4,480 results
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252. Stability and Bifurcation Analysis of a Nonlinear Rotating Cantilever Plate System.
- Author
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Chen, Shuping, Zhang, Danjin, and Qian, Youhua
- Subjects
NONLINEAR analysis ,CANTILEVERS ,HOPF bifurcations ,CENTRIFUGAL force ,NONLINEAR systems ,TORUS - Abstract
This paper investigates the bifurcation behavior and the stability of the rotating cantilever rectangular plate that is subjected to varying speed and centrifugal force. The local stability of the degenerated equilibrium of nonlinear system with symmetry is observed after analyzing the corresponding characteristic equation. In addition to complex phenomena such as static bifurcation and Hopf bifurcation, the 2-D torus bifurcation is investigated in this paper. Thereafter, the steady-state solutions and stability region are obtained using the center manifold theory and normal form method. Finally, numerical simulations are conducted to show the nonlinear dynamical behaviors of the rotating cantilever rectangular plate. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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253. A nonsmooth van der Pol-Duffing oscillator (II): The sum of indices of equilibria is \begin{document}$ 1 $\end{document}.
- Author
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Wang, Zhaoxia and Chen, Hebai
- Subjects
LIMIT cycles ,HOPF bifurcations ,BIFURCATION diagrams ,EQUILIBRIUM - Abstract
We continue to study the nonsmooth van der Pol-Duffing oscillator x ˙ = y x ˙ = y , y ˙ = a 1 x + a 2 x 3 + b 1 y + b 2 | x | y y ˙ = a 1 x + a 2 x 3 + b 1 y + b 2 | x | y , where a i , b i a i , b i are real and a 2 b 2 ≠ 0 a 2 b 2 ≠ 0 , i = 1 , 2 i = 1 , 2. Notice that the sum of indices of equilibria is − 1 − 1 for a 2 > 0 a 2 > 0 and 1 1 for a 2 < 0 a 2 < 0. When a 2 > 0 a 2 > 0 , the nonsmooth van der Pol-Duffing oscillator has been studied completely in the companion paper. Attention goes to the bifurcation diagram and all global phase portraits in the Poincaré disc of the nonsmooth van der Pol-Duffing oscillator for a 2 < 0 a 2 < 0 in this paper. The bifurcation diagram is more complex, which includes two Hopf bifurcation surfaces, one pitchfork bifurcation surface, one homoclinic bifurcation surface, one double limit cycle bifurcation surface and one bifurcation surface for equilibria at infinity. When b 2 > 0 b 2 > 0 is fixed, this nonsmooth van der Pol-Duffing oscillator cannot be changed into a near-Hamiltonian system for small a 1 , b 1 a 1 , b 1. Moreover, the global dynamics of the nonsmooth van der Pol-Duffing oscillator and the van der Pol-Duffing oscillator are different. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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254. Bifurcations due to different delays of high-order fractional neural networks.
- Author
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Huang, Chengdai and Cao, Jinde
- Subjects
HOPF bifurcations ,LEAKAGE - Abstract
This paper expounds the bifurcations of two-delayed fractional-order neural networks (FONNs) with multiple neurons. Leakage delay or communication delay is viewed as a bifurcation parameter, stability zones and bifurcation conditions with respect to them are commendably established, respectively. It declares that both leakage delay and communication delay immensely influence the stability and bifurcation of the developed FONNs. The explored FONNs illustrate superior stability performance if selecting a lesser leakage delay or communication delay, and Hopf bifurcation generates once they overstep their critical values. The verification of the feasibility of the developed analytic results is implemented via numerical experiments. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
255. Global dynamical behavior of a delayed cytokine-enhanced viral infection model with nonlinear incidence.
- Author
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Xu, Jinhu, Huang, Guokun, Zhang, Suxia, Hao, Mengli, and Wang, Aili
- Subjects
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HOPF bifurcations , *VIRUS diseases , *IMMUNE system , *FUNCTIONALS , *IMMUNE response , *COMPUTER simulation - Abstract
In this paper, a delayed cytokine-enhanced viral infection model incorporating nonlinear incidence and immune response is proposed and studied. The global stabilities of the equilibria of the model are characterized by constructing suitable Lyapunov functionals. The existence and properties of local Hopf bifurcation are discussed by regarding the immunity delay as the bifurcation parameter. Moreover, the global existence of Hopf bifurcation has been proved by regarding the immune delay as the bifurcation parameter. Numerical simulations are carried out to validate the obtained results. The results show that ignoring the cytokine-enhanced effect makes the infection risk underevaluated. The simulations show that increasing the immune delay destabilizes the model and generates a Hopf bifurcation and stability switches occurs. Moreover, it shows that immune delay may dominate the intracellular delays in such a viral infection model which means that the immune system of the host itself is complicated during virus infection. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
256. Hopf bifurcation in a memory-based diffusion predator-prey model with spatial heterogeneity.
- Author
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Liu, Di and Jiang, Weihua
- Subjects
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HOPF bifurcations , *HETEROGENEITY , *DIFFUSION coefficients , *COGNITIVE ability , *NONLINEAR oscillators - Abstract
In this paper, we present a memory-based diffusion predator-prey model that incorporates spatial heterogeneity and is subject to homogeneous Dirichlet boundary conditions. Prey species lack memory or cognitive abilities, exhibiting only random diffusion. In contrast, predators utilize memory-based self-diffusion. For the proposed model, we establish the existence and explicit expression of a spatially non-constant positive steady-state. Furthermore, we demonstrate that memory-based diffusion and the averaged memory period can lead to richer dynamics. Specifically, when the memory-based diffusion coefficient is not dominant, the averaged memory period has no impact on the non-constant steady-state. However, when the memory-based diffusion coefficient takes precedence, the averaged memory period can destabilize the non-constant steady-state, resulting in Hopf bifurcation and the emergence of spatially non-homogeneous periodic solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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257. Normal forms of a class of partial functional differential equations.
- Author
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Fan, Yanhui and Wang, Chuncheng
- Subjects
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PARTIAL differential equations , *FUNCTIONAL differential equations , *HOPF bifurcations , *NONLINEAR equations , *NONLINEAR theories , *PHASE space - Abstract
In this paper, we investigate the normal form a class of partial functional differential equations, which takes account of delays in the diffusion terms as well as a wider scope of nonlinear terms. We first study the associated linear theory, mainly including the spectral properties of infinitesimal generator, formal adjoint and decomposition of phase space. Based on these results, the normal form theory for nonlinear equation is established, which can be used to study the local dynamics near the steady state for such equations. As an application, we consider the Hopf bifurcation problem of a scalar diffusive equation with delay not only involved in the diffusive term but also in reaction terms. The normal form, depending on the original coefficients, up to the third order term is calculated, which allows us to determine the direction of Hopf bifurcation and stability of bifurcated periodic solutions. The results are then applied to study the scalar population model with memory-based diffusion for modeling the movement of highly-developed animals. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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258. The Dynamics and Theoretical Analysis Underlying Periodic Bursting in the Nonsmooth Murali–Lakshmanan–Chua Circuit.
