finally, a feedback control method is suggested, utilizing feedback of system states and perturbation of feedback parameters, to efficiently manage the bifurcations and chaotic attractors of the discrete predator-prey model. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
48. NONPOWER LAW CONSTANT FLUX SOLUTIONS FOR THE SMOLUCHOWSKI COAGULATION EQUATION.
- Author
-
FERREIRA, MARINA A., LUKKARINEN, JANI, NOTA, ALESSIA, and VELÁZQUEZ, JUAN J. L.
- Subjects
COAGULATION ,EQUATIONS ,HOPF bifurcations - Abstract
It is well known that for a large class of coagulation kernels, Smoluchowski coagulation equations have particular power law solutions which yield a constant flux of mass along all scales of the system. In this paper, we prove that for some choices of the coagulation kernels there are solutions with a constant flux of mass along all scales which are not power laws. The result is proved by means of a bifurcation argument. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
49. The dynamics of a delayed predator-prey model with square root functional response and stage structure.
- Author
-
Peng, Miao, Lin, Rui, Zhang, Zhengdi, and Huang, Lei
- Subjects
POPULATION ecology ,SQUARE root ,HOPF bifurcations ,MANIFOLDS (Mathematics) ,MATHEMATICAL models - Abstract
In recent years, one of the most prevalent matters in population ecology has been the study of predator-prey relationships. In this context, this paper investigated the dynamic behavior of a delayed predator-prey model considering square root type functional response and stage structure for predators. First, we obtained positivity and boundedness of the solutions and existence of equilibrium points. Second, by applying the stability theory of delay differential equations and the Hopf bifurcation theorem, we discussed the system's local stability and the existence of a Hopf bifurcation at the positive equilibrium point. Moreover, the properties of the Hopf bifurcation were deduced by using the central manifold theorem and normal form method. Analytical results showed that when the time delay was less than the critical value, the two populations will coexist, otherwise the ecological balance will be disrupted. Finally, some numerical simulations were also included to verify the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
50. LOCAL PERCEPTION AND LEARNING MECHANISMS IN RESOURCE-CONSUMER DYNAMICS.
- Author
-
QINGYAN SHI, YONGLI SONG, and HAO WANG
- Subjects
HOPF bifurcations ,ANIMAL mechanics ,SPATIAL memory ,STABILITY constants ,IDENTIFICATION of animals - Abstract
Spatial memory is key in animal movement modeling, but it has been challenging to explicitly model learning to describe memory acquisition. In this paper, we study novel cognitive consumer-resource models with different consumer learning mechanisms and investigate their dynamics. These models consist of two PDEs in composition with one ODE such that the spectrum of the corresponding linearized operator at a constant steady state is unclear. We describe the spectra of the linearized operators and analyze the eigenvalue problems to determine the stability of the constant steady states. We then perform bifurcation analysis by taking the perceptual diffusion rate as the bifurcation parameter. It is found that steady-state and Hopf bifurcations can both occur in these systems, and the bifurcation points are given so that the stability region can be determined. Moreover, rich spatial and spatiotemporal patterns can be generated in such systems via different types of bifurcation. Our effort establishes a new approach to tackling a hybrid model of PDE-ODE composition and provides a deeper understanding of cognitive movement-driven consumer-resource dynamics. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
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