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In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rock–scissors–paper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game. [ABSTRACT FROM AUTHOR]

• The high-order fractional complex-valued BAM neural networks are proposed and their sufficient stability conditions are given. • Taking time delay as a bifurcation parameter, the general Hopf bifurcation conditions of fractional complex-valued BAM neural networks are obtained. • The relationship between fractional order and critical bifurcation point is also discussed. In this paper dynamical behaviors of a class of high–order fractional complex–valued bidirectional associative memory neural networks with multiple time delays are investigated. Firstly, they are reduced to real–valued systems by separating the real and imaginary parts. Then, stability criteria of fractional complex–valued bidirectional associative memory neural networks without delay are obtained. Concerning the delay case, the time delay is set as a bifurcation parameter and the condition of Hopf bifurcation is given by analyzing roots of characteristic equations. Finally, two numerical examples are presented and illustrate that Hopf bifurcation does happen when time delay exceeds the critical value. [ABSTRACT FROM AUTHOR]

• The three-strategy game model with continuous distributed time delay is analyzed. • The two-community three-strategy games with two delays is discussed. • The existence of Hopf bifurcation in replicator equation is proved. • The rock-paper-scissors game is simulated as an example. In this paper, we combine evolutionary game with dynamics, and discuss two kinds of game models with time delay. First, based on continuous distributed (kernel function) stochastic delay, we give the asymptotic stability condition of general three-strategy game model. Second, we study the replicator dynamics of two-community three-strategy game model with two fixed time delays (i.e. τ 1 within the community and τ 2 between different communities). Based on Hopf bifurcation theory, by calculating the critical conditions for the existence of bifurcation, the influence of bifurcation on the stability of equilibrium points is analyzed. Finally, we take rock-paper-scissors game as an example to verify the correctness of the theoretical results, when the delay τ meets certain conditions or doesn't exceed the critical value, the stability of the system will not change. [ABSTRACT FROM AUTHOR]

Abstract This paper considers Hopf bifurcation of complex network with two independent delays. By analyzing the eigenvalue equations, the local stability of the system is studied. Taking delay as parameter, the change of system stability with time is studied and the emergence of inherent bifurcation is given. By changing the value of the delay, the bifurcation of a given system can be controlled. Numerical simulation results confirm the validity of the results found. [ABSTRACT FROM AUTHOR]

In this paper we investigate the dynamical properties of a two prey-one predator system with quadratic self interaction represented by a three-dimensional system of differential equations by using tools of computer algebra. We first investigate the stability of the singular points. We show that the trajectories of the solutions approach to stable singular points under given conditions by numerical simulation. Then, we determine the conditions for the existence of the invariant algebraic surfaces of the system and we give the invariant algebraic surfaces to study the flow on the algebraic invariants which is a useful approach to check if Hopf bifurcation exists. [ABSTRACT FROM AUTHOR]

In this paper, a predator-prey model for exploited fish populations is considered, where the prey and the predator both show schooling behavior. Due to this coordinated behavior, predator-prey interaction occurs only at the outer edge of the schools formed by the populations. Positivity and boundedness of the model are discussed. Analysis of the equilibria is presented. A criterion for Hopf bifurcation is obtained. The optimal harvest policy is also discussed using Pontryagin’s maximum principle, where the effort is used as the control parameter. Numerical simulations are carried out to validate our analytical findings. Implications of our analytical and numerical findings are discussed critically. [ABSTRACT FROM AUTHOR]