- Author
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Song, Jinchen, Ma, Nan, Zhang, Zhengdi, and Yu, Yue
- Subjects
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JACOBIAN matrices , *HOPF bifurcations , *TIME series analysis , *SYSTEM dynamics , *OSCILLATIONS - Abstract
Motivated by the study of multiple time scale system dynamics, we analyze the Murali–Lakshmanan–Chua (MLC) dissipative circuit exhibiting periodic bursting with different types of oscillations in this paper. Such patterns can be analyzed by treating the slow variable as a parameter of the nonsmooth fast subsystem. The discontinuous bifurcation structure of the subsystem can be described by the generalized differential of Clarke and the Jacobian matrix of the system. Bursting patterns as well as the oscillation mechanisms can be clarified by the discontinuous bifurcations, time series and phase portraits. For the chosen circuit parameter values, this designed circuit admits both MLC type bursting attractors and Duffing–van der Pol circuit type bursting attractors. It is found that not only the features combined with the circuit parameters of the system, but also the two switching boundaries may have an important impact on the origin of the bursting behaviors. In particular, the explicit analytic solutions of the proposed circuit are investigated in detail. We also show that the bursting can arise from either discontinuous saddle-node bifurcation or slow passage through a discontinuous Hopf bifurcation. Finally, the validity of the theoretical analysis has been well verified by the numerical simulations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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259. Impact of hunting cooperation in predator and anti‐predator behaviors in prey in a predator–prey model.
- Author
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Li, Yan and Ding, Mengyue
- Subjects
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ANTIPREDATOR behavior , *PREDATION , *LIMIT cycles , *HOPF bifurcations , *HUNTING , *BIRTH rate , *LOTKA-Volterra equations - Abstract
In this paper, we propose a predator–prey model with hunting cooperation in predator and anti‐predator behaviors in prey. The conditions for the existence and the stability of the unique positive constant equilibrium are given. It is found that with the increasing of the birth rate r0$$ {r}_0 $$ of the prey, the trivial solution loses its stability, and the semi‐trivial solution emerges and also loses its stability. For the positive constant solution, we find that as the hunting cooperation b$$ b $$ in predator increases or the fear k0$$ {k}_0 $$ decreases, the positive constant equilibrium loses its stability, and Hopf bifurcation occurs. We also derive the existence of limit cycles by Poincaré‐Bendixson theorem. We also study a diffusive model and derive that self‐diffusion can induce Turing instability. Finally, we conduct numerical simulations to present our conclusions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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260. JOINT IMPACT OF MATURATION DELAY AND FEAR EFFECT ON THE POPULATION DYNAMICS OF A PREDATOR-PREY SYSTEM.
- Author
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XIAOKE MA, YING SU, and XINGFU ZOU
- Subjects
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PREDATION , *HOPF bifurcations , *POPULATION dynamics , *FISCAL year , *SYSTEM dynamics - Abstract
In this paper, taking into account the maturation period of prey, we propose a predator-prey model with time delay and fear effect. We confirm the well-posedness of the model system, explore the stability of the equilibria and uniform persistence of the model, and investigate Hopf bifurcations. Moreover, we also numerically explore the global continuation of the Hopf bifurcation. Interestingly, our results show that as the delay increases, the stable and unstable periodic solutions may both disappear and the unstable positive equilibrium may regain its stability. These results reveal how the maturation delay and the fear effect jointly impact the population dynamics of the predator-prey system. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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261. Threshold dynamics and bifurcation analysis of an SIS patch model with delayed media impact.
- Author
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Zhang, Hua and Wei, Junjie
- Subjects
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BASIC reproduction number , *HOPF bifurcations - Abstract
In this paper, an susceptible–infected–susceptible (SIS) epidemic patch model with media delay is proposed at first. Then the basic reproduction number R0$\mathcal {R}_0$ is defined, and the threshold dynamics are studied. It is shown that the disease‐free equilibrium is globally asymptotically stable if R0<1$\mathcal {R}_0<1$ and the disease is uniformly persistent if R0>1$\mathcal {R}_0>1$. When the dispersal rates of susceptible and infected populations are identical and less than a critical value, it is proved that the limiting model has a unique positive equilibrium. Furthermore, the stability of the positive equilibrium and the existence of local and global Hopf bifurcations are obtained. Finally, some numerical simulations are performed. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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262. Bifurcations of a Laminated Circular Cylindrical Shell.
- Author
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Zhang, D. M. and Li, F.
- Subjects
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CYLINDRICAL shells , *HOPF bifurcations , *LAMINATED materials , *EIGENVALUES , *STRUCTURAL shells , *LAMINATED composite beams , *POLYMERIC membranes - Abstract
In this paper, the stability and local bifurcations of a composite laminated circular cylindrical shell with radially pre-stretched membranes are explored by using analytical and numerical methods. On the basis of the four-dimensional averaged equation in the case of 1:1 internal resonance, three types of critical points are studied in detail. They are characterized as (1) one pair of purely imaginary eigenvalues and two negative eigenvalues; (2) a simple zero and one pair of purely imaginary eigenvalues; (3) two pairs of purely imaginary eigenvalues in nonresonant case. With the aid of normal form theory and Maple software, the steady-state solutions and the stability regions of the initial equilibrium solutions are obtained. The explicit expressions for the critical bifurcation curves leading to static bifurcation and Hopf bifurcation are also presented. The presence of Hopf bifurcation indicates that the circular cylindrical shell will flutter. The results contribute to the design of reasonable structure parameters to avoid flutter. Finally, numerical simulations are also presented to demonstrate the good agreement with the analytical predictions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
263. Bifurcation and Instability of a Spatial Epidemic Model.
- Author
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Yuan, Hailong, Zhou, You, Yang, Xiaoyi, Lv, Yang, and Guo, Gaihui
- Subjects
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PARTIAL differential equations , *HOPF bifurcations , *ORDINARY differential equations , *EPIDEMICS , *BIFURCATION diagrams - Abstract
This paper is concerned with a spatial S − I epidemic model with nonlinear incidence rate. First, the existence of the equilibrium is discussed in different conditions. Then the main criteria for the stability and instability of the constant steady-state solutions are presented. In addition, the effect of diffusion coefficients on Turing instability is described. Next, by applying the normal form theory and the center manifold theorem, the existence and direction of Hopf bifurcation for the ordinary differential equations system and the partial differential equations system are given, respectively. The bifurcation diagrams of Hopf and Turing bifurcations are shown. Moreover, a priori estimates and local steady-state bifurcation are investigated. Furthermore, our analysis focuses on providing specific conditions that can determine the local bifurcation direction and extend the local bifurcation to the global one. Finally, the numerical results demonstrate that the intrinsic growth rate, denoted as r , has significant influence on the spatial pattern. Specifically, different patterns appear, with the increase of r. The obtained results greatly expand on the discovery of pattern formation in the epidemic model. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
264. Combined Impact of Multidelay Feedback Number and Interval: A Novel Mechanism for Controlling the Stability of Stochastic Duffing Systems.
