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Summary: The unstable operation of a direct‐drive permanent magnet synchronous generator (DPMSG) is exacerbated by periodic oscillation, having a significant impact on the safe and stable functioning of wind energy generation systems. This paper proposes an approximate solution method for analyzing the periodic oscillation of the Hopf bifurcation and a method for suppressing its bifurcation through H∞ robust control. Firstly, a three‐dimensional nonlinear dynamic model of the DPMSG is constructed. Secondly, the Hopf bifurcation of the system with changes in internal parameters is solved using the time domain and frequency domain methods, and the harmonic balancing method is used to approximate the periodic solution at the point of Hopf bifurcation. The order of the Hopf bifurcation neighborhood is then decreased using the central manifold theorem. Finally, this study suggests a H∞ output feedback control employing bifurcation parameters. The Nyquist criteria are used for evaluating small signal stability and convergence speed of the response system in DPMSG before and after dimensionality reduction. The simulation results suggest that the proposed strategy helps to tackle the periodic solution expression and instability difficulties caused by Hopf bifurcation. This paper provides theoretical suggestions for the future reliable operation of new energy generation systems. [ABSTRACT FROM AUTHOR]

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]

In this paper, we consider a diffusive predator–prey model with spatial memory of prey and gestation delay of predator. For the system without delays, we study the stability of the positive equilibrium in the case of diffusion and no diffusion, respectively. For the delayed model without diffusions, the existence of Hopf bifurcation is discussed. Further, we investigate the stability switches of the model with delays and diffusions when two delays change simultaneously by calculating the stability switching curves and obtain the existence of Hopf bifurcation. We also calculate the normal form of Hopf bifurcation to determine the direction of Hopf bifurcation and the stability of bifurcation periodic solutions. Finally, numerical simulations verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper explores the dynamics analysis of a human immunodeficiency virus (HIV) model with saturated cytotoxic T lymphocyte (CTL) immune response and Beddington–DeAngelis infection rate. There are two time delays in the model to describe the time needed for infection of cell and CTL immune response generation, respectively. We obtain two thresholds and three possible equilibria from the model. By analyzing the corresponding characteristic equations, we study the stabilities of equilibrium and the effect of delays on CTL immune response. The results indicate that when immune delay is present, the steady state of equilibrium is disrupted and leads to a Hopf bifurcation. Finally, we use sensitivity analyses to show the effect of parameters on thresholds and numerical simulations to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to build and analyze a model of labor market slack considering unemployment along with employment in which the number of hours is limited to a level below that preferred by employees. We thus have "underemployment" along with unemployment. We analyze the need for potential policy action directed at the reduction of unemployment, simultaneous with an autonomous process of labor market job creation and migration. We model delays in labor market responses to past unemployment and vacancies creation and capture the effect on unemployment through a non‐linear dynamic system. We observe not only job separation and matching but also movement into and out of underemployment. The model allows for migration in an open economy context. We analyze the stability behavior of the resulting equilibria for our dynamic system, including Dirac and weak kernels. [ABSTRACT FROM AUTHOR]

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]

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]

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]

In this paper, a diffusive predator–prey system with fear factor response subject to Neumann boundary conditions is considered. Bifurcations at the boundary equilibria of the corresponding ODE are confirmed by Sotomayor's theorem. Detailed bifurcation analysis shows that the reaction–diffusion system undergoes Hopf bifurcation and steady‐state bifurcation. The bifurcation direction and the stability of the bifurcating periodic solutions are also given. Finally, numerical simulation results are carried out to verify the theoretical results. [ABSTRACT FROM AUTHOR]

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]

In this paper, we provide the normal form for the Hopf bifurcation of a class of the reaction‐diffusion equation with memory‐based diffusion and nonlocal effect, where the delay is present in the differential term, similar to the chemotaxis model with time delay. The eigenvalue problems and the decomposition of the phase space are discussed in detail. Through a series of variable transformations, we obtain the third‐order truncated normal form of the model constrained on the central manifold and its equivalent equation in polar coordinates. Then, with the help of the dynamic analysis for the finite dimensional equations, the key parameters for determining the direction and stability of the Hopf bifurcation are given. These theoretical results are applied to the Bazykin's model, the stability, Turing bifurcation and Hopf bifurcation of the equilibrium are demonstrated through both theoretical and numerical methods. [ABSTRACT FROM AUTHOR]

