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This paper is concerned with the sustained periodic oscillation phenomenon in a three-species delayed predation system with Holling type II functional response and age structure in top predator. The top predator fertility function is regarded as a piecewise continuous smooth function with regard to their maturation period τ 2 . The complicated dynamic behavior is proved by integrated semigroup theory and Hopf bifurcation theorem for semilinear equations with non-dense domain. Through qualitative analysis and bifurcation study of the system, we yield that this system has a nontrivial periodic solution that bifurcates from the positive equilibrium age distribution when bifurcation parameter τ passes through some critical values. Numerical simulations are also provided to illustrate theoretically analytical results. [ABSTRACT FROM AUTHOR]

The paper discusses the nutrient-plankton system with effect of time delay in nutrient recycling and toxin-determined function response (TDFR). The designed model system explores the delay-induced system dynamics. We present the local stability analysis of interior equilibrium points in absence as well as in presence of time delay. Further, the direction of Hopf bifurcation is obtained. We perform the numerical computation and observe that time delay in nutrient recycling can generate the periodic solution in a stable nutrient-plankton system. Some other essential parameters, such as input concentration of nutrients and natural removal rate of nutrients, also regulate the dynamical system. The system shows Hopf and double-Hopf bifurcation in the presence of time delay. Our study shows that the delay in the nutrient recycling causes instability transition phenomenon. The delay-induced nutrient recycling and different input concentrations of nutrients can regulate the estuarine system. Finally, the stability switching is observed for delayed system. [ABSTRACT FROM AUTHOR]

Purpose: This paper mainly studies the double Hopf bifurcation for coupled van der Pol–Rayleigh system with time delay. It aims to see how time delay and coupling factors affect the system, then avoid unstable or harmful vibrations and look for steady state in engineering. Methods: This paper mainly uses the normal form and center manifold theory and the Runge–Kutta method. First, we analyze the stability switching of the coupled van der Pol–Rayleigh system with time delay and the dynamical behavior caused by the double Hopf bifurcation. Secondly, we derive the normal form of double Hopf bifurcation by the normal form and center manifold theory. Then the dynamical phenomena of the system are investigated. Finally, we perform numerical simulations to verify the theoretical analysis by the Runge–Kutta method. Results: The stability switching region of the system with respect to the delay is obtained. And the parameter regions of amplitude death, stable periodic motion and quasi-periodic motion are obtained. And as parameter values change, the steady state of the periodic solution follows. Conclusions: In the field of engineering, it is more practical to consider systems with time delay. So we consider system with time delay, obtain the stability switching region about the time delay and discuss the dynamics of each bifurcation region. When doing some system design, it is necessary to avoid the occurrence of bifurcation behavior, because it will lead to changes in system stability. We can adjust the system parameters to avoid bifurcation according to stability region. Or some engineering problem requires the avoidance of harmful vibrations and the pursuit of steady state. With the change of time delay and coupling factors, we can find the amplitude death region and the region of stable periodic motion and quasi-periodic motion, thus avoiding harmful vibration in practical engineering by adjusting the parameters. [ABSTRACT FROM AUTHOR]

In this paper, we consider the long-term sustainability of the northeast Korean pine. We propose a class of natural Korean pine population system with time delay and diffusion term. First, by analyzing the roots distribution of the characteristic equation, we study the stability of the model system with diffusion terms and prove the occurrence of Hopf bifurcation. Second, we introduce lactation time delay into a population model with a diffusion term, based on stability theory of ordinary differential equation, norm form methods and center manifold theorem, the stability of bifurcating periodic solutions and the relevant formula for the direction of Hopf bifurcation are given. Finally, some numerical simulations are given. [ABSTRACT FROM AUTHOR]

In this paper, the stability and bifurcation of a two–dimensional p53 gene regulatory network without and with time delay are taken into account by rigorous theoretical analyses and numerical simulations. In the absence of time delay, the existence and local stability of the positive equilibrium are considered through the Descartes' rule of signs, the determinant and trace of the Jacobian matrix, respectively. Then, the conditions for the occurrence of codimension–1 saddle–node and Hopf bifurcation are obtained with the help of Sotomayor's theorem and the Hopf bifurcation theorem, respectively, and the stability of the limit cycle induced by hopf bifurcation is analyzed through the calculation of the first Lyapunov number. Furthermore, codimension-2 Bogdanov–Takens bifurcation is investigated by calculating a universal unfolding near the cusp. In the presence of time delay, we prove that time delay can destabilize a stable equilibrium. All theoretical analyses are supported by numerical simulations. These results will expand our understanding of the complex dynamics of p53 and provide several potential biological applications. [ABSTRACT FROM AUTHOR]