- Author
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Hu, Zhouyu, Han, Zikun, Yang, Yanling, and Wang, Qiubao
- Subjects
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STOCHASTIC systems , *DUFFING equations , *HOPF bifurcations , *PSYCHOLOGICAL feedback - Abstract
In this paper, we study a class of nonlinear stochastic Duffing oscillators with multidelay feedback. We propose an effective reduction approach with the help of center manifold theory and stochastic averaging method. Taking the initial time-delay τ as the parameter, we reduce the original system to a one-dimensional averaged Itô equation. Our analysis reveals that the original system exhibits stochastic bifurcations, including stochastic D and P bifurcations. Once we have a clear understanding of the bifurcation structure, we can use this knowledge to choose appropriate system parameters and place the system in the desired state. For instance, by adjusting the initial time-delay τ of the control system, we can stabilize the system and achieve the desired outcome. Numerical simulations also verify the theoretical results. With appropriate parameter choices, multiple time delays can destabilize the equilibrium and promote chaotic behaviors, and can also lead to more stable dynamical behavior. Remarkably, we discovered that increasing the interval of time delays and feedback numbers can enhance system stability. It may potentially serve as a novel mechanism for stabilizing stochastic systems. The study provides a solid theoretical foundation for exploring stochastic systems subject to complex time-delay feedback control, and offers a valuable framework for related fields. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
265. DYNAMICS ANALYSIS OF HIV-1 INFECTION MODEL WITH CTL IMMUNE RESPONSE AND DELAYS.
- Author
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TING GUO and FEI ZHAO
- Subjects
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IMMUNE response , *BASIC reproduction number , *HIV , *HOPF bifurcations - Abstract
In this paper, we rigorously analyze an HIV-1 infection model with CTL immune response and three time delays which represent the latent period, virus production period and immune response delay, respectively. We begin this model with proving the positivity and bound-edness of the solution. For this model, the basic reproduction number R0 and the immune reproduction number R1 are identified. Moreover, we have shown that the model has three e-quilibria, namely the infection-free equilibrium E0, the infectious equilibrium without immune response E1 and the infectious equilibrium with immune response E2. By applying uctuation lemma and Lyapunov functionals, we have demonstrated that the global stability of E0 and E1 are only related to R0 and R1. The local stability of the third equilibrium is obtained under four situations. Further, we give the conditions for the existence of Hopf bifurcation. Finally, some numerical simulations are carried out for illustrating the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
266. A non-linear restatement of Kalecki's business cycle model with non-constant capital depreciation.
- Author
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De Cesare, Luigi and Sportelli, Mario
- Subjects
DEPRECIATION ,BUSINESS cycles ,INTEGRO-differential equations ,HOPF bifurcations ,DIFFERENTIAL equations - Abstract
This paper deals with Kalecki's 1935 business cycle model, where a finite time lag in the investment dynamics is assumed. The time lag is the gestation period elapsing between orders for capital goods and deliveries of finished industrial equipment. Including the actual mainstream theory, the economic literature agrees on the consequences that time lag has on the economic activity. It is a cause of persistent economic fluctuations. Following some recent research lines on this model, here we restate the Kalecki approach, assuming sigmoidal functions in addition to Kalecki's linear treatment and further considering a non-constant capital depreciation. Never made until now, this last assumption is such that to yield, in place of a delayed differential equation, a Volterra delayed integro-differential equation. Taken the time delay and the rate of capital depreciation as critical parameters, a qualitative study of that equation is carried out. We proved that with a small-time lag stable equilibria arise. But, when the delay increases, equilibria are destabilized through Hopf bifurcations and stability switches occur. Consequently, a variety of cyclical behaviors appear. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
267. Hopf and Zero-Hopf Bifurcation Analysis for a Chaotic System.
- Author
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Husien, Ahmad Muhamad and Amen, Azad Ibrahim
- Subjects
- *
HOPF bifurcations , *LIMIT cycles , *ORDINARY differential equations , *NONLINEAR differential equations , *ELECTRONIC circuits , *ORBITS (Astronomy) - Abstract
In this paper, we investigate a quadratic chaotic system modeling self-excited and hidden attractors which is described by a system of three nonlinear ordinary differential equations with three real parameters. The primary goal is to establish the existence of two limit cycles that bifurcate based on the system's nature as an electronic circuits model, specifically via Hopf bifurcation. Notably, the application of the first and second Lyapunov coefficients is utilized to demonstrate the bifurcation of two limit cycles from an equilibrium point near a Hopf critical point. Furthermore, employing the first-order averaging theory enables us to confirm the presence of unstable periodic orbits originating from the zero-Hopf equilibrium. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
268. Rich Dynamics in a Hindmarsh–Rose Neuronal Model with Time Delay.
- Author
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Jiao, Xubin and Liu, Xiuxiang
- Subjects
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HOPF bifurcations , *EQUILIBRIUM , *COMPUTER simulation - Abstract
This paper presents the stability and bifurcation of the Hindmarsh–Rose neuronal model with time delay. First, we discuss the existence and stability of the equilibria, revealing that the system can have a maximum of three equilibria which exhibit always stable, always unstable, and stability switching dynamics, depending on the stability of the equilibria when there is no time delay. Furthermore, an explicit formula for the time delay at which Hopf bifurcation occurs is derived, and the direction and stability of the Hopf bifurcation are also given. Finally, numerical simulations are carried out to support our theoretical results. We find that due to the existence of time delay, our system can exhibit many interesting dynamical behaviors depending on the number of equilibria and the increase in time delay. For instance, with increasing time delay, bistability transitions to the coexistence of a periodic solution and a stable equilibrium, followed by the emergence of quasi-periodic motion alongside stable equilibria, and eventually leading to chaotic motion coexisting with a stable equilibrium. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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269. Dynamics of a High-Speed Railway Wheelset with Two Time Delays in Primary Suspension Dampers.
- Author
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Li, Yue, Huang, Caihong, Shi, Huailong, Zeng, Jing, and Cao, Hongjun
- Subjects
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LIMIT cycles , *HIGH speed trains , *HOPF bifurcations , *PERIODIC motion - Abstract
A two-degree-of-freedom nonlinear high-speed railway wheelset model with two time delays in the lateral and yaw dampers is studied. The aim is to investigate the effect of time delays on stability and Hopf bifurcation characteristics of the wheelset model. The local stability of the trivial equilibrium under different time delay conditions is qualitatively analyzed. Analytical studies reveal that the wheelset model undergoes stability switches with the variation of the time delays. The stability switches correspond to Hopf bifurcations that occur when the time delays cross critical values. Furthermore, properties of Hopf bifurcation including direction and stability of bifurcating limit cycles are studied by using the normal form theory and the center manifold theorem. Our findings indicate that time delays in both lateral dampers and yaw dampers influence the stability and direction of Hopf bifurcation. Additionally, the numerical results show that time delays in the lateral and yaw dampers not only affect the amplitude of the hunting motion of the wheelset but also the periodic and chaotic motions. If the time delays gradually increase, the wheelset will vibrate irregularly with large lateral displacements. The analytical results presented in this paper offer a theoretical reference for the stability design of wheelsets. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
270. Hopf Bifurcation and Turing Instability of a Delayed Diffusive Zooplankton–Phytoplankton Model with Hunting Cooperation.