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]

In this paper, an SIR model with a nonlinear incidence rate is studied. A disease‐free equilibrium E0$$ {E}_0 $$, an endemic equilibrium E1$$ {E}_1 $$, and the basic reproduction number of the model R0$$ {R}_0 $$ are obtained. If R0<1,E0$$ {R}_0&lt;1,{E}_0 $$ is locally asymptotically stable and if R0>1,E1$$ {R}_0&gt;1,{E}_1 $$ is locally asymptotically stable. By Barbalat's lemma, we study the global stability of the model. Transcritical bifurcation analysis is investigated by using the Sotomayor theorem. As the infection rate increases, the asymptotic behavior of the system near E0$$ {E}_0 $$ approaches E1$$ {E}_1 $$ and the system has a transcritical bifurcation. Also, we check the existence of Hopf bifurcation for the given system. In addition, a sensitivity analysis is provided for the basic reproduction number. Our results are supported with numerical simulations. [ABSTRACT FROM AUTHOR]

In most HIV models, the emergence of backward bifurcation means that the control for basic reproduction number less than one is no longer effective for HIV treatment. In this paper, we study an HIV model with CTL response and cell‐to‐cell transmission by using the dynamical approach. The local and global stability of equilibria is investigated, the relations of subcritical Hopf bifurcation and supercritical bifurcation points are revealed, especially, the so‐called new type bifurcation is also found with two Hopf bifurcation curves meeting at the same Bogdanov–Takens bifurcation point. Forward and backward bifurcation, Hopf bifurcation, saddle‐node bifurcation, Bogdanov–Takens bifurcation are investigated analytically and numerically. Two limit cycles are also found numerically, which indicates that the complex behavior of HIV dynamics. Interestingly, the role of cell‐to‐cell interaction is fully uncovered, it may cause the oscillations to disappear and keep the so‐called new type bifurcation persist. Finally, some conclusions and discussions are also given. [ABSTRACT FROM AUTHOR]

In this paper, a class of respiratory disease model is established by taking into account simultaneously two situations that susceptible individuals directly got sick by inhaling air pollutants and indirectly fallen ill from infection by patients with respiratory diseases. The sufficient conditions for equilibria existence of the system and locally asymptotically stability of endemic equilibria are obtained. The existence of saddle‐node bifurcation is derived by using the Sotomayor theorem. The stability of periodic solution of Hopf bifurcation is determined by calculating the first Lyapunov number. The direct conversion rate and indirect infection rate in the system are selected as bifurcation parameters; the existence of Bogdanov–Takens bifurcation of codimension 2 of the system is proved by calculating the universal unfolding near the cusp. This research shows that direct and indirect ways of getting sick have a significant influence on dynamical behaviors of the respiratory disease system. [ABSTRACT FROM AUTHOR]

This study is devoted to considering dynamic properties of an environmental protection expenditures model for protected areas (PAs) with two delays. It presents an improved understanding of relationship among tourism, environment, and capital stock in PAs and provides a sustainable strategy to keep the balance between them. The model considers the interactions among environmental resource E$$ E $$, visitors V$$ V $$, and capital stock K$$ K $$ in PAs. The tourism income is used partly to increase the capital stock and partly to protect the environment. In this paper, time delays τ1$$ {\tau}_1 $$ and τ2$$ {\tau}_2 $$ are introduced into the system. The technique of stability crossing curves for two delays is proposed to analyze the occurrence of Hopf bifurcation. Numerical simulations are performed for understanding the analytical results. The results show that under some parameter conditions, the share η$$ \eta $$ of total revenue generated by tourism user fees for environmental protection is related to delays τ1$$ {\tau}_1 $$ and τ2$$ {\tau}_2 $$. Moreover, the environment and the capital stock can be maintained persistently if time delay τ1$$ {\tau}_1 $$ is not too large. [ABSTRACT FROM AUTHOR]