This paper studies an Internet congestion control system with two time delays, which are accessed by a single resource and considers both discrete and distributed delays of the system. By designing a new hybrid controller containing negative feedback control and time delay feedback control to control the system, and taking discrete time delay variables as bifurcation parameters, the local stability and Hopf bifurcation of the system are studied. The results show that the Hopf bifurcation can be effectively delayed or avoided by adjusting the value of the feedback control parameter β . The global asymptotically stable dynamic characteristics of the system are ideal, which has important functional significance for optimizing network congestion control. Finally, a large number of simulation examples verify the correctness of the conclusions. [ABSTRACT FROM AUTHOR]

This paper will explore a fractional-order hepatitis B virus (HBV) infection model that takes into account cell-cell and virus-cell transmission and multiple treatment modalities. The desired control strategy is realized by means of a fractional order PD controller. Firstly, we calculated the basic regeneration number and equilibrium point of the HBV model. Afterwards, for the uncontrolled HBV virus system, the adequate conditions for both stability and Hopf bifurcation are systematically investigated via choosing the appropriate time delay as a parameter for bifurcation. Subsequently, under fractional order PD controller, the effect of a proposed controller on system stability and Hopf bifurcation is studied. The desired dynamic characteristics can be obtained afterwards. Finally, numerical simulations show that all three treatments significantly reduce R 0. The onset of oscillations can be delayed by decreasing the order of the fractional order. There are three control pathways for fractional order PD control, and the generation of bifurcation can also be delayed by changing the gain parameter. Using the above methods, the diffusion of HBV virus particles in the body can be effectively controlled. The conclusions drawn in this paper are extremely novel and have potential theoretical value for the future treatment of hepatitis B illness. [ABSTRACT FROM AUTHOR]

Time delay profoundly impacts reaction-diffusion systems, which has been considered in many areas, especially infectious diseases, neurodynamics, and chemistry. This paper aims to investigate the pattern dynamics of the reaction-diffusion model with time delay. We obtain the condition in which the system induced the Hopf bifurcation and Turing instability as the parameter of the diffusion term and time delay changed. Meanwhile, the amplitude equation of the reaction-diffusion system with time delay is also derived based on the Friedholm solvability condition and the multi-scale analysis method near the critical point of phase transition. We discussed the stability of the amplitude equation. Theoretical results demonstrate that the delay can induce rich pattern dynamics in the Brusselator reaction-diffusion system, such as strip and hexagonal patterns. It is evident that time delay causes steady-state changes in the spatial pattern under certain conditions but does not cause changes in pattern selection under certain conditions. However, diffusion and delayed feedback affect pattern formation and pattern selection. This paper provides a feasible method to study reaction-diffusion systems with time delay and the development of the amplitude equation. The numerical simulation well verifies and supports the theoretical results. [ABSTRACT FROM AUTHOR]

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]

In this paper, based on the earlier research, a new fractional-order chaotic Genesio-Tesi model is established. The chaotic phenomenon of the fractional-order chaotic Genesio-Tesi model is controlled by designing two suitable time-delayed feedback controllers. With the aid of Laplace transform, we obtain the characteristic equation of the controlled chaotic Genesio-Tesi model. Then by regarding the time delay as the bifurcation parameter and analyzing the characteristic equation, some new sufficient criteria to guarantee the stability and the existence of Hopf bifurcation for the controlled fractional-order chaotic Genesio-Tesi model are derived. The research shows that when time delay remains in some interval, the equilibrium point of the controlled chaotic Genesio-Tesi model is stable and a Hopf bifurcation will happen when the time delay crosses a critical value. The effect of the time delay on the stability and the existence of Hopf bifurcation for the controlled fractional-order chaotic Genesio-Tesi model is shown. At last, computer simulations check the rationalization of the obtained theoretical prediction. The derived key results in this paper play an important role in controlling the chaotic behavior of many other differential chaotic systems. [ABSTRACT FROM AUTHOR]