• To demonstrate the Galerkin method's validity, via the accuracy of its predictions using DPDE models. • To structure a process for applying the theoretical framework to find theoretical results. • To show the impacts of diffusion and immune response delay on the model, through bifurcation diagrams. • To determine the steady-state solutions, Hopf points, and the stable and unstable zones. • To illustrate comparisons of the DDE and DPDE approaches through a selection of 2-D and 3-D solution maps. This paper studies a diffusive viral infection system with delayed immune response in the one-domain system. A system of DDE equations was explored, both analytically and numerically, using the Galerkin method. A condition that helps to find Hopf bifurcation points is determined. Full maps of the Hopf bifurcation points as well regions of stability are constructed and considered in detail. It is shown that the time delay of cytotoxic T lymphocyte (CTL) response and the diffusion parameter can significantly impact upon the stability regions. Furthermore, the influences of the other free values have been examined for their effects on stability. It is found that, as diffusion increases, the CTL response delay increases, and also as the CTL response delay is increased, the Hopf points for both generation rate and activate rate are decreased, whereas the Hopf points for the infection and death rates increased. Moreover, an increase diffusion results in an increase in the Hopf points for growth rate and activation rate, while the Hopf bifurcations are decreased for the death rate of infected cells. Bifurcation diagrams are plotted to show selected examples of limit cycle behavior (periodic oscillation), and 3-D solutions for the three concentrations in the model have been plotted to corroborate all analytical results from the theoretical section. [ABSTRACT FROM AUTHOR]

This paper investigates an issue of bifurcation control for a novel incommensurate fractional-order predator–prey system with time delay. Firstly, the associated characteristic equation is analyzed by taking time delay as the bifurcation parameter, and the conditions of creation for Hopf bifurcation are established. It is demonstrated that time delay can heavily effect the dynamics of the proposed system and each order has a major influence on the creation of bifurcation simultaneously. Then, a linear delayed feedback controller is introduced to successfully control the Hopf bifurcation for such system. It is shown that the control effort is markedly influenced by feedback gain. It is further found that the onset of the bifurcation can be delayed as feedback gain decreases. Finally, two illustrative examples are exploited to verify the validity of the obtained newly results. [ABSTRACT FROM AUTHOR]

Complex-valued neural networks (CVNNs) with integer-order have attracted much attention, and which have been well discussed. Fractional complex-valued neural networks (FCVNNs) are more suitable to describe the dynamical properties of neural networks, but have rarely been studied. It is the first time that the stability and bifurcation of a class of delayed FCVNN is investigated in this paper. The activation function can be expressed by separating into its real and imaginary parts. By using time delay as the bifurcation parameter, the dynamical behaviors that including local asymptotical stability and Hopf bifurcation are discussed, the conditions of emergence of bifurcation are obtained. Furthermore, it reveals that the onset of the bifurcation point can be delayed as the order increases. Finally, an illustrative example is provided to verify the correctness of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we proposed and analyzed a prey predator system with stage-structured for the predator population. A time delay is incorporated due to the gestation for the matured predator. All the possible non-negative equilibria are obtained and their local as well as global behavior are studied. Choosing delay as a bifurcation parameter, the existence of the Hopf bifurcation of the system has been investigated. Moreover, we use the normal form method and the center manifold theorem to examine the direction of the Hopf bifurcation and the nature of the bifurcating periodic solution. Some numerical simulations are given to support the analytical results. Some interesting conclusions are obtained from our analysis and it is given at the end of the paper. [Copyright &y& Elsevier]

Based on Lorenz system, a new four-dimensional quadratic autonomous hyper-chaotic attractor is presented in this paper. It can generate double-wing chaotic and hyper-chaotic attractors with only one equilibrium point. Several properties of the new system are investigated using Lyapunov exponents spectrum, bifurcation diagram and phase portraits. Using the center manifold and normal form theories, the local dynamics, the stability and Hopf bifurcation at the equilibrium point are analyzed. To obtain the ellipsoidal ultimate bound, the ultimate bound of the proposed system is theoretically estimated using Lagrange multiplier method, Lagrangian function and local maximizer point. By properly choosing P and Q matrices, an estimation of the ultimate bound region, as a function of the Lagrange coefficient, is obtained using local maximizer point and reduced Hessian matrix. To demonstrate the evolution of the bifurcation and to show the ultimate bound region, numerical simulations are provided. [ABSTRACT FROM AUTHOR]