In this paper, we investigate bifurcations of stationary solutions of a cross‐diffusion prey–predator system with Ivlev functional response and Neumann boundary conditions. First, we analyze the local stability and existence of a Hopf bifurcation at a coexistence stationary solution. We show that the bifurcating periodic solutions are asymptotically orbitally stable and the bifurcation direction is supercritical when the ratio of the conversion of prey captured by predator to the death rate of predator is in the interval (1,2]$$ \left(1,2\right] $$. Next, we derive sufficient conditions for the existence of a steady‐state bifurcation from a simple eigenvalue. We establish the existence of two intersecting C1$$ {C}^1 $$ curves of steady‐state solutions near the coexistence stationary solution. To illustrate our theoretical results, we give some numerical examples. From numerical simulations we observe that the coexistence stationary solution loses its stability via the Hopf bifurcation and a periodic solution emerges after a short time. Also by taking the bifurcation parameter value in the stable region, the effect of the initial condition disappears over time and the solution returns to the coexistence stationary solution. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a cannibalistic predator‐prey system with Beddington‐DeAngelis type response function when the infection only exists in the predator species. The predator species are subject to cannibalistic interaction and the infection spreads among the predators via cannibalism. The model is studied in presence of multiple delays for the maturation of prey and predator. Local stability analysis of the predator‐prey model around the biologically realistic two essential steady states is investigated. We also investigate the bifurcation analysis of the proposed model around the co‐existing steady states. Using the normal form method and center manifold theory, the direction of Hopf bifurcation and the Hopf bifurcating periodic solutions are explored. In presence of time delays, we derive the criteria for the permanence of the model. Our findings demonstrate that maturation delays destabilize the system and can produce high periodic oscillations. Cannibalism can act as a self‐regulatory scenario controlling the transmission of disease among the predator species and stabilizing the oscillations in the predator‐prey system. Our theoretical analysis is well supported with numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a predator–prey model with Beddington–DeAngelis functional response and predator competition, which is a five‐parameter family of planar vector field. It is shown that the model can undergo a sequence of bifurcations including focus type degenerate Bogdanov–Takens bifurcation of codimension 3 and Hopf bifurcation of codimension at least 2 as the parameters vary. Our theoretical results indicate that predator competition can cause richer dynamics such as two limit cycles enclosing one or three hyperbolic positive equilibria and three kinds of homoclinic orbits (homoclinic to hyperbolic saddle, saddle‐node, or neutral saddle). Moreover, there exists a threshold value m0$$ {m}_0 $$ for predator capturing rate m$$ m $$, below or equal to which the predators always tend to extinction, above which the predators and preys will coexist in the form of multiple steady states or periodic oscillations for all positive initial populations. Numerical simulations are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a multiple scales method (MMS) is employed to analyze Hopf bifurcations in differential equations with two linearly state‐dependent time delays. Firstly, the linear stability of the linearized equation near the only equilibrium (the trivial equilibrium) is performed analytically. Then, the case for which the coefficients of the delayed terms are small, the method of multiple scales (MMS) bypassing the need to use center manifold reduction allows the normal form to be easily obtained. Furthermore, the stability and bifurcation analysis are undertaken for the normal form to determine the types of the Hopf bifurcation. The proposed method can not only determine the direction of Hopf bifurcation but also its type. The numerical simulation results agree well with the analytical predictions. This suggests that the MMS employed in this paper provides a simple, accurate and effective means of analyzing Hopf bifurcations in the state‐dependent delayed differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we improve a new mathematical model associated with glue applying of particleboard. First, we study the existence and stability of the equilibria and the existence of fold, Hopf, and Bogdanov–Takens bifurcations in above system. Second, the normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation are derived, and the classifications of local dynamics near above bifurcation critical values are analyzed. Then, numerical simulation results show that the glue flow system associated with glue applying of particleboard exists stable equilibrium, stable periodic‐1, periodic‐2, and periodic‐4 solutions, and chaotic attractor phenomenon from a sequence of period‐doubling bifurcations. Finally, we compare the dynamical phenomena of glue flow system with frequency water supply system, showing that cubic terms which the latter lacks, could influence the dynamical behaviors of glue flow system. [ABSTRACT FROM AUTHOR]