In this paper, we study a classic mutualistic relationship between the leaf cutter ants and their fungus garden, establishing a time delay mutualistic system with stage structure. We investigate the stability and Hopf bifurcation by analyzing the distribution of the roots of the associated characteristic equation. By means of the center manifold theory and normal form method, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some numerical simulations are carried out for illustrating the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a diffusive predator-prey system with fear response delay, where a benefit from the ani-predation response in addition to the cost is taken into account. As a step toward understanding the underlying mechanism of the spatiotemporal pattern formations by rigorous mathematical analysis, we are interested in how the joint effects of time delay, the fear effect strength, the carrying capacity, as well as the diffusion rates could affect the complex spatiotemporal pattern formations of the system. In particular, Turing instability of the Hopf bifurcating periodic solutions is investigated in details. To that end, a general formula is derived to determine Turing instability of the Hopf bifurcating periodic solutions for general 2-component delayed-diffusive system. This extends our earlier general results for non-delayed diffusive system to its delayed-diffusive counterparts. Our general formula tends to be new and can be applied to a variety of 2-component delayed-diffusive systems. [ABSTRACT FROM AUTHOR]

Ecotoxicological models play a vital role in understanding the inuence of toxicants on population dynamics in contaminated aquatic ecosystems. Traditional differential equation models describing interactions between populations and toxicants typically assume instantaneous population growth, overlooking potential time delays associated with reproductive and maturation processes. In this paper, we introduce two models with time delays to investigate the interaction between a population and a toxicant, where the population growth is governed by a delayed logistic equation. We mainly focus on the stability analysis of the steady states of the models. Our fndings indicate that high toxicant concentrations result in population extinction, whereas moderate toxicant levels can potentially induce bistability, where the population's fate, whether persistence or extinction, depends on the initial densities of the population and toxicant. Furthermore, both our theoretical analysis and numerical simulations demonstrate that the time delay can lead to the destabilization of the coexistence steady states and the appearance of periodic solutions through Hopf bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, the asymptotical behaviour of an age-structured Human Immunodeficiency Virus infection model with general non-linear infection function and logistic proliferation term is studied. Based on the existence of the equilibria and theory of operator semigroups, linearized stability/instability of the disease-free and endemic equilibria is investigated through the distribution of eigenvalues of the linear operator. Then persistence of the solution semi-flow of the considered system is studied by showing the existence of a global attractor and the obtained result shows that the solution semi-flow is persistent as long as the basic reproduction number |$R_{0}>1$|⁠. Moreover, the Hopf bifurcations problem around the endemic equilibrium is also considered for the situation with a specific infection function. Since the system has two different delays, four cases are discussed to investigate the influence of the time delays on the dynamics of system around the endemic equilibrium including stability and Hopf bifurcations. At last, some numerical examples with concrete parameters are provided to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This paper proposes a partial differential equation model based on the model introduced by V. A. Kuznetsov and M. A. Taylor, which explains the dynamics of a tumor–immune interaction system, where the immune reactions are described by a Michaelis–Menten function. In this work, time delay and diffusion process are considered in order to make the studied model closer to reality. Firstly, we analyze the local stability of equilibria and the existence of Hopf bifurcation by using the delay as a bifurcation parameter. Secondly, we use the normal form theory and the center manifold reduction to determine the normal form of Hopf bifurcation for the studied model. Finally, some numerical simulations are provided to illustrate the analytic results. We show how diffusion has a significant effect on the dynamics of the delayed interaction tumor–immune system. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the impact of the varying population sizes on the dynamic behavior of the age-structured epidemic model. A age-structured SIRS epidemic model with the varying population sizes is established and investigated to take into account time delay. The non-negativity of the solution, the existence and stability of the steady states, and the existence of the Hopf bifurcation are discussed. The numerical simulations show that the varying population sizes can cause the age-structured SIRS model to produce multiple stability switches. [ABSTRACT FROM AUTHOR]

In this paper, we propose a cancer virotherapy model with virus lytic cycle and diffusion term. Spatiotemporal dynamic properties of the cancer virotherapy system are studied. First, by analyzing the roots distribution of the characteristic equation and transcendental equation, the conditions for the local stability of the constant equilibria of system are given. Second, we select delay as the bifurcation parameter, the existence conditions of Hopf bifurcation are given. By using the center manifold theory and normal form method of partial functional differential equation, the detailed formulae for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are given. Finally, some numerical simulations are given. [ABSTRACT FROM AUTHOR]