• A model for HTLV-I infection that includes uninfected & infected CD4+T cells and HTLV-I-specific CD8+T cells. • To better understand the dynamics of HTLV-I infection, we incorporate a CTL immune response delay. • Applying Lyapunov functional procedures to determine global stability by the reproduction numbers R0 and R1. • The model with delay experiences a destabilization of the infected steady state leading to Hopf bifurcation and periodic solutions. • We estimate the length of delay to preserve the stability of bifurcating limit cycle using Nyquist criteria. This paper deals with a four dimensional mathematical model for Human T-cell leukemia virus type-I (HTLV-I) infection that includes delayed CD8+ cytotoxic T-cells (CTLs) immune response. The proposed system has three biologically feasible steady states, namely disease-free steady state, CTL-inactive steady state and an interior steady state. Our theoretical analysis demonstrates that local and global stability analysis are established by the two critical parameters R 0 and R 1 , basic reproduction numbers due to viral infection and due to CTLs immune response, respectively. The disease-free steady state E 0 is globally stable if R 0 ≤ 1 , and the HTLV-I infections are eliminated. The asymptotic-carrier steady state E 1 is globally stable if R 1 ≤ 1 < R 0 , which indicates HTLV-I infection is chronic without persistence of CTLs immune response. The interior steady state E 2 is globally asymptotic stable if R 1 > 1 , which implies that the HTLV-I infection is choric in persistence of CTLs immune response. Due to immune response delay, our proposed model undergoes a destabilization of the interior steady state leading to Hopf bifurcation and periodic oscillations. We estimate the length of time delay that preserve the stability of period-1 limit cycle. We also derived the direction and stability of Hopf bifurcation around the interior steady state by center manifold theory and normal form method. To determine the robustness of the model, we performed normalized forward sensitivity analysis with reference to R 0 and R 1. Our proposed model undergoes Hopf bifurcation with respect to the production rate of uninfected CD4+T cells h , removal rate of virus-specific CTLs d 4 , spontaneous infected CD4+T cell activation d 2 and transmissibility coefficient β. Implications of our numerical illustrations to the pathogenesis of HTLV-I infection and the development of HTLV-I related HAM/TSP are explored. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a neutral neural network model with two delays is investigated. The condition to ensure the stability of the zero solution of the system is decided by choosing τ 1 and τ 2 as parameters, respectively. Then the Hopf bifurcation is discussed by using the center manifold theory and normal form method introduced by Hassard and Kazarinoff. Global existence of periodic solution is studied by using the global Hopf bifurcation theory. Finally, some numerical simulations are carried out to illustrate the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, a three species dynamical system is explored. The system consisting of two logistically growing competing species and the third species acts as a predator as well as host. It is predating over second species with Holling type II functional response, while first species is benefited from the third species. In addition, the prey species move into a refuge to avoid high predation. The essential mathematical features of the proposed model are studied in terms of boundedness, persistence, local stability and bifurcation. The existence of transcritical bifurcations have been established about two axial points. It has been observed that survival of all three species may be possible due to commensalism. Numerical simulations have been performed to show the Hopf bifurcation about interior equilibrium point. The existence of period-2 solution is observed. Further, the bifurcations of codimension-2 have also been investigated. [ABSTRACT FROM AUTHOR]

This paper proposes and investigates a delayed model for the dynamics and control of a mosquito population which is subject to an integrated strategy that includes pesticide release, the use of mechanical controls and the use of the sterile insect technique (SIT). The existence of positive equilibria is characterized in terms of two threshold quantities, being observed that the “richer” equilibrium (with more mosquitoes in the aquatic phase) has better chances to be stable, while a longer duration of the aquatic phase has the potential to destabilize both equilibria. It is also found that the stability of the trivial equilibrium appears to be mostly determined by the value of the maturation rate from the aquatic phase to the adult phase. A nonstandard finite difference (NSFD) scheme is devised to preserve the positivity of the approximating solutions and to keep consistency with the continuous model. The resulting discrete model is transformed into a delay-free model by using the method of augmented states, a necessary condition for the existence of optimal controls then determined. The particularities of different control regimes under varying environmental temperature are investigated by means of numerical simulations. It is observed that a combination of all three controls has the highest impact upon the size of the aquatic population. At higher environmental temperatures, the oviposition rate is seen to possess the most prominent influence upon the outcome of the control measures. [ABSTRACT FROM AUTHOR]