In this paper, we propose a reaction–diffusion predator–prey model with fear effect under a predator‐poisoned environment. First, we analyze the existence and stability of constant positive steady states. Second, taking the cost of minimum fear as the bifurcation parameter, the direction and stability of spatially homogeneous/inhomogeneous periodic solutions are investigated. Then, some properties of nonconstant positive steady states are investigated, such as nonexistence, existence, and steady state bifurcation. Applying the fixed point index theory, the existence of the nonconstant positive steady state is discussed, indicating that the proper diffusion rate of prey and larger diffusion rate of predator are beneficial to survival of populations. Moreover, taking the diffusion rate of predator as the bifurcation parameter, the steady state bifurcations from simple and double eigenvalues are intensively established. At last, the validity of the theoretical analysis is verified by numerical simulations. Biological interpretation reveals that the fear effect and the diffusion rate of populations can lead to the emergence of Hopf and steady state bifurcation, thereby destroying the stability of populations and promoting the benign evolution of populations. [ABSTRACT FROM AUTHOR]

Bifurcation theory has been highly popular in the analysis of mathematical models. However, stability and bifurcation analyses are only for asymptotic dynamics while applied scientists care more about transient dynamics. In this paper, we first rigorously analyze Holling–Tanner model with generalist predators who have alternative food sources, and then discuss transient dynamics via a changing environment. For a constant environment, we provide a complete bifurcation analysis with high codimension. It is shown that the highest codimension of a nilpotent cusp is 3, and the model can undergo degenerate Bogdanov–Takens bifurcation of codimension 3. Moreover, by using resultant elimination to solve the semialgebraic varieties of Lyapunov coefficients, we show that a center‐type equilibrium is a weak focus with order at most 2, and the model can exhibit Hopf bifurcation of codimension 2. Our results indicate that generalist predators can cause not only richer dynamics and bifurcations, but also the extinction of prey for some positive initial densities. Numerical simulations, including the coexistence of a limit cycle and a homoclinic cycle, tristability, two limit cycles, are presented to illustrate the theoretical results. In a changing environment, the populations start along one stable state but can track unstable states or oscillations when the system crosses a bifurcation point, and then tend to another stable state or oscillations. This tracking on transient dynamics predicts regime shifts under environmental changes. When environmental conditions vary, the populations can track unstable states in the constant environment. The rate of environmental change determines how long the system tracks an unstable state although finally the solution under environmental change is attracted to a stable steady state or limit cycle. Finally, we focus on a periodic environment and find that the populations converge to a periodic solution or an invariant torus depending on both the initial environmental capacity and the amplitude of periodic fluctuation. [ABSTRACT FROM AUTHOR]

This paper is concerned with a diffusive predator–prey model with prey‐taxis and prey‐stage structure under the homogeneous Neumann boundary condition. The stability of the unique positive constant equilibrium of the predator–prey model is derived. Hopf bifurcation and steady‐state bifurcation are also concluded. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a fractional differential‐algebraic model with two discrete delays and double linear species harvesting. One of the delays denotes the time taken for digestion of the prey, and the other reflects the negative feedback of the predator's density. Taking into account of the economic profit, linear predator harvesting and prey harvesting have been incorporated into the proposed predator–prey system. From the biological viewpoint, we discuss the sufficient conditions for the existence of nontrivial positive equilibrium point. By jointly using the stability theory of fractional differential‐algebraic equations and the bifurcation theory, we study the delay‐induced instability and Hopf bifurcations. Finally, the correctness of the theoretical analysis is verified by numerical simulations, and the effects of delays, economic profit, and fractional exponents on the stability of the system are explored. [ABSTRACT FROM AUTHOR]