- Author
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Meng, Xin-You and Xiao, Li
- Subjects
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HOPF bifurcations , *PARTIAL differential equations , *COMPETITION (Biology) , *BIFURCATION theory , *HUNTING , *COOPERATION - Abstract
In this paper, a diffusive zooplankton–phytoplankton model with time delay and hunting cooperation is established. First, the existence of all positive equilibria and their local stability are proved when the system does not include time delay and diffusion. Then, the existence of Hopf bifurcation at the positive equilibrium is proved by taking time delay as the bifurcation parameter, and the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are investigated by using the center manifold theorem and the normal form theory in partial differential equations. Next, according to the theory of Turing bifurcation, the conditions for the occurrence of Turing bifurcation are obtained by taking the intraspecific competition rate of the prey as the bifurcation parameter. Furthermore, the corresponding amplitude equations are discussed by using the standard multi-scale analysis method. Finally, some numerical simulations are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
271. Slow–Fast Dynamics of a Piecewise-Smooth Leslie–Gower Model with Holling Type-I Functional Response and Weak Allee Effect.
- Author
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Wu, Xiao and Xie, Feng
- Subjects
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ALLEE effect , *HOPF bifurcations , *PREDATION , *FOOD quality , *PARAMETERS (Statistics) , *GLOBAL analysis (Mathematics) , *DIFFERENTIABLE dynamical systems , *LIMIT cycles , *LOTKA-Volterra equations - Abstract
The slow–fast Leslie–Gower model with piecewise-smooth Holling type-I functional response and weak Allee effect is studied in this paper. It is shown that the model undergoes singular Hopf bifurcation and nonsmooth Hopf bifurcation as the parameters vary. The theoretical analysis implies that the predator's food quality and Allee effect play an important role and lead to richer dynamical phenomena such as the globally stable equilibria, canard explosion phenomenon, a hyperbolically stable relaxation oscillation cycle enclosing almost two canard cycles with different stabilities and so on. Moreover, the predator and prey will coexist as multiple steady states or periodic oscillations for different positive initial populations and positive parameter values. Finally, we present some numerical simulations to illustrate the theoretical analysis such as the existence of one, two or three limit cycles. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
272. STABILITY AND BIFURCATION OF A PREDATOR–PREY SYSTEM WITH MULTIPLE ANTI-PREDATOR BEHAVIORS.
- Author
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XIA, YUE, HUANG, XINHAO, CHEN, FENGDE, and CHEN, LIJUAN
- Subjects
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ANTIPREDATOR behavior , *PREDATION , *HOPF bifurcations , *POPULATION density , *CUSP forms (Mathematics) , *EQUILIBRIUM - Abstract
In this paper, a predator–prey system with multiple anti-predator behaviors is developed and studied, where not only the prey may spread between patches but also the fear effect and counter-attack behavior of the prey are taken into account. First, the stability and existence of coexistence equilibria are presented. The unique positive equilibrium may be a saddle-node or a cusp of codimension 2. Then, various transversality conditions of bifurcations such as saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation are obtained. Moreover, compared with a single strategy, the multiple anti-predator strategies are more beneficial to the persistence and the population density of prey. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
273. DYNAMICS AND BLOW-UP CONTROL OF A LESLIE–GOWER PREDATOR–PREY MODEL WITH GROUP DEFENCE IN PREY.
- Author
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PATRA, RAJESH RANJAN, MAITRA, SARIT, and KUNDU, SOUMEN
- Subjects
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LOTKA-Volterra equations , *LATIN hypercube sampling , *HOPF bifurcations , *LYAPUNOV functions , *SENSITIVITY analysis - Abstract
In this paper, we designed a population model that shows how a prey species defends itself against a generalist predator by exhibiting group defence. A non-monotonic functional response is used to represent the group defence functionality. We have demonstrated the model's local stability in the vicinity of the coexisting equilibrium solution employing a local Lyapunov function. Condition for existence of Hopf bifurcation is obtained along with its normal form. The suggested model has been validated by numerical simulations, which have also been used to verify the acquired analytical results. The parameters are subjected to sensitivity analysis by utilizing partial rank correlation coefficient (PRCC) and Latin hypercube sampling (LHS). The Z-type dynamic method is used to prevent population blow-up. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
274. Dynamical complexity of a fractional‐order neural network with nonidentical delays: Stability and bifurcation curves.
- Author
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Mo, Shansong, Huang, Chengdai, Li, Huan, and Wang, Huanan
- Subjects
- *
BIFURCATION diagrams , *HOPF bifurcations , *FUNCTIONAL equations , *FRACTIONAL calculus , *STABILITY constants - Abstract
Recently, many scholars have discovered that fractional calculus possess infinite memory and can better reflect the memory characteristics of neurons. Therefore, this paper studies the Hopf bifurcation of a fractional‐order network with short‐cut connections structure and self‐delay feedback. Firstly, we use the Laplace transform to obtain the characteristic equation of the model, which is the transcendental equation containing four times transcendental item. Secondly, by selecting the communication delay as the bifurcation parameter and the other delay as the constant in its stability interval, the conditions for the occurrence of Hopf bifurcation are established; the bifurcation diagrams are provided to ensure that the derived bifurcation findings are accurate. Thirdly, in the case of identical neurons, the crossing curves method is exploited to the fractional‐order functional function equation to extract the Hopf bifurcation curve. Finally, two numerical examples are employed to confirm the efficiency of the developed theoretical outcomes. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
275. Bifurcation Analysis of an eco-epidemiological model involving prey refuge, fear impact and hunting cooperation.
- Author
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Mohammed, Alaa khadim and Majeed, Salam Jasim
- Subjects
HOPF bifurcations ,LOTKA-Volterra equations ,NUMERICAL analysis ,EPIDEMIOLOGICAL models ,FEAR - Abstract
This paper centers on an eco-epidemiological predator-prey model that accounts for hunting cooperation among predators and prey shelter and fear in afflicted prey. The objective is to investigate the effects of parameter factors on the model's bifurcation behavior. Theoretical part of this study demonstrate that a transcritical bifurcation can result from infection rate and refuge rate. Furthermore, fear rate can cause a Hopf-bifurcation to arise close to the positive equilibrium point. The presence of local bifurcations close to the non-trivial equilibrium points is verified by numerical analysis, which also guarantees the veracity of the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