Due to the finite speed of signal transmission, time delay is a common phenomenon in neuronal systems. The spatiotemporal dynamics of the FitzHugh–Rinzel model with time delay and diffusion in a random network are investigated in this paper. The conditions for Turing instability and Hopf bifurcation are obtained by linear stability analysis. It is found that the stability of the system changes with the time delay. Then the critical time delay for the state transition of the system is derived. Moreover, it is shown that Turing pattern is related to the network diffusion and connection probability. The increase of the diffusion coefficient will change the spatiotemporal pattern of the system. In addition, the system will achieve firing synchronization as the connection probability increases. Finally, numerical simulation verifies the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a toxin producing plankton-fish model with three-dimensional patch and time delay is proposed. We assume that a small part of phytoplankton gathers to form a patch to release toxins for self-defense and introduce a discrete-time delay in fish populations due to gestation. The sufficient conditions for the stability of the solution and the existence of Hopf bifurcation and transcritical bifurcation are obtained. Furthermore, the conditions of stability and direction of Hopf bifurcation in the time delay system are derived by using normal form theory and center manifold theorem. Finally, numerical simulation demonstrates the dynamical outcome such as periodic and chaotic solutions of the system, which validate our analytical findings. [ABSTRACT FROM AUTHOR]

Mixed-mode oscillations (MMOs) consisting of spikes alternating with a series of subthreshold oscillations have been observed in various neurons related to some physiological functions. In the present paper, inhibitory-autapse-induced MMOs are simulated by using the Hodgkin-Huxley neuron model, and the underlying dynamical mechanism is identified to be related to dynamics of unstable behaviors near subcritical Hopf bifurcation. For the monostable spiking, a delayed inhibitory current pulse activated by a spike can suppress the phase trajectory corresponding to depolarization phase of the next spike to the unstable focus nearby or the neighborhood outside of unstable limit cycle, respectively. Then the trajectory rotates multiple cycles away and converges to the stable limit cycle, resulting in an evolution process of membrane potential from small-amplitude subthreshold oscillations to a large-amplitude spike, i.e., MMOs. For the spiking coexisting with the resting state, inhibitory autapse induces MMOs and resting state from the spiking. The difference in the MMOs from those induced by the excitatory autapse is identified. The result presents the underlying nonlinear mechanisms of inhibitory autapse to suppress the neuronal firing and reveals the potential role to control the neuronal firing patterns near subcritical Hopf bifurcation. [ABSTRACT FROM AUTHOR]

Time delay is very common and inevitable in engineering. In this paper, a time delay duffing system with combining parametric and external excitation has been analyzed. First, the system with only parametric excitation shows pitchfork bifurcation. When the slow variable experiences the pitchfork bifurcation, the system will not jump immediately but moves along the unstable bifurcation for a while, and then shows the bursting oscillation. Time delay makes the system randomly choose to go up or down into the bursting oscillation, which may be due to the influence of different regions of attraction. Second, the system with parametric and external excitation shows Fold/Fold bursting. We can construct different paths to make the system bursting including linear, vertical parabolic and horizontal parabolic. At the same time, we find that the phase difference between parametric excitation and external excitation has a vital effect on the realization of bursting oscillation. Finally, we analyze that the positive time delay will not occur Hopf bifurcation, therefore the dynamical behavior of systems with time delay is similar to that without time delay. [ABSTRACT FROM AUTHOR]

• Frequency response of a strongly nonlinear time delay system is obtained. • Optimal time delays of control system are given. • First statement about the feedback control relationship between displacement and velocity. • Appropriate time delay parameters can make the system softening or hardening. In this paper, the strongly nonlinear vibration of a cantilever beam system with concentrated mass at an intermediate position controlled by displacement and velocity time delay is investigated. The nonlinear governing equation is studied using the homotopy analysis method. The effects of the time delay, displacement and velocity feedback gain coefficients, as well as frequency on the amplitude of the system were studied in detail. The results indicate that the velocity feedback control is not necessarily superior to displacement feedback control. Reasonable selection of the displacement and velocity time delay parameters can avoid the Hopf bifurcation, adjust the number, amplitude and stability of periodic solutions, and exhibit the softening behavior at low frequency and hardening spring characteristic at high frequency. The theoretical research in this paper will promote the application of homotopy analysis method in the field of time delay control, and serve as a theoretical reference for the design and optimization of cantilever beam control systems with concentrated mass. [ABSTRACT FROM AUTHOR]