We deal with a fractional‐order nonlinear financial model with time delay in this paper. The quantitative analysis of the model has been done in which the asymptotic stability of the equilibrium points of the model have been discussed. Furthermore, the Hopf bifurcation analysis of the model has been done under the impact of time delay. The stability of the model has been studied with the reproduction number less than or greater than 1. The analysis of the model has been done by using the Laplace transformation technique. The analysis shows that the fractional‐order model with a time delay can adequately enhance the elements and fortify the results for either stable or unstable criteria. All unstable cases in the nonlinear financial system are converted to stable cases for fractional order under neighborhood points. The reproduction ranges of all parameters have been discussed. The paper examines the impact of time delay and the importance of determining essential parameters such as saving amount by using Hopf bifurcation. Finally, numerical simulations have been carried out by MATLAB to illustrate our derived results. The financial system is verified under the theoretical outcomes from parameter estimation. [ABSTRACT FROM AUTHOR]

This paper studies the nonlinear modeling and operation stability of variable speed pumped storage power station (PSPS). Firstly, basic equations of variable speed PSPS are established. Nonlinear state equation in the form of relative deviation value is derived by considering supplementary conditions. Then, Hopf bifurcation analysis of variable speed PSPS is carried out. Stable domain of variable speed PSPS is studied. Finally, influence of factors on stability is analyzed. Dynamic response of variable speed PSPS is investigated. The results indicate that variable speed PSPS is a five‐order nonlinear dynamic system. Nonlinear characteristics of variable speed PSPS reflect the nonlinear coupling effect between unit speed and rotor current. The emerged Hopf bifurcation is supercritical. Stable domain locates at the lower side of bifurcation line. Stable domain intuitively reflects and evaluates the stability of variable speed PSPS. When state point locates at stable domain, variable speed PSPS is stable and dynamic response of state variables is damped oscillation. A smaller flow inertia of penstock, a greater head loss of penstock, and a greater slip ratio are favorable for stability of variable speed PSPS. Stable domain under positive disturbance of active power is larger than that under negative disturbance of active power. [ABSTRACT FROM AUTHOR]

In this paper, we develop an age‐structured model with two infectious stages to study the effect of infection age on dynamic properties. By reformulating the model as a non‐densely defined Cauchy problem and applying theorem related with Hille–Yosida operator, the threshold dynamics are investigated. Theoretical analysis shows that destabilization of the age‐dependent endemic equilibrium can occur due to the perturbation of critical infection age that separates the two infectious stages. Furthermore, numerical simulations are conducted to illustrate the dynamical behaviors of stability switching. [ABSTRACT FROM AUTHOR]

With desertification becoming more and more serious all over the world, it is crucial to understand the intrinsic relationship between vegetation and water (soil water and surface water). In this paper, a cross‐diffusion model with mixed delays is established to describe the dynamics of vegetation, soil water, and surface water in the case of water shortage. The local stability conditions of positive equilibrium point is established by using Routh–Hurwitz criterion. Then, the conditions for generating Hopf bifurcation are studied. In the end, numerical simulations are carried out to illustrate the effectiveness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

Since population behaviors possess the characteristic of history memory, we, in this paper, introduce time fractional‐order derivatives into a diffusive Gause‐type predator‐prey model, which is time fractional‐order reaction‐diffusion equations and a generalized form of its corresponding first‐derivative model. For this kind of model, we prove the existence and uniqueness of a global positive solution by using the theory of evolution equations and the comparison principle of time fractional‐order partial differential equations. Besides, we obtain the stability and Hopf bifurcation of the Gause‐type predator‐prey model in the forms of the time fractional‐order ordinary equations and of the time fractional‐order reaction‐diffusion equations, respectively. Our results show that the stable region of the parameters in these 2 models can be enlarged by the time fractional‐order derivatives. Some numerical simulations are made to verify our results. [ABSTRACT FROM AUTHOR]

In this paper, the stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment is studied. It is shown that there exist 3 equilibria. The sufficient conditions for local asymptotic stability of the infection-free equilibrium and no-immune equilibrium are given. We also discussed the local stability of positive equilibrium and the existence of Hopf bifurcation. Moreover, the direction and stability of Hopf bifurcation is obtained by using standard form theory and the center manifold theorem. Finally, numerical simulations are performed to verify the theoretical conclusions. [ABSTRACT FROM AUTHOR]

