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The movements of the plankton in the ocean are driven by random diffusion and cognitive judgement with herd-taxis. In this paper, we formulate a phytoplankton–zooplankton model with time delay in the herd-taxis effect diffusion and homogeneous Neumann boundary conditions. The conditions to guarantee the existence of the coexistence equilibrium of the model are given. By analyzing the distribution of the eigenvalues of the characteristic equation, the local asymptotic stability of the coexistence equilibrium is achieved under certain condition. When there is no time delay in the herd-taxis effect, the model can possess the Turing bifurcation when we consider the nonlinear diffusion term, which leads to instability. When taking the time delay into account, the Hopf bifurcation occurs instead as the time delay varies. Furthermore, we investigate the situation without the fact of time, that is the steady-state bifurcation and the stability of bifurcating solution. Finally, the stability of the coexistence equilibrium, the Turing bifurcation and the Hopf bifurcation of the system are modeled by numerical simulation. The simulations shown are coordinated with the theoretical results which we arrive at in the former part of the paper. The results illustrate that the time delay in the herd-taxis effect of the zooplankton influence the dynamics of the plankton system. [ABSTRACT FROM AUTHOR]

This paper explores the stability and bifurcation of a Caputo fractional-order BAM neural network with time delay and inertia terms. Subsequently, the limitation in bifurcation characteristics of Caputo fractional-order delayed inertial BAM neural network (CFODIBAMNN) is surpassed. By analyzing the stability of the system without time delay, the direct method is applied that involves solving the eigenvalues to determine its stability. Ulteriorly, analyzing the system in the presence of time delays and choosing the time delay as the bifurcation parameter to identify the Hopf bifurcation properties of the system. Eventually, during the verifications, the critical values of bifurcation are accurately calculated, the fluctuation and phase diagrams for the ranges of different fractional order are simulated and the impact of fractional orders on system stability is analyzed. [ABSTRACT FROM AUTHOR]

The benefits of self-excited motion include the capacity to draw energy directly from the external environment, as well as autonomy and portability, and the creation of additional self-oscillating systems holds the promise of finding applications in various fields, including medical devices, autonomous robotics, energy harvesting, sensors, and more. Inspired by the magnetic perpetual motion, this paper introduces a conceptual perpetual motion machine utilizing liquid crystal elastomer (LCE) and steady illumination. The system consists of a turntable and multiple motion ducts, each containing a mass block. A nonlinear dynamics model for the LCE-based perpetual motion machine operating under steady illumination is established, considering the dynamic LCE model. Theoretical analysis and numerical simulations, employing the classical fourth-order Runge-Kutta method, are employed to examine the system's dynamic behaviors. The system exhibits a supercritical Hopf bifurcation between static and self-rotation patterns. Notably, the external environment's energy input counteracts damping dissipation, preventing the system from halting due to damping and other factors. Theoretical condition for Hopf bifurcation in the perpetual motion machine's self-rotation is derived and validated via numerical computations. Extensive quantitative investigations are conducted to determine the influence of system parameters on self-rotation frequency. The results illustrate that through parameter adjustments, the motion pattern and frequency of the system can be controlled effectively. The proposed system offers continuous rotation in a zero-energy mode, presenting remarkable benefits, including energy efficiency, reduced noise, decreased wear, and robust stability. These attributes position it for significant future applications in power generation, energy harvesting, sensors, and various other fields. [Display omitted] • A light-powered liquid crystal elastomer-based perpetual motion machine was conceptualized. • The perpetual motion machine has two typical motion patterns: static and self-rotation pattern. • Theoretical Hopf bifurcation condition was derived and validated via numerical computations. • The self-rotation frequency can be designed by several parameters. [ABSTRACT FROM AUTHOR]

To study the current trend in the society of online gaming addiction. A two delayed social networking game addiction model with categorized contagion rate is presented in this paper. The influence of the two delays and sufficient prerequisites for the reality of Hopf bifurcation are explored by means of involving the feasible combos of the double delays as a bifurcation limitation. Furthermore, specific formulae for deciding direction of the Hopf bifurcation and permanence of the bifurcating periodic options are obtained by exhausting normal form theory and centre manifold theory. To end with, some numerical replications are conceded out with the assist of MAT LAB software program in directive to interpretation for the approve of theoretical results. • A two delayed social networking game addiction model. • Sufficient prerequisites for the reality of Hopf bifurcation are explored. • Specific formulae for deciding direction of the Hopf bifurcation and permanence of the bifurcating periodic options are obtained. • Numerical replications are conceded. [ABSTRACT FROM AUTHOR]

In the classical evolutionary game theory, mutation is usually considered as a constant, however strategy mutation is affected by strategies in the real game process. Therefore, the main purpose of this paper is to study the effects of mutation feedback and time delays on strategy dynamics, where mutation is a linear feedback related to strategy. First, we construct a co-evolutionary game model with two time delays induced by mutation feedback and analyze the existence and stability of the equilibria of the non-time delay system. The conditions for the coexistence of the two strategies and the mutation rate are obtained. Second, Sotomayor's theorem is used to explore the transcritical bifurcation of the system. Then, the existence of Hopf bifurcation is investigated by using feedback delay and payoff delay as bifurcation parameters in the time-delay system. Furthermore, we discuss the direction of the Hopf bifurcation, stability and periodic change of the periodic solution in detail. Finally, a series of numerical simulations are used to describe the theoretical analysis. The main results are as follows: (i) Mutation causes negative feedback to cooperative strategy. (ii) As the time delay increases, the stable equilibrium point becomes unstable, and which branches a stable limit cycle. When the time delay continues to increase sufficiently, the stable limit cycle becomes unstable and produces irregular oscillation and chaos. (iii) When the two time delays are large enough, the coexistence of the two strategies becomes that defective strategy is dominant, and the mutation rate also reaches the maximum. • A co-evolutionary game model with time delays and mutation feedback is proposed. • Mutation is not a constant but is affected by strategy. • Mutation causes negative feedback to cooperative strategy. • The time delay is large enough, strategies produce irregular oscillations and chaos. • Two delays are