A modified high-temperature superconducting maglev model is studied in this paper, mainly considering the influence of time delay on the dynamic properties of the system. For the original model without time delay, there are periodic equilibrium points. We investigate its stability and Hopf bifurcation and study the bifurcation properties by using the center manifold theorem and the normal form theory. For the delayed model, we mainly study the co-dimension two bifurcations (Bautin and Hopf–Hopf bifurcations) of the system. Specifically, we prove the existence of Bautin bifurcation and calculate the normal form of Hopf–Hopf bifurcation through the bifurcation theory of functional differential equations. Finally, we numerically simulate the abundant dynamic phenomena of the system. The two-parameter bifurcation diagram in the delayed model is given directly. Based on this, some nontrivial phenomena of the system, such as periodic coexistence and multistability, are well presented. Compared with the original ordinary differential equation system, the introduction of time delay makes the system appear chaotic behavior, and with the increase in delay, the variation law between displacement and velocity becomes more complex, which provides further insights into the dynamics of the high-temperature superconducting maglev model. [ABSTRACT FROM AUTHOR]

In this paper, time-delayed sampled-data feedback control technique is used to asymptotically stabilize a class of unstable delayed differential systems. Through the analysis for the distribution change of eigenvalues, an effective interval of the control parameter is obtained for a given sampling period. Here an indirect strategy is taken. Specifically, the system of continuous-time delayed feedback control is studied first by Hopf bifurcation theory. And then, the result and implicit function theorem are used to analyze the system of time-delayed sampled-data feedback control with a sufficiently small sampling period. Considering the practical criterion for the size of sampling period, the upper bound of sampling period is estimated. Finally, an application example, an unstable Mackey–Glass model, is asymptotically stabilized by introducing a blood transfusion item with time-delayed sampled-data feedback control. The blood transfusion speed and blood collection test period are derived from the main results. Some simulations and comparisons show the correctness and advantages of the main 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]

This paper investigates the problem of stabilization and Hopf bifurcation for a fractional order disease spreading model simulated by small-world networks via two types of time delays, that is, the linear and nonlinear time delays. It is worth mentioning that the study of making delay parameter pairs as the bifurcation parameter is unprecedented. Firstly, we build a fractional disease transmission network model based on the Caputo fractional derivative, and fix one type of time delay and use another one as the bifurcation parameter to get criterions of stability and Hopf bifurcation. Then, numerical fitting obtains clear stable region and bifurcation boundary curve, and Hopf bifurcation occurs at the equilibrium when select time delay parameter pairs are in the area through the bifurcation boundary curve. Finally, some numerical examples verify the effectiveness of theories. In addition, simulation examples of regular lattices show that the new model covers the extreme conditions of regula and random networks, and it presents more flexible internal nonlinear interactions than previous models. [ABSTRACT FROM AUTHOR]

In this paper, we establish a delayed semilinear plankton system with habitat complexity effect and Neumann boundary condition. Firstly, by using the eigenvalue method and geometric criterion, the stability of the equilibria and some conditions for determining the existence of Hopf bifurcation are studied. Through analyzing the stability of positive equilibrium, we found that at the positive equilibrium the system may switch finitely many times from stable to unstable, then from unstable to stable, finally becoming unstable, i.e., the time delay induces a "stability switch" phenomenon. Secondly, the properties of Hopf bifurcation are derived by applying the normal form method and center manifold theory, including the bifurcation direction and the stability of bifurcating periodic solutions. Finally, some numerical simulations are given to illustrate the theoretical results, and a biological explanation is given. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study the oscillatory behaviors of gene expression networks in quorum-sensing system with time delay. The stability of the unique positive equilibrium and the existence of Hopf bifurcation are investigated by choosing the time delay as the bifurcation parameter and by applying the bifurcation theory. The explicit criteria determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are developed based on the normal form theory and the center manifold theorem. Numerical simulations demonstrate good agreements with the theoretical results. Results of this paper indicate that the time delay plays a crucial role in the regulation of the dynamic behaviors of quorum-sensing system. [ABSTRACT FROM AUTHOR]