In this paper, an SEIR epidemiological model with saturation incidence and a time delay describing the latent period of the disease is investigated, where it is assumed that the susceptible population is subject to logistic growth in the absence of the disease. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is discussed. The existence of Hopf bifurcations at the endemic equilibrium is established. By means of Lyapunov functionals and LaSalle’s invariance principle, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable and the disease dies out; if the basic reproduction number is greater than unity, sufficient conditions are obtained for the global stability of the endemic equilibrium. Numerical simulations are carried out to illustrate some theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a delayed ratio-dependent predator–prey model with Holling type III functional response and stage structure for the predator is considered. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of Hopf bifurcations at the coexistence equilibrium is established. By utilizing normal form method and center manifold theorem, the explicit formulas which determine the direction of Hopf bifurcation and the stability of bifurcating period solutions are derived. Finally, numerical simulations supporting the theoretical analysis are given. [ABSTRACT FROM AUTHOR]

In this paper, a delayed biological economic system which considers a plankton system with harvest effort on phytoplankton is proposed. By using the theory of partial functional differential equations, Hopf bifurcation of the proposed system with delay as the bifurcation parameter is investigated. It reveals that the discrete time delay has a destabilizing effect in the plankton dynamics, and a phenomenon of Hopf bifurcation occurs as the delay increases through a certain threshold. Then by numerical simulations the impact of delay, diffusion and economic interest on plankton system are explored. It is found that delay can cause system into chaos and can trigger the emergence of irregular spatial patterns via a Hopf bifurcation. Moreover, diffusion and economic profit can also affect the dynamic behavior of the system. [ABSTRACT FROM AUTHOR]

In this paper, a BAM neural network with multiple time delays and diffusion under homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of the trivial uniform steady state and the existence of Hopf bifurcation under two different cases are established, respectively. By using the normal form theory and the center manifold reduction of partial functional differential equations (PFDEs), explicit formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results. [ABSTRACT FROM AUTHOR]

Hopf bifurcation is one of the important dynamical behaviors. It could often cause some phenomena, such as quasiperiodicity and intermittency. Consequently, chaos will happen due to such dynamical behaviors. Since chaos appears in the Lorenz-like system, to understand the dynamics of such system, Hopf bifurcation will be explored in this paper. First, the stability of equilibrium points is presented. Then Hopf bifurcation of the Lorenz-like system is investigated. By applying the normal form theory, the conditions guaranteeing the Hopf bifurcation are derived. Further, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also presented. Finally, numerical simulations are given to verify the theoretical analysis. It is found that Hopf bifurcation could happen when conditions are satisfied. The stable bifurcating periodic orbit is displayed. Chaos will also happen when parameter further increases. [ABSTRACT FROM AUTHOR]

In this paper, we aim to investigate the dynamics of a gene regulatory network which is a time-delayed version of the model proposed by Elowitz and Leibler [Nature 403 (2000) 335–338] . Based on the normal form theory and center-manifold reduction, Hopf bifurcations including the bifurcation direction and stability of the bifurcated periodic orbits are investigated. We also discuss effects of transcriptional rate and time delay on the amplitude and period of the oscillation of the network. It shows that variations of time delay or transcriptional rate can change the period and amplitude of the oscillation. More precisely, (i) the amplitude increases with small time delay, while the change of amplitude is not sensitive to relatively large time delay. However, the robustness of amplitudes is not true any more for the case of using the transcriptional rate as parameter, where amplitude always increases quickly and linearly with the transcriptional rate; (ii) the period of oscillation increases as the time delay increases, but it grows up initially as the transcriptional rate increases and then keeps unchanged to certain constant value, which implies that the robustness of period to the transcriptional rate variations occurs. Our numerical simulations also support the theoretical conclusions, namely both suggest that time delay and transcriptional rate can be used as control parameters in genetic regulatory networks. [ABSTRACT FROM AUTHOR]