276. Hopf-Hopf bifurcation in a predator-prey model with nonlocal competition and refuge in prey.
- Author
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Ma, Yuxin and Yang, Ruizhi
- Subjects
HOPF bifurcations ,BIFURCATION diagrams ,COMPUTER simulation - Abstract
In this paper, a diffusive predator-prey model with nonlocal competition and prey refuge is considered. The influence of parameters on the existence, multiplicity and stability of nonhomogeneous steady-state solutions is studied. It is obtained that an unstable positive nonconstant steady state exists in the neighborhood of the positive constant steady state. Compared with the model without nonlocal competition, the model with nonlocal competition can generate Hopf-Hopf bifurcation under some conditions. Through the qualitative analysis, the normal form at the Hopf-Hopf bifurcation singularity is calculated to analyze the different dynamic properties exhibited by the model in different parameter regions. In order to illustrate the feasibility of the obtained results and the dependence of the dynamic behavior on the nonlocal competition, numerical simulations are carried out. Through the numerical simulations, it is further shown that under certain conditions, the nonlocal competition will lead to the stablly spatially non-homogeneous periodic solutions and stablly spatially non-homogeneous quasi-periodic solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
277. Delayed nonmonotonic immune response in HIV infection system.
- Author
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Shanshan Wang and Shaoli Wang
- Subjects
- *
HIV infections , *IMMUNE response , *HOPF bifurcations , *IMMUNE system - Abstract
In this paper, we study a delayed HIV infection model with nonmonotonic immune response and perform stability and bifurcation analysis. Our results show that the delayed HIV infection system with nonmonotonic immune response has bistability and stable periodic solution appear. We find that both the uninfected and immune-free equilibria are globally asymptotically stable under certain conditions which are not affected by time delay. However, the time delay makes one immune equilibrium always unstable for τ ≥ 0 and also makes another immune equilibrium appear stability switches; meanwhile, the system will exhibit local Hopf bifurcation, global Hopf bifurcation, and saddle-node-Hopf bifurcation. Numerical simulations are carried out to verify our results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
278. Stability and bifurcation analysis of fractional-order tumor-macrophages interaction model with multi-delays.
- Author
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Padder, Ausif, Mokkedem, Fatima Zahra, and Lotfi, El Mehdi
- Subjects
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HOPFIELD networks , *BIOLOGICAL systems , *CELL populations , *HOPF bifurcations , *DIFFERENTIAL equations , *DELAY differential equations - Abstract
In the realm of modeling biological systems with memory, particularly those involving intricate interactions like tumor-immune responses, the utilization of multiple time delays and Caputo-type fractional-order derivatives represents a cutting-edge approach. In this research paper, we introduce a novel fractional-order model to investigate the dynamic interplay between tumors and macrophages, a key component of the immune system, while incorporating multiple time delays into our framework. Our proposed model comprises a system of three Caputo-type fractional-order differential equations, each representing distinct cell populations: tumor cells, anti-tumor cells (specifically M1-type macrophages with pro-inflammatory properties), and pro-tumor cells (M2-type macrophages with immune-suppressive characteristics). The stability of equilibria is discussed by analyzing the characteristic equations for each case, and the existence conditions for the Hopf bifurcation are obtained according to the critical values of delay parameters. Furthermore, numerical simulations are presented in order to verify the analytical results obtained for stability and Hopf-bifurcation with respect to the two-time delay parameters τ1 and τ2. The analysis shows the rich dynamics of the model according to the fractional-order parameter and the time delay parameters. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
279. Global stability analysis and Hopf bifurcation due to memory delay in a novel memory-based diffusion three-species food chain system with weak Allee effect.
- Author
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Tingting Ma and Xinzhu Meng
- Subjects
- *
ALLEE effect , *HOPF bifurcations , *FOOD chains , *LYAPUNOV functions , *MEMORY - Abstract
This paper presents a detailed study on the dynamics of a three-species food chain system with memory-based delay under weak Allee effect and middle predator refuge. The local stability analyses of the non-diffusive and memory-based diffusion systems are investigated. Moreover, we give a priori estimates and obtain the existence of the positive constant steady state by applying the comparison theorem. Sufficient conditions for the global stability are established by Barbalat lemma and the Lyapunov function. The theoretical results suggest the joint effect of cross-diffusion and memory-based delay can lead to Hopf bifurcation, which cannot appear in the system with self-diffusion or a small cross-diffusion coefficient. Numerical results verify the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
280. Bifurcations and global dynamics of a predator--preymitemodel of Leslie type.
- Author
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Yue Yang, Yancong Xu, Libin Rong, and Shigui Ruan
- Subjects
- *
HOPF bifurcations , *LIMIT cycles , *BIFURCATION diagrams , *PREDATORY animals - Abstract
In this paper, we study a predator--prey mite model of Leslie type with generalized Holling IV functional response. The model is shown to have very rich bifurcation dynamics, including subcritical and supercritical Hopf bifurcations, degenerate Hopf bifurcation, focustype and cusp-type degenerate Bogdanov--Takens bifurcations of codimension 3, originating from a nilpotent focus or cusp of codimension 3 that acts as the organizing center for the bifurcation set. Coexistence of multiple steady states, multiple limit cycles, and homoclinic cycles is also found. Interestingly, the coexistence of two limit cycles is guaranteed by investigating generalized Hopf bifurcation and degenerate homoclinic bifurcation, and we also find that two generalized Hopf bifurcation points are connected by a saddle-node bifurcation curve of limit cycles, which indicates the existence of global regime for two limit cycles. Our work extends some results in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
281. Equivariant Hopf Bifurcation in a Class of Partial Functional Differential Equations on a Circular Domain.
- Author
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Chen, Yaqi, Zeng, Xianyi, and Niu, Ben
- Subjects
- *
PARTIAL differential equations , *HOPF bifurcations , *REACTION-diffusion equations , *BOUNDARY value problems , *TIME delay systems , *STANDING waves , *FUNCTIONAL differential equations - Abstract
Circular domains frequently appear in mathematical modeling in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations at equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction–diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
282. Bifurcation Analysis of a Holling–Tanner Model with Generalist Predator and Constant-Yield Harvesting.
- Author
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Wu, Hongqiuxue, Li, Zhong, and He, Mengxin
- Subjects
- *
LIMIT cycles , *HOPF bifurcations , *PREDATORY animals , *DYNAMIC models , *COMPUTER simulation - Abstract
In this paper, we introduce constant-yield prey harvesting into the Holling–Tanner model with generalist predator. We prove that the unique positive equilibrium is a cusp of codimension 4. As the parameter values change, the system exhibits degenerate Bogdanov–Takens bifurcation of codimension 4. Using the resultant elimination method, we show that the positive equilibrium is a weak focus of order 2, and the system undergoes degenerate Hopf bifurcation of codimension 2 and has two limit cycles. By numerical simulations, we demonstrate that the system exhibits homoclinic bifurcation and saddle–node bifurcation of limit cycles as the parameters are varied. The main results show that constant-yield prey harvesting and generalist predator can lead to complex dynamic behavior of the model. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