Mathematical modeling plays a vital role in the epidemiology of infectious diseases. Policy makers can provide the effective interventions by the relevant results of the epidemic models. In this paper, we build a fractional-order SIRS epidemic model with time delay and logistic growth, and we discuss the dynamical behavior of the model, such as the local stability of the equilibria and the existence of Hopf bifurcation around the endemic equilibrium. We present the numerical simulations to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a nitrogen fixation game system with two time delays under nitrogen limitation is investigated. Firstly, we discuss the existence and local stability of the equilibrium points for the nondelay system. Then the nitrogen fixation delay and strategy-dependent delay are used as bifurcation parameters to analyze the local stability and the Hopf bifurcation. In addition, we obtain the direction of Hopf bifurcation, the change of the period for periodic solution and the stability of periodic solution via center manifold theory and normal form method. Finally, numerical simulation is employed to visualize the theoretical analysis results and we find that the nitrogen fixation strategy is the dominant strategy when the values of the two time delays are large enough. This study promotes the investigation of the effects of two time delays on the dynamics of evolutionary games with environmental feedback, especially on stability and bifurcation. [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]

The phenomenon of coherence resonance (CR) in a delayed noisy Van der Pol system with supercritical Hopf bifurcation, which is influenced by a recycled noise, is numerically studied. Different from the traditional CR theory, in this paper, the characteristics of CR is affected by the time delay in the input noise. Namely, the CR is weakened or enhanced by the time delay feedback. Moreover, we find that several characteristics of this particular system vary periodically and its period has some certain relation with the natural frequency. By using the results given by the paper, we can control the noise-induced motion by modulating the time delay in noise. [ABSTRACT FROM AUTHOR]

In this paper, we study time-delayed reaction–diffusion systems with advection subject to Lotka–Volterra competition dynamics over one-dimensional domains. These systems model the population dynamics of two groups of competing species, with one dispersing randomly and the other a combination of random and biased dispersal (to avoid competition). We show that time-delay(s) in the interspecific competition mechanism can induce instability of the homogeneous equilibrium to the reaction–advection–diffusion systems, and further promote the appearance of time-oscillating spatially inhomogeneous distributions of the species. Our results indicate that these time-delayed systems (both single and double time-delays) can be used to model the well-observed time-periodic distributions of interacting species in natural fields, compared to the systems without time-delay(s). [ABSTRACT FROM AUTHOR]

In this paper, we focus on investigating the effects of time delay on burst synchronization transitions of a neuronal network which is locally modeled by Hindmarsh–Rose neurons. Here, neurons inside the neuronal network are connected through electrical synapses or chemical synapses. With the numerical results, it is revealed that burst synchronization transitions of both electrically and chemically coupled neuronal networks could be induced by time delay just when the coupling strength is large enough. Meanwhile, it is found that, in electrically and excitatory chemically coupled neuronal networks, burst synchronization transitions are observed through change of spiking number per burst when coupling strength is large enough; while in inhibitory chemically coupled neuronal network, burst synchronization transitions are observed for large enough coupling strength through changing fold-Hopf bursting activity to fold-homoclinic bursting activity and vice versa. Namely, two types of burst synchronization transitions are observed. One type of burst synchronization transitions occurs through change of spiking numbers per burst and the other type of burst synchronization transition occurs through change of bursting types. [ABSTRACT FROM AUTHOR]

A delayed ecoepidemic model with ratio-dependent transmission rate has been proposed in this paper. Effects of the time delay due to the gestation of the predator are the main focus of our work. Sufficient conditions for local stability and existence of a Hopf bifurcation of the model are derived by regarding the time delay as the bifurcation parameter. Furthermore, properties of the Hopf bifurcation are investigated by using the normal form theory and the center manifold theorem. Finally, numerical simulations are carried out in order to validate our obtained theoretical results. [ABSTRACT FROM AUTHOR]