The present paper concerns with the dynamics of a fractional neural network involving three neurons. Firstly, the bifurcation point is identified for which Hopf bifurcations may occur by taking the system parameter as a bifurcation parameter via the stability analysis of fractional systems. It is indicated that the system parameter can significantly affect the dynamical properties of such network. Secondly, the impact of the order on the bifurcation point is carefully examined. It is found that the occurrence of bifurcation is delayed as the order increases as long as the other system parameters are established. Finally, a numerical example is exploited to verify the efficiency of theoretical results. [ABSTRACT FROM AUTHOR]

In studying small limit cycles of finite‐dimensional systems, one of the central problem is the computation of focus quantities. In practice, the computation is a challenging problem even for some simple low‐dimensional systems. This paper is devoted to the computation of focus quantities of all orders and to the study of Hopf bifurcations in some chaotic systems. A recursive formula for computing focus quantities is presented for a K + 2‐dimensional system. The formula is a generalization of previous results on low‐dimensional systems with K = 0 and K = 1. For a four‐dimensional hyper‐chaotic system, according to the sign of the first focus quantity, we prove that the simple Hopf bifurcation of the system is supercritical. For a five‐dimensional chaotic system with four equilibria of Hopf type, according to the signs of the first focus quantities, we prove that the simple Hopf bifurcations of the system are subcritical. [ABSTRACT FROM AUTHOR]

Summary: This paper introduces a new hybrid control strategy to control the bifurcation of the delayed congestion model. The delay or gain is taken as the bifurcation parameter to analyze the characteristic root distribution of the linearized system. The stability and existence conditions of Hopf bifurcation are obtained. The results show that by selecting the appropriate method, the dynamics of the controlled system can be changed, and the nonzero equilibrium constant control parameters are maintained. It reveals that when such bifurcation is undesirable or desirable, the hybrid controller can delay or advance the onset of the intrinsic bifurcation. Finally, the case study illustrates and visualizes the effectiveness of the new control strategy in controlling Hopf bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, a delayed with Holling type II functional response (Beddington‐DeAngelis) and Allee effect predator‐prey model is considered. The growth of the prey is affected by the parameter M, which defines the Allee effect. In addition, the delay τ also influences the logistic growth of the prey, which can be interpreted as the maturity time or the gestation period. In the study of the characteristic equation, we observe that the delay τ also depends on the parameter M, which affects the dynamics in the prey population. Considering the delay as a bifurcation parameter, the local asymptotic stability of the positive equilibrium is investigated. On the other hand, we find that the system can also suffer a Hopf bifurcation in the positive equilibrium when the delay passes through a sequence of critical values. In particular, we study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions, an explicit algorithm is provided applying the normal form theory and center manifold reduction for the functional differential equations. Finally, numerical simulations that support the theoretical analysis are included. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate the stability and transition of the Navier‐Stokes‐alpha model. By using the continued‐fraction method, combining with the dynamic transition theory, we show the existence of a Hopf bifurcation in this model as Reynolds number crosses a critical value. Upon deriving the explicit expression of a non‐dimensional number P called transition number, which is a function of the critical Reynolds number and the aspect ratio, we further analyze the transition associated with the Hopf bifurcation. More precisely, it is shown that the modeled flow exhibits either a continuous or catastrophic transition at the critical Reynolds number, whose specific type of the transition is determined by the sign of the real part of P at the critical Reynolds number, and the spatio‐temporal structure of the limit cycle bifurcated that corresponds to a wave that propagates slowly westward and is symmetric about the mid‐axis of the channel. [ABSTRACT FROM AUTHOR]

In this paper, Hopf bifurcation and center problem for a generic three‐dimensional Chua's circuit system are studied. Applying the formal series method of computing singular point quantities to investigate the two cases of the generic circuit system, we find necessary conditions for the existence of centers on a local center manifold for the systems, then Darboux method is applied to show the sufficiency. Further, we determine the maximum number of limit cycles that can bifurcate from the corresponding equilibrium via Hopf bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate with a time fractional‐order derivative in a three‐species predator‐prey model with the presence of prey social behavior. A new approximation for predator‐prey interaction in the presence of prey social behavior has been considered. For the model analysis, the study has been divided into two principal parts. First of all, we study the local stability of the equilibria and the existence of Hopf bifurcation. Then, for the numerical analysis, the Caputo fractional derivative operator is utilized to approximate the numerical solution of the model. An excellent agreement is seen between the numerical results and the theoretical predictions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dynamics of a diffusive predator‐prey system with strong Allee effect and delayed Ivlev‐type functional response. At first, we apply the method of upper‐lower solutions and the comparison principle in proving the nonnegativity of the solutions. Then by analyzing the distribution of the eigenvalues, we obtain the bistability of the system and the existence of Hopf bifurcation. Furthermore, by using the center manifolds theory and normal form method, we study the properties of the Hopf bifurcations. Finally, some numerical simulations are carried out for illustrating the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the problem of bifurcation control for a small‐world network model with time delay is studied. We first put forward a Proportional‐Integral‐Derivative (PID) feedback scheme to control the Hopf bifurcation of the network. The time delay is selected as the bifurcation parameter. The conditions of the stability and Hopf bifurcation are given for the controlled network. By using the center manifold theorem and the normal form theory, the direction and stability of bifurcating periodic solutions are confirmed. The feasible region of the parameters of the controller is determined. It is found that the bifurcation dynamics of the small‐world network are optimized by adjusting the parameters of the PID controller. Finally, a numerical example verifies the effectiveness of the designed PID controller, and the relationships between the onset of the Hopf bifurcation and the control parameters are obtained. [ABSTRACT FROM AUTHOR]

In this paper, a stock‐effort dynamical model with two fishing zones is discussed. The nonlinear harvesting function is assumed depending upon stock size as well as fishing effort. The migration of fish is considered between two zones. The harvesting vessels also move between zones to increase their revenue. The movements of fish and fishing vessels between zones are assumed to take place at a faster time scale as compared with processes involving growth and harvesting occurring at a slow time scale. The aggregated model is obtained for total fish stock and fishing effort. This aggregated (reduced) model is analyzed analytically as well as numerically. Biological and bionomic equilibria of the system are obtained, and criteria for local stability or instability of the system are derived. The impact of levels of taxation T on the fish population and on the revenue earned by the fishery is investigated. An optimal harvesting policy is also discussed using the Pontryagin's maximum principle. The aggregated model also exhibits Hopf and transcritical bifurcation with respect to the bifurcation parameter tax T. Numerical simulations are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

In the paper entitled 'A novel chaotic system and its topological horseshoe' in [Nonlinear Analysis: Modelling and Control 18 (1) (2013) 66-77], proposed the 3D chaotic system [ABSTRACT FROM AUTHOR]

In this paper, an age‐structured population model with the form of neutral functional differential equation is studied. We discuss the stability of the positive equilibrium by analyzing the characteristic equation. Local Hopf bifurcation results are also obtained by choosing the mature delay as bifurcation parameter. On the center manifold, the normal form of the Hopf bifurcation is derived, and explicit formulae for determining the criticality of bifurcation are theoretically given. Moreover, the global continuation of Hopf bifurcating periodic solutions is investigated by using the global Hopf bifurcation theory of neutral equations. Finally, some numerical examples are carried out to support the main results. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a multidelayed in‐host HIV model to represent the interaction between human immunodeficiency virus and immune response. One delay was considered to incorporate the time required by the virus for various intracellular events to make a host cell productively infective, and the second delay was introduced to take into account the time required for adaptive immune system to respond against infection. We extensively analyzed this multidelayed model analytically and numerically. We show that delay may have both destabilizing and stabilizing effects even when the system contains a single immune response delay. It happens when there exists two sequences of critical values of this delay. If the system starts with stable state in absence of delay, then the smallest value of these critical delays causes instability and the next higher value causes stability. The system may also show stability switching for different values of the virus replication factor. These results demonstrate the possible reasons of intrapatients and interpatients variability of CD4+ T cells and virus counts in HIV‐infected patients. [ABSTRACT FROM AUTHOR]

This paper deals with a diffusive toxin producing phytoplankton‐zooplankton model with maturation delay. By analyzing eigenvalues of the characteristic equation associated with delay parameter, the stability of the positive equilibrium and the existence of Hopf bifurcation are studied. Explicit results are derived for the properties of bifurcating periodic solutions by means of the normal form theory and the center manifold reduction for partial functional differential equations. Numerical simulations not only agree with the theoretical analysis but also exhibit the complex behaviors such as the period‐3, 5, 6, 7, 8, 11, and 12 solutions, cascade of period‐doubling bifurcation in period‐2, 4, quasi‐periodic solutions, and chaos. The key observation is that time delay may control harmful algae blooms (HABs). Moreover, numerical simulations show that the chaotic states induced by the period‐doubling bifurcation are purely temporal, which is stationary in space and oscillatory in time. The investigations may provide some new insights on harmful phytoplankton blooms. [ABSTRACT FROM AUTHOR]

"Delay" has been considered as one of destabilizing factors in economic dynamics for a long time. Dynamic macroeconomics is concerned with explaining growth and fluctuations. This paper shows how various dynamics involving cyclic fluctuations can emerge in the standard neoclassical growth model when two distinct delays, a delay in production and a delay in depreciation, are explicitly taken into account. We first confirm that the production delay has a stabilizing effect and the depreciation delay has a destabilizing effect in a one‐delay model. We then determine the stability switching curve analytically in the two delay model. It is shown that cyclic fluctuations emerge via Hopf bifurcation when stability is lost. It is also found that stability loss and gain repeatedly occur. Numerical examples verify the theoretical results when the Cobb‐Douglas production function is adopted. [ABSTRACT FROM AUTHOR]

In this paper, a planar system of ordinary differential equations is considered, which is a modified Leslie‐Gower model, considering a Beddington‐DeAngelis functional response. It generates a complex dynamics of the predator‐prey interactions according to the associated parameters. From the system obtained, we characterize all the equilibria and its local behavior, and the existence of a trapping set is proved. We describe different types of bifurcations (such as Hopf, Bogdanov‐Takens, and homoclinic bifurcation), and the existence of limit cycles is shown. Analytic proofs are provided for all results. Ecological implications and a set of numerical simulations supporting the mathematical results are also presented. [ABSTRACT FROM AUTHOR]

In this paper, a dynamic state feedback is applied to control Hopf bifurcations arising from a fractional-order Van Der Pol oscillator. The degree parameter indicating the strength of the nonlinear damping is chosen as the bifurcation parameter. It is shown that in the absences of the dynamic state feedback controller, the fractional-order Van Der Pol oscillator loses the stability via the Hopf bifurcation early, and can maintain the stability only in a certain domain of the degree parameter. When applying the state feedback controller to the fractional-order Van Der Pol oscillator, the onset of the undesirable Hopf bifurcation is postponed. Thus, the stability domain is extended, and the system possesses the stability in a larger parameter range. Numerical simulations are given to justify the validity of the dynamic state feedback controller in bifurcation controls. [ABSTRACT FROM AUTHOR]

Recent empirical contributions highlight the negative impact of pollution on labor supply. This relationship is explained by two mechanisms: first, pollution modifies agents' work–leisure trade‐off as it deteriorates their working conditions (incentive effect); and second, a polluted environment is likely to generate more frequent epidemic outbreaks and to affect agents' immune systems (health effect). Bosi et al. () explore the aggregate consequences of the incentive effect and show that it can generate endogenous fluctuations in the economic activity. The present paper focuses on the health effect as we study a Ramsey model augmented with the spread of infectious disease. We find that industrial pollution may generate limit cycles around an endemic steady state. More precisely, the economic system may undergo a transcritical bifurcation followed by two Hopf bifurcations near this steady state. [ABSTRACT FROM AUTHOR]

This paper is concerned with Hopf bifurcation and its stability for the model related to the disease of human T‐cell lymphotropic virus type I (HTLV‐I). In fact, we are considering Hopf bifurcation of the system because it plays an important role in indicating when the disease can return to a steady state condition. The model consists of CD4+ T‐cell subsets of healthy, latently infected, and actively infected cells, as well as for leukemia cells. Latently infected T‐cells may harbor the virus for several years until they become activated and able to infect susceptible T‐cells. In this model, we apply techniques such as the center manifold and normal‐form theorem to simplify bifurcating and the dimension of the dynamical system. The results are applied to clinical data as an example. [ABSTRACT FROM AUTHOR]