283. Multiscale Effects of Predator–Prey Systems with Holling-III Functional Response.
- Author
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Zhang, Kexin, Yu, Caihui, Wang, Hongbin, and Li, Xianghong
- Subjects
- *
PREDATION , *HOPF bifurcations , *BIFURCATION theory , *LOTKA-Volterra equations , *OSCILLATIONS , *BIFURCATION diagrams - Abstract
In this paper, we proposed a Holling-III predator–prey model considering the perturbation of slow-varying, carrying capacity parameters. The study aims to address how the slow changes in carrying capacity influence the dynamics of the model. Based on the bifurcation theory and the slow–fast analysis method, the existence and the equilibrium of the autonomous system are explored, and then, the critical condition of Hopf bifurcation and transcritical bifurcation is established for the autonomous system. The slow–fast coupled nonautonomous system has quasiperiodic oscillations, single Hopf bursting oscillations, and transcritical–Hopf bursting oscillations within a certain range of perturbation amplitude variation if the carrying capacity perturbation amplitude crosses some critical values, such that the predator–prey management is challenging for the extinction of predator populations under the critical value. The motion pattern of the nonautonomous system is closely related to the transcritical bifurcation, Hopf bifurcation and attractor type of the autonomous system. Finally, the effects of changes in parameters related to predator aggressiveness on system behavior are investigated. These results show how crucial the predator–prey control is for varying carrying capacities. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
284. Dynamics of a New Delayed Glucose–Insulin Model with Obesity.
- Author
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Gao, Chunyan, Chen, Fangqi, and Yu, Pei
- Subjects
- *
INSULIN , *LOW-calorie diet , *MULTIPLE scale method , *BLOOD sugar , *OBESITY , *HOPF bifurcations , *TYPE 2 diabetes - Abstract
In this work, a new glucose–insulin model incorporating time delay and obesity is developed to gain insights of its dynamical mechanisms. Through the method of multiple scales, we theoretically demonstrate that time delay can drive the system to yield Hopf bifurcation, thereby producing oscillating solutions that are consistent with the simulation results. Moreover, obesity changes the level of glucose, but cannot induce oscillations. In particular, it is found that under the combined effect of obesity and time delay, obesity delays the appearance of Hopf bifurcation which is induced by time delay. Results show that a low calorie diet can achieve therapeutic effects including reducing blood glucose fluctuations and insulin resistance, which can be used as an adjuvant for the treatment of diabetes. In addition, our results indicate that the delay, together with an optimal rate of model parameters can cause a variety of dynamics and induce glucose oscillations. The result obtained in this paper may help to better understand the obesity, diabetes, and the interaction between glucose and insulin, so that control strategies can be designed to better regulate blood glucose levels and fluctuations and mitigate the occurrence of type-2 diabetes. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
285. Spatiotemporal dynamics of prey–predator model incorporating Holling-type II functional response with fear and its carryover effects.
- Author
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Dubey, Balram, Singh, Anand, and Anshu
- Subjects
- *
PREDATION , *HOPF bifurcations , *TWO-dimensional models , *SPECIES distribution , *MATHEMATICAL models , *COMPUTER simulation - Abstract
The recent focus in the fields of biology and ecology has centered on the significant attention given to the mathematical modeling and analyzing the spatiotemporal population distribution among species engaged in interactions. This paper explores the dynamics of the temporal and spatiotemporal delayed Bazykin-type prey–predator model, incorporating fear and its carryover effect. In our model, we incorporated a functional response of the Holling-type II. In the temporal model, a detailed dynamic analysis was carried out, investigating the positivity and boundedness of solutions, establishing the uniqueness and existence of positive interior equilibria, and examining both local and global stability. Additionally, we explored the presence of saddle-node, transcritical, and Hopf bifurcations varying attack rate parameter. The delayed system shows highly periodic behavior. Additionally, for the spatiotemporal model, we provide a complete analysis of local and global stability, and we derive the conditions for the existence of Turing instability for both self-diffusion and cross-diffusion, respectively. The two-dimensional diffusive model is further discussed, highlighting various Turing patterns, including holes, stripes, and hot and cold spots, along with their biological significance. Numerical simulations are executed to validate the analytical findings in both temporal and spatiotemporal models. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
286. Spatial memory drives spatiotemporal patterns in a predator-prey model describing intraguild predation.
- Author
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Pu, Xiaoyan, Zhang, Guohong, and Wang, Xiaoli
- Subjects
PREDATION ,SPATIAL memory ,HOPF bifurcations ,STABILITY constants ,COMPUTER simulation ,CONSUMERS - Abstract
In this paper, we consider a predator-prey model describing intraguild predation with memory-based diffusion. We first investigate the stability of two semi-trivial constant steady states and find that they are unstable if the positive steady state exists and can not be stable at the same time. Then we study the stability of the positive constant steady state and find that system has no Turing bifurcation when the predator is assumed to forage the prey and multiple stability switches can arise through Hopf bifurcation with joint effects of the memory-based diffusion and delay. Numerical simulations also show that the competition between two consumers has also important influence on complex spatiotemporal patterns formation of system. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
287. Complex Dynamics Analysis and Chaos Control of a Fractional-Order Three-Population Food Chain Model.
- Author
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Cui, Zhuang, Zhou, Yan, and Li, Ruimei
- Subjects
FOOD chains ,PREDATION ,JACOBI operators ,STABILITY theory ,FRACTIONAL differential equations ,HOPF bifurcations ,POLYNOMIAL chaos - Abstract
The present study investigates the stability analysis and chaos control of a fractional-order three-population food chain model. Previous research has indicated that the predation relationship within a long-established predator–prey system can be influenced by factors such as the prey's fear of the predator and its carry-over effects. This study examines the state evolution of fractional-order systems and compares their dynamic behavior with integer-order systems. By utilizing the Routh–Hurwitz condition and the stability theory of fractional differential equations, this paper establishes the local stability conditions of the model through the application of the Jacobi matrix and eigenvalue method. Furthermore, the conditions for the Hopf bifurcation generation are determined. Subsequently, chaos control techniques based on the Lyapunov stability theory are employed to stabilize the unstable trajectory at the equilibrium point. The theoretical findings are validated through numerical simulations. These results enhance our understanding of the stability properties and chaos control mechanisms in fractional-order three-population food chain models. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
288. Dynamic Behavior of a Class of Six-Neuron Fractional BAM Neural Networks.
- Author
-
Li, Weinan, Liao, Maoxin, Li, Dongsheng, Xu, Changjin, and Li, Bingbing
- Subjects
HOPF bifurcations - Abstract
In this paper, the stability and Hopf bifurcation of a six-neuron fractional BAM neural network model with multiple delays are considered. By transforming the multiple-delays model into a fractional-order neural network model with a delay through the variable substitution, we prove the conditions for the existence of Hopf bifurcation at the equilibrium point. Finally, our results are verified by numerical simulations. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
289. Stability and bifurcation of a delayed prey-predator eco-epidemiological model with the impact of media.
- Author
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Meng, Xin-You and Lu, Miao-Miao
- Subjects
PREDATION ,HOPF bifurcations ,STABILITY criterion ,HOPFIELD networks ,COMPUTER simulation - Abstract
In this paper, a delayed prey-predator eco-epidemiological model with the nonlinear media is considered. First, the positivity and boundedness of solutions are given. Then, the basic reproductive number is showed, and the local stability of the trivial equilibrium and the disease-free equilibrium are discussed. Next, by taking the infection delay as a parameter, the conditions of the stability switches are given due to stability switching criteria, which concludes that the delay can generate instability and oscillation of the population through Hopf bifurcation. Further, by using normal form theory and center manifold theory, some explicit expressions determining direction of Hopf bifurcation and stability of periodic solutions are obtained. What's more, the correctness of the theoretical analysis is verified by numerical simulation, and the biological explanations are also given. Last, the main conclusions are included in the end. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
290. Complicate dynamical analysis of a discrete predator-prey model with a prey refuge.
- Author
-
Khan, A. Q. and Alsulami, Ibraheem M.
- Subjects
STATE feedback (Feedback control systems) ,HOPF bifurcations ,BIFURCATION theory ,LOTKA-Volterra equations ,PREDATORY animals - Abstract
In this paper, some complicated dynamic characteristics are formulated for a discrete predator-prey model with a prey refuge. After studying the local dynamical properties about fixed points, our main purpose is to investigate condition(s) for the occurrence of flip and hopf bifurcations, respectively. Further, by the bifurcation theory, we have studied flip bifurcation at boundary fixed point, and flip and hopf bifurcations at interior fixed point of the discrete model. We have also studied chaos by state feedback control strategy. Furthermore, theoretical results are numerically verified. Finally, we have also discussed the influence of prey refuge in the discrete model. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
291. Tanh-like models for analysis and prediction of time-dependent flow around a circular cylinder at low Reynolds numbers.
- Author
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Wang, Libao, Xu, Min, Zheng, Boda, and An, Xiaomin
- Subjects
REYNOLDS number ,FLUID flow ,VORTEX shedding ,PREDICTION models ,HOPF bifurcations ,COORDINATE transformations - Abstract
When employing traditional low-order approximation equations to forecast the Hopf bifurcation phenomenon in the wake of a circular cylinder at low Reynolds numbers, inaccuracies may arise in estimating the phase. This is due to the fact that, in this transition process, the frequency varies with time. In this paper, we propose a method for analyzing and predicting the vortex shedding behind a cylinder at low Reynolds numbers. The proposed method is based on coordinate transformation and description function and is demonstrated using data from computational fluid dynamics simulation of flow around a cylinder at Reynolds number 100. The resulting governing equations explicitly contain the flow amplitude and implicitly contain the flow frequency. The proposed method is found to have higher accuracy compared to other methods for nonlinear identification and order reduction. Finally, the method is extended to predict nonlinear vortex shedding in the Reynolds number range of 80–200. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
292. ON THE DYNAMICS OF A FRACTIONAL-ORDER RICCATI DIFFERENTIAL EQUATION WITH PERTURBED DELAY.
- Author
-
EL-SAYED, A. M. A., SALMAN, S. M., and ABDELFATTAH, A. A. F.
- Subjects
RICCATI equation ,DELAY differential equations ,HOPF bifurcations ,LYAPUNOV exponents ,BIFURCATION diagrams ,DISCRETE systems - Abstract
This paper studies the dynamics of a fractional-order Riccati differential equation with perturbed delay and introduces a novel concept of perturbed delay. The study focuses on understanding the behaviour of the solution through the application of analytical techniques to investigate the existence and uniqueness of the solution and its continuous dependence on initial conditions. Analyses of Hopf bifurcations and the local stability of fixed points are studied. The discrete system is generated by piecewise constant arguments in order to simulate the behaviour of the system under consideration. The local stability analysis of the fixed points of the discrete system is presented. Numerical simulations using bifurcation diagrams, Lyapunov exponents and phase diagrams are illustrated. This helps confirm our research and unearth more complex dynamics. The theoretical results of the fractional order Riccati differential equation with delay and its perturbed equation are compared. Our results show that, under specific conditions, the fractional-order Riccati differential equation with perturbed delay exhibits equivalent dynamical properties to the fractional-order Riccati differential equation with delay. [ABSTRACT FROM AUTHOR]
- Published
- 2023
293. Hopf Bifurcation Analysis of a Housefly Model with Time Delay.
- Author
-
Chang, Xiaoyuan, Gao, Xu, and Zhang, Jimin
- Subjects
HOPF bifurcations ,BIFURCATION theory ,HOUSEFLY ,EIGENVALUES - Abstract
The oscillatory dynamics of a delayed housefly model is analyzed in this paper. The local and global stabilities at the non-negative equilibria are obtained via analyzing the distribution of eigenvalues and Lyapunov–LaSalle invariance principle, and the model undergoes the supercritical Hopf bifurcation and the transient oscillation. Based on Wu's global Hopf bifurcation theory, the existence of the global bifurcation is established under certain conditions. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
294. Complex bifurcations and noise-induced transitions: A predation model with fear effect in prey and crowding effect in predators.
- Author
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Wang, Cuihua, Wang, Hao, and Yuan, Sanling
- Subjects
PREDATION ,HOPF bifurcations ,PREDATORY animals ,STOCHASTIC models ,STOCHASTIC analysis ,LIMIT cycles - Abstract
It has been well established in the literature that the predator-induced fear has indirect impact on prey but can have comparable effects on prey population as direct killing, and that the crowding effect or self-limitation of population plays pivotal roles in determining the dynamics and interactions among populations. In this paper, we first propose and investigate a deterministic prey-predator model incorporating simultaneously fear effect in prey and crowding effect in predators. The model has rich dynamics, including one up to three positive equilibria, complex bifurcations (saddle-node, Hopf and Bogdanov-Takens bifurcations), and two types of bistability (between two interior equilibria or between an interior equilibrium and an interior limit cycle). Thus the model is easily affected by external environmental fluctuations. When environmental noises are involved, some new dynamics can be observed for the developed stochastic model. Especially, for the scenarios when the deterministic model exhibits bistability, we can observe noise-induced frequent transitions between two different interior attractors (two interior equilibria or an interior equilibrium and an interior limit cycle). The tipping points of noise intensities for the occurrence of such transitions are estimated by constructing the confidence ellipse/band for the equilibrium/limit cycle. These indicate that the predators and prey can coexist in two different modes and switch randomly between them. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
295. Stabilization and Chaos Control of an Economic Model via a Time-Delayed Feedback Scheme.
- Author
-
Hu, Yang and Hu, Guangping
- Subjects
ECONOMIC models ,HOPF bifurcations ,MATHEMATICAL models ,COMPUTER simulation - Abstract
This paper addresses the problem of chaos control in an economic mathematical dynamical model. By regarding the control variables as the bifurcation parameters, the stability of equilibria and the existence of Hopf bifurcations of the relevance feedback system are investigated, and the criterion of controllability for the chaotic system is obtained based on a time-delayed feedback control technique. Furthermore, numerical simulations are provided to demonstrate the feasibility of our methods and results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
296. Canard-induced mixed mode oscillations as a mechanism for the Bonhoeffer-van der Pol circuit under parametric perturbation.