This paper basically concentrates on the bifurcation control of a fractional delayed predator–prey system via feedback control methodology. Firstly, time delay is regarded as a bifurcation parameter, and the delay-induced bifurcation conditions for such uncontrolled predator–prey system are established. Secondly, state feedback controllers are implemented to suitably control the Hopf bifurcation for the proposed system. Surprisingly, it is discovered that bifurcation control cost can be highly cut down by a single controller as compared with three. Particularly, the effects of control for the first feedback gain surpass the other two. The desirable dynamics can be acquired by cautiously selecting advisable feedback gains. Finally, numerical examples exhibit the theoretical analysis which coincides with the schemed methods. [ABSTRACT FROM AUTHOR]

In this paper, a dynamic model of piecewise nonlinear system with fractional-order time delay is simplified. The amplitude frequency response equation of the dynamic model of piecewise nonlinear system with fractional-order time delay under periodic excitation is obtained by using the average method. It is found that the amplitude of the system changes when the external excitation frequency changes. At the same time, the amplitude frequency response characteristics of the system under different time delay parameters, different fractional-order parameters, and coefficient are studied. By analyzing the amplitude frequency response characteristics, the influence of time delay and fractional-order parameters on the stability of the system is analyzed in this paper, and the bifurcation equations of the system are studied by using the theory of continuity. The transition sets under different piecewise states and the constrained bifurcation behaviors under the corresponding unfolding parameters are obtained. The variation of the bifurcation topology of the system with the change of system parameters is given. [ABSTRACT FROM AUTHOR]

This paper deals with reduction of oscillations of a continuous spinning shaft using proportional-derivative time delay controller. The stable and unstable zones of the system are obtained using the Routh-Hurwitz criteria. For small values of proportional coefficients, the Hopf bifurcation occurs in the high values of time delays while in the higher values of proportional coefficients instability occurs in the lower values. Also, in small values of derivative coefficients the Hopf bifurcation happens in the lower values of time delays, while in the higher values of derivative coefficients mentioned bifurcation occurs in the higher values. [ABSTRACT FROM AUTHOR]

In this paper, we studied a delayed host-generalist parasitoid model with Holling II functional response and diffusion term. The Turing instability and local stability are studied. The existence of Hopf bifurcation is investigated, and some explicit formulas for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived by the theory of center manifold and normal form method. Some numerical simulations are carried out. [ABSTRACT FROM AUTHOR]

Pattern formations in reaction–diffusion (RD) systems with time delay constitute a vital class of dynamical mechanisms extensively investigated for biological and chemical processes, where Hopf bifurcation usually occurs. Recent studies show that pattern formations differ significantly between RD systems with large-time and small-time delay. Therefore, in this paper, we aim to explore the exact role of the time delay in RD systems based on complex networks, which would affect the form of patterns. Depicting networked dynamics of the predator–prey system by a set of RD equations, it is found that boundaries of Hopf bifurcation are decided by diffusion coefficients, as well as the Eigen-spectra of networks. We also obtain mathematical expressions of the boundaries in both large-time and small-time delay cases. Through extensive simulations, it is unveiled that the connectivity structures of networks hardly have impact on the trend of evolutionary processes. Compared to large-time delay cases, the oscillation cycle of average prey density becomes shorter red with small-time delay, and the oscillation amplitude also decreases. We finally reveal the evolution process of the prey density and discover the thick-tailed phenomenon in large-time delay cases. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dynamics of delayed Gierer--Meinhardt system, which is used as a classic example to explain the mechanism of pattern formation. The conditions for the occurrence of Turing, Hopf and Turing--Hopf bifurcation are established by analyzing the characteristic equation. For Turing--Hopf bifurcation, we derive the truncated third-order normal form based on the work of Jiang et al. [11], which is topologically equivalent to the original equation, and theoretically reveal system exhibits abundant spatial, temporal and spatiotemporal patterns, such as semistable spatially inhomogeneous periodic solutions, as well as tristable patterns of a pair of spatially inhomogeneous steady states and a spatially homogeneous periodic solution coexisting. Especially, we theoretically explain the phenomenon that time delay inhibits the formation of heterogeneous steady patterns, found by S. Lee, E. Gaffney and N. Monk [The influence of gene expression time delays on Gierer--Meinhardt pattern formation systems, Bull. Math. Biol., 72(8):2139--2160, 2010.] [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model to study the impact of Twitter in controlling infectious disease is proposed. The model includes the dynamics of “tweets” which may enhance awareness of the disease and cause behavioral changes among the public, thus reducing the transmission of the disease. Furthermore, the model is improved by introducing a time delay between the outbreak of disease and the release of Twitter messages. The basic reproduction number and the conditions for the stability of the equilibria are derived. It is shown that the system undergoes Hopf bifurcation when time delay is increased. Finally, numerical simulations are given to verify the analytical results. [ABSTRACT FROM AUTHOR]

This paper investigates Turing instability and Hopf bifurcation for a diffusive plankton system with time delay and cell size. To determine the effects of diffusion and cell size on the dynamics of the system, we first study the system without time delay, where the conditions of stability of coexisting equilibrium and Turing instability are obtained through Routh-Hurwitz criterion. Then we give the existence of Hopf bifurcation using time delay as bifurcation parameter by analyzing the distribution of eigenvalues, and derive the property of Hopf bifurcation by applying the centre manifold theory. Finally, numerical simulation shows that different cell size increases the variety of dynamics for diffusive plankton system. [ABSTRACT FROM AUTHOR]

In this paper, the problems of stability and Hopf bifurcation in a class of fractional-order complex-valued single neuron model with time delay are addressed. With the help of the stability theory of fractional-order differential equations and Laplace transforms, several new sufficient conditions, which ensure the stability of the system are derived. Taking the time delay as the bifurcation parameter, Hopf bifurcation is investigated and the critical value of the time delay for the occurrence of Hopf bifurcation is determined. Finally, two representative numerical examples are given to show the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper deals with a three-species ecological system with time delay and harvesting. Sufficient conditions guaranteeing the local stability and the occurrence of Hopf bifurcation for the system are obtained. Further, the properties of Hopf bifurcation are investigated using the center manifold theorem and normal form theory. Computer simulations are carried out to illustrate the theoretical predictions. Finally, biological meaning and a conclusion are presented. [ABSTRACT FROM AUTHOR]

This paper studies the containment control issue of multi-agent systems with time delay. By adopting continuous-time and sampled-data-based control protocols, we establish some necessary and sufficient containment control criteria for multi-agent systems with time delay. For the continuous-time case, our results show that both topology structure and time delay play an important role in the containment control of multi-agent systems, no matter whether the leaders are stationary (first-order system) or dynamic with constant velocity and time-varying position (second-order system). Furthermore, the maximum value of time delay to guarantee the achievement of containment control is given by the Hopf bifurcation theory. Moreover, we extend the results to the sampled-data case. Finally, numerical simulations are given to illustrate our main results. [ABSTRACT FROM PUBLISHER]

Recently Dalgaard and Strulik have proposed (in Resour. Energy Econ. 33:782–797, 2011 ) an energy model of capital accumulation based on the mathematical framework developed by Solow-Swan and coupled with Cobb-Douglas production function (Solow in Q. J. Economics 70:65–94, 1956 ; Swan in Econ. Rec. 32(63):334–361, 1956 ). The model is based on a constant rate of population growth assumption. The present paper, according to the analysis performed by Yukalov et al. (Physica D 238:1752–1767, 2009 ), improves the Dalgaard-Strulik model by introducing a logistic-type equation with delayed carrying capacity which alters the asymptotic stability of the relative steady state. Specifically, by choosing the time delay as a bifurcation parameter, it turns out that the steady state loses stability and a Hopf bifurcation occurs when time delay passes through critical values. The results are of great interest in the applied and theoretical economics. [ABSTRACT FROM AUTHOR]

In this paper, a fractional-order delayed SIQR model with nonlinear incidence rate is investigated. Two time delays are incorporated in the model to describe the incubation period and the time caused by the healing cycle. By analyzing the associated characteristic equations, the stability of the endemic equilibrium and the existence of Hopf bifurcation are obtained in three different cases. Besides, the critical values of time delays at which a Hopf bifurcation occurs are obtained, and the influence of the fractional order on the dynamics behavior of the system is also investigated. Numerically, it has been shown that when the endemic equilibrium is locally stable, the convergence rate of the system becomes slower with the increase of the fractional order. Besides, our studies also imply that the decline of the fractional order may convert a oscillatory system into a stable one. Furthermore, we find in all these three cases, the bifurcation values are very sensitive to the change of the fractional order, and they decrease with the increase of the order, which means the Hopf bifurcation gradually occurs in advance. [ABSTRACT FROM AUTHOR]