In this paper we study a model of HCV with saturation and delay, we stablish the local and global stability of system also we stablish the occurrence of a Hopf bifurcation. We will determine conditions for the permanence of model, and the length of delay to preserve stability. We present a sensitivity analysis for the basic reproductive number. [ABSTRACT FROM AUTHOR]

In this paper, we consider a delayed diffusive predator–prey model with herd behavior. Firstly, by choosing the appropriate bifurcation parameter, the stability of the positive equilibria and the existence of Hopf bifurcations, induced by diffusion and delay respectively, are investigated by analyzing the corresponding characteristic equation. Then, applying the normal form theory and the center manifold argument for partial functional differential equations, the formula determining the properties of the Hopf bifurcation are obtained. Furthermore, the instability of the Hopf bifurcation leads to the emergence of spatial patterns. Finally, some numerical simulations are also carried out to illustrate and expand the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical behavior of a tumor–immune system (T–IS) interaction model with two discrete delays, namely the immune activation delay for effector cells (ECs) and activation delay for helper T cells (HTCs). By analyzing the characteristic equations, we establish the stability of two equilibria (tumor-free equilibrium and immune-control equilibrium) and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter. Our results exhibit that both delays do not affect the stability of tumor-free equilibrium. However, they are able to destabilize the immune-control equilibrium and cause periodic solutions. We numerically illustrate how these two delays can change the stability region of the immune-control equilibrium and display the different impacts to the control of tumors. The numerical simulation results show that the immune activation delay for HTCs can induce heteroclinic cycles to connect the tumor-free equilibrium and immune-control equilibrium. Furthermore, we observe that the immune activation delay for HTCs can even stabilize the unstable immune-control equilibrium. [ABSTRACT FROM AUTHOR]

We propose a predator–prey ecoepidemic model with parasitic infection in the prey. We assume infection time delay as the time of transmission of disease from susceptible to infectious prey. We examine the effects of supplying additional food to predator in the proposed model. The essential theoretical properties of the model such as local and global stability and in addition bifurcation analysis is done. The parameter thresholds at which the system admits a Hopf bifurcation are investigated in presence of additional food with non-zero time lag. The conditions for permanence of the system are also determined in this paper. Theoretical analysis results are verified through numerical simulations. By supplying additional food we can control predator population in the model. Most important observation is that we can control parasitic infection of prey species by supplying additional food to predator. Eliminating the most infectious individuals from the prey population, predator quarantine the infected prey and prevent the spreading of disease. [ABSTRACT FROM AUTHOR]

In this paper we study the nonlinear dynamics of a modified van der Pol oscillator. More precisely, we study the local codimension one, two and three bifurcations which occur in the four parameter family of differential equations that models an extension of the classical van der Pol circuit with cubic nonlinearity. Aiming to contribute to the understand of the complex dynamics of this system we present analytical and numerical studies of its local bifurcations and give the corresponding bifurcation diagrams. A complete description of the regions in the parameter space for which multiple small periodic solutions arise through the Hopf bifurcations at the equilibria is given. [ABSTRACT FROM AUTHOR]

In this paper, we consider the effect of distributed time delays on dynamics of a mathematical model of gene expression. Both the weak and strong delay kernels are discussed. Sufficient conditions for the local stability of the unique equilibrium are obtained. Taking the average delay as a bifurcation parameter, we investigate the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the method of multiple time scales. Finally, numerical simulation is carried out to illustrate our theoretical results. It shows both subcritical and supercritical Hopf bifurcations can happen. [ABSTRACT FROM AUTHOR]

This paper deals with a three-dimensional plant-herbivore-predator model that incorporates explicitly the plant toxicity in plant-herbivore interactions. The existence and stability conditions of all the feasible equilibria are established. Our results indicate that plant toxicity may play a key role in the dynamical behavior of the system. By adding another plant species with a different toxicity level to this system, we derive threshold conditions on the invasion of the second plant species. The analysis indicates that several parameters may be critical to determine successful invasion. Numerical simulations are also provided to reinforce the theoretical conclusions. [ABSTRACT FROM AUTHOR]

• This paper is concerned with the detailed dynamics analysis of the Lengyel-Epstein system with a discrete delay. • Under the assumption that the positive equilibrium of the model is locally asymptotically stable in the absence of delay, the effect of the increase of delay on the stability of the unique positive equilibrium is analyzed in detail. • The phenomenon that the equilibrium becomes ultimately unstable after passing through multiple stability switches and Hopf bifurcations at some certain critical values of delay is found. • By means of the normal form method and the center manifold reduction for retarded functional differential equations, the explicit formulae determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions are obtained. • To verify our theoretical conclusions, some numerical simulations for specific examples are also included at the end of this article. This paper examines the dynamical analysis of the Lengyel-Epstein system with a discrete delay in detail. Under the assumption that the unique positive equilibrium of the model is locally asymptotically stable in the absence of the delay, the effect of the increase of delay on the stability of the unique positive equilibrium is analyzed in detail. It is found that under suitable conditions on the other parameters, the delay doesn't affect the stability of the equilibrium, namely, the equilibrium is absolutely stable while under the other conditions on the other parameters, the equilibrium will become ultimately unstable after passing through multiple stability switches and Hopf bifurcations at some certain critical values of delay. Particularly, by means of the normal form method and the center manifold reduction for retarded functional differential equations, the explicit formulae determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions are obtained. To verify our theoretical conclusions, some numerical simulations for specific examples are also included at the end of this article. [ABSTRACT FROM AUTHOR]

Abstract: This paper develops the residue harmonic balance solution procedure to predict the bifurcated periodic solutions of some autonomous delay differential systems at and after Hopf bifurcation. In this solution procedure, the zeroth-order solution employs just one Fourier term. The unbalanced residues due to Fourier truncation are considered by solving linear equation iteratively to improve the accuracy. The number of Fourier terms is increased automatically. The well-known sunflower equation and van der Pol equation with unit delay are given as numerical examples. Their solutions are verified for a wide range of system parameters. Comparison with those available shows that the residue harmonic balance method is effective to solve the autonomous delay differential equations. Moreover, the present method works not only in determining the amplitude but also the frequency at bifurcation. [Copyright &y& Elsevier]

Abstract: In this paper, an HIV-1 infection model with saturation incidence and time delay due to the CTL immune response is investigated. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcation at the CTL-activated infection equilibrium are established, respectively. By means of Lyapunov functionals and LaSalle’s invariance principle, it is shown that the infection-free equilibrium is globally asymptotically stable when the basic reproduction ratio is less than unity. When the immune response reproductive ratio is less than unity and the basic reproductive ratio is greater than unity, the CTL-inactivated infection equilibrium of the system is globally asymptotically stable. [Copyright &y& Elsevier]

Abstract: The ratio-dependent theory is favored by researchers since it is more suitable for describing the relationship between predator and its prey. In this paper, a ratio-dependent predator–prey system with Holling type II functional response, two time delays and stage structure for the predator is investigated. Firstly, by choosing the two time delays as the bifurcation parameter, the sufficient conditions for the local stability and the existence of Hopf bifurcation with respect to both delays are established. Furthermore, based on the normal form method and center manifold theorem, explicit formulas are derived to determine the direction of Hopf bifurcation and stability of the bifurcating periodic solution. Finally, numerical simulations are given to verify the theoretical analysis. [Copyright &y& Elsevier]

Abstract: In this paper we propose an extension to the classic Solow model by introducing a non-concave production function and a time-to-build assumption. The capital accumulation equation is given by a delay differential equation that has two non-trivial stationary equilibria. By choosing time delay as the bifurcation parameter, we demonstrate that the “high” stationary solution may lose its stability and a Hopf bifurcation occurs when the delay passes through critical values. By applying the center manifold theorem and the normal form theory, we obtain formulas for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. In addition, the Lindstedt–Poincaré method is used to calculate the bifurcated periodic solution, the direction of the bifurcation, and the stability of the periodic motion resulting from the bifurcation. The Hopf bifurcation is found to be supercritical. Finally, numerical simulations are given to justify the validity of the theoretical analysis. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the Hopf bifurcation analysis and amplitude control of the modified Lorenz system. The Hopf bifurcation of system is investigated by utilizing the Hopf bifurcation theory and the center manifold theorem firstly. Then the direction and stability of limit cycle emerging from Hopf bifurcation are determined by the designed controller. Moreover, the amplitude of limit cycle emerging from Hopf bifurcation is controlled by a nonlinear feedback controller. Finally, numerical simulations are given to verify theoretical analysis. [Copyright &y& Elsevier]

Abstract: In this paper, we study a predator–prey model with a transmissible disease spreading in the predator population and a time delay representing the gestation period of the predator. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease-free equilibrium and the coexistence equilibrium are established, respectively. By means of Lyapunov functionals and LaSalle’s invariance principle, sufficient conditions are derived for the global stability of the predator-extinction equilibrium and the disease-free equilibrium and the global attractiveness of the coexistence equilibrium of the system, respectively. Numerical simulations are carried out to support the theoretical analysis. [Copyright &y& Elsevier]

• New fractional-order six-neuron bi-directional associative memory (BAM) neural networks involving multiple delays are investigated. • Existence, uniqueness and boundedness of solution for the involved neural networks are studied. • The stability and the existence of Hopf bifurcation of the involved neural networks are analyzed. • Up to now, there are few papers that deal with fractional-order high-dimensional BAM neural networks. This study mainly explores fractional-order six-neuron bi-directional associative memory (BAM) neural networks involving multi-delays. Taking advantage of contraction mapping principle, we prove that the solution of the addressed BAM neural networks exists and is unique. Utilizing a acceptable function, we confirm that the solution of the addressed BAM neural networks is bounded. By applying a suitable variable substitution, new fractional order six-neuron BAM neural networks involving mult-delays are converted to a class of fractional order six-neuron BAM neural networks with single delay. Using the stability criterion and bifurcation theory of fractional order differential dynamical systems, we carry out a detailed discussion on the stability and the onset of Hopf bifurcation of the established BAM neural networks. The study shows that the time delay is an important factor which affects the stability behavior and Hopf bifurcation of the involved neural networks. Numerical simulation plots are presented to illustrate our derived key conclusions. The derived analytical findings of the study play a vital role in optimizing and designing neural networks. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we have considered a system of delay differential equations. The system without delayed arises in the Lengyel–Epstein model. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. Linear stability is investigated and existence of Hopf bifurcation is demonstrated via analyzing the associated characteristic equation. For the Hopf bifurcation analysis, the delay parameter is chosen as a bifurcation parameter. The stability of the bifurcating periodic solutions is determined by using the center manifold theorem and the normal form theory introduced by Hassard et al. (1981) [7]. Furthermore, the direction of the bifurcation, the stability and the period of periodic solutions are given. Finally, the theoretical results are supported by some numerical simulations. [Copyright &y& Elsevier]

Abstract: This paper considers flexible guideway system for maglev train with time-delayed velocity feedback control gain. Taking the time delay as a bifurcation parameter, the parameter condition that Hopf bifurcation occurs is deduced. The stability and direction of the bifurcation periodic solution are analyzed by applying the normal form theory and the center manifold theorem. Numerical simulation and experimental result demonstrate the complex behavior of maglev system and support the theoretical analysis. [Copyright &y& Elsevier]

Abstract: In this paper, a modified Leslie-Gower predator–prey system with time delays is investigated, where the time delays are regarded as bifurcation parameters. By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium is considered. Moreover, we show that Hopf bifurcations occur when time delay crosses some critical values. By deriving the equation describing the flow on the center manifold, we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. In addition, we also try on the global existence of periodic solutions by using the global Hopf bifurcation result of Wu [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799–4838.] for functional differential equations. Numerical simulations are carried out to illustrate the theoretical results and they show that the time delays in the system under consideration can destroy the stability of the system. [Copyright &y& Elsevier]

• Propose a general single species animal movement model with distributed delays in diffusion and reaction. • Investigate the impact of spatiotemporal delays on stability of memory-based model. • Obtain conditions for the occurrence of Hopf and steady state bifurcations. • Explore dynamics due to the interaction of Hopf and steady state bifurcations. In this paper, we investigate the spatiotemporal dynamics of a single-species model with spatiotemporal delays characterizing spatial memory and maturation. Through stability and bifurcation analysis, we find that the spatial memory-based diffusion coefficient, the spatiotemporal diffusive delay and spatiotemporal reaction delay have important effects on the dynamics of the model and their combined impact can cause the destabilization of the positive constant steady state and give rise to steady state and Hopf bifurcations. Taking the coefficient of spatial memory diffusion as the bifurcation parameter, the critical values of steady state and Hopf bifurcations are rigorously determined. Furthermore, we apply the theoretical results to a modified diffusive logistic model with predation and obtain spatially inhomogeneous steady states and spatially homogeneous and inhomogeneous periodic solutions via numerical simulations. [ABSTRACT FROM AUTHOR]

• The Lotka-Volterra competition model is reconstructed with maturation delays. • Under the identical delays, the steady state loses stability if the delays are larger than some critical value. • The stability switching curve is analytically constructed and numerically verified in the case of dint delays. • The unstable system can generate various dynamics including Hopf cycles, multistability and chaos. This paper reexamines the Lotka-Volterra competition model with two delays. The steady state is shown to be locally asymptotically stable without delay. If the two delays are identical, then the model becomes a one-delay system. The critical value of the delay is determined when stability might be lost. If the delays are different, then the stability switching curves are analytically defined and numerically verified. It is demonstrated that the unstable two-delay system may exhibit periodic behavior, multistability, quasi period-doubling cascade and even complicated dynamics depending on model parameters. [ABSTRACT FROM AUTHOR]

This paper investigates a three-dimensional mixing competitive system with one exponential growth rate and two rational growth rates, whose nullclines are linearly determined. In total, 33 stable nullcline classes exist. Hopf bifurcations are studied in classes 26-31. We provide examples to prove the existence of at least two limit cycles in each of the classes 27-31. [ABSTRACT FROM AUTHOR]

• Propose for the first time a network organized semiarid vegetation model. • Study the stability and direction of Hopf bifurcation at the coexistence equilibrium. • Compare different variants of modeling vegetation and water. • Find that fragmented habitats more likely induce variation in plant density. Spatiotemporal dynamics of vegetation models are traditionally investigated on a spatially continuous domain. The increasingly fragmented agricultural landscape necessitates network-organized models. In this paper, we develop a semiarid vegetation model to describe the spatiotemporal dynamics between plant and water on a network accounting for fragmented habitats which are connected by dispersal of seeds. Time delay is introduced to account for time lag in water uptake. By linear stability analysis we show that the coexistence equilibrium is asymptotically stable in the absence of time delay, but loses its stability via Hopf bifurcation when time delay is beyond a threshold. Applying the center manifold theory, we derive the explicit formulas that determine the stability and direction of the Hopf bifurcation. Numerical simulations demonstrate the emergence of spatial patterns on a network. Comparing our network-organized model to other model variants, we find that increasing landscape fragmentation is more likely to generate the variation of plant density among different habitats. [ABSTRACT FROM AUTHOR]