- Author
-
Yu, Yue, Zhang, Cong, Chen, Zhenyu, and Zhang, Zhengdi
- Subjects
OSCILLATIONS ,HOPF bifurcations ,BIFURCATION theory ,BIFURCATION diagrams ,PERTURBATION theory ,DIFFERENTIAL equations - Abstract
Purpose: This paper aims to investigate the singular Hopf bifurcation and mixed mode oscillations (MMOs) in the perturbed Bonhoeffer-van der Pol (BVP) circuit. There is a singular periodic orbit constructed by the switching between the stable focus and large amplitude relaxation cycles. Using a generalized fast/slow analysis, the authors show the generation mechanism of two distinct kinds of MMOs. Design/methodology/approach: The parametric modulation can be used to generate complicated dynamics. The BVP circuit is constructed as an example for second-order differential equation with periodic perturbation. Then the authors draw the bifurcation parameter diagram in terms of a containing two attractive regions, i.e. the stable relaxation cycle and the stable focus. The transition mechanism and characteristic features are investigated intensively by one-fast/two-slow analysis combined with bifurcation theory. Findings: Periodic perturbation can suppress nonlinear circuit dynamic to a singular periodic orbit. The combination of these small oscillations with the large amplitude oscillations that occur due to canard cycles yields such MMOs. The results connect the theory of the singular Hopf bifurcation enabling easier calculations of where the oscillations occur. Originality/value: By treating the perturbation as the second slow variable, the authors obtain that the MMOs are due to the canards in a supercritical case or in a subcritical case. This study can reveal the transition mechanism for multi-time scale characteristics in perturbed circuit. The information gained from such results can be extended to periodically perturbed circuits. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
297. Dynamics of a three-molecule autocatalytic Schnakenberg model with cross-diffusion: Turing patterns of spatially homogeneous Hopf bifurcating periodic solutions.
- Author
-
Li, Weiyu and Wang, Hongyan
- Subjects
HOPF bifurcations ,AUTOCATALYSIS ,DIFFUSION ,ORDINARY differential equations ,DYNAMICAL systems - Abstract
In this paper, a three-molecule autocatalytic Schnakenberg model with cross-diffusion is established, the instability of bifurcating periodic solutions caused by diffusion is studied, that is, diffusion can destabilize the stable periodic solutions of the ordinary differential equation (ODE) system. First, utilizing the local Hopf bifurcation theory, the central manifold theory, the normal form method and the regular perturbation theory of the infinite dimensional dynamical system, the stability of periodic solutions for the ODE system is discussed. Second, for this model, according to the implicit function existence theorem and Floquet theory, the Turing instability of spatially homogeneous Hopf bifurcating periodic solutions is studied. It is proved that the otherwise stable Hopf bifurcating periodic solutions in the ODE system produces Turing instability in the Schnakenberg model with cross-diffusion. Finally, through numerical simulations, it is verified that Turing instability of periodic solutions is determined by cross-diffusion rates. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
298. Fractional-order PID control of tipping in network congestion.
- Author
-
He, Jiajin, Xiao, Min, Lu, Yunxiang, Wang, Zhen, and Zheng, Wei Xing
- Subjects
PID controllers ,HOPF bifurcations ,TELECOMMUNICATION systems - Abstract
Tracing the rapid progress of communication network, the control of dynamic evolution of network has become a central issue. There are a lot of tipping phenomena in network congestion systems. Therefore, tipping control principally centres on traditional control policies, and some advanced control approaches need to be supplemented. In this paper, a fractional-order proportional-integral-derivative (PID) controller is introduced to a network congestion model to ponder corresponding bifurcation-induced tipping regulation. First, a fractional-order congestion model with fractional-order PID controller is constructed. Then the onset of the tipping induced by Hopf bifurcation of the uncontrolled model is studied. By contrast, the tipping point can be delayed under the controller for the controlled model. Some conditions under which Hopf bifurcation occurs are given. The stable and unstable ranges of control parameters for the controlled model are also deduced. At last, some simulated examples are given to verify the theoretical results and demonstrate the superiority of the controller in tipping regulation. Moreover, the bidirectional effects of the controller are displayed by manipulating the control parameters. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
299. Bifurcation and stability of a diffusive predator–prey model with the fear effect and time delay.
- Author
-
Wang, Huatao, Zhang, Yan, and Ma, Li
- Subjects
HOPF bifurcations ,LYAPUNOV-Schmidt equation ,BIFURCATION theory ,PREDATION ,POPULATION density ,EIGENVALUES - Abstract
The predator–prey system can induce wealth properties with fear effects. In this paper, we propose a diffusive predator–prey model where the influence of fear effects and time delay is considered, under the Dirichlet boundary condition. It follows from the Lyapunov–Schmidt reduction method that there exists a non-homogeneous steady-state solution of the system and the specific expressions are also given. By the aid of bifurcation theory and eigenvalue theory, we also investigate the existence/non-existence and the stability of Hopf bifurcation under three different conditions of bifurcation parameters. Furthermore, the effects of the fear on population density, stability, and Hopf bifurcation are also considered and the results show that the increase of fear effects will reduce the population density, and Hopf bifurcation is more likely difficult to undergo as k increases under some conditions. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
300. Kinetic analysis of spike and wave discharge in a neural mass model.
- Author
-
Wang, Zhihui and Xie, Manhong
- Subjects
WAVE analysis ,LIMIT cycles ,INTERNEURONS ,PYRAMIDAL neurons ,NEURAL inhibition ,HOPF bifurcations - Abstract
In the brain, feedforward inhibition is a fundamental regulator of balancing neural excitation and inhibition, and its dysfunction may cause epilepsy. In this paper, we investigate the influence of feedforward inhibition on absence seizures and its dynamical mechanisms based on a neural mass model. On the one hand, pyramidal neurons (PY), fast kinetic interneurons ( I fast ) and slow kinetic interneurons ( I slow ) form two feedforward inhibition pathways, which are the PY– I fast – I slow pathway and the PY– I slow – I fast pathway, respectively. When changing the synaptic strength in different pathways, the system shows simple oscillation state, spike and wave discharge (SWD), and multi-spike and wave discharge (m-SWD). On the other hand, the one-parameter bifurcation analysis reveals that the system develops fold bifurcations, Hopf bifurcations, fold bifurcations on limit cycle, and period doubling bifurcations when state transitions occur. In particular, when the system is in the bistable region, the dynamic state of this region is closely related to the stable limit cycle and stable fixed point. Therefore, feedforward inhibition pathways are indeed involved in the regulation of absence seizures, excitatory connections in both feedforward inhibition pathways are more effective than the inhibitory connections in the control of absence seizures. More interestingly, the feedforward inhibition pathway PY– I slow – I fast has a stronger regulation of absence seizures than PY– I fast – I slow . These results provide a theoretical basis for a more detailed understanding of the underlying mechanisms of neurological disorders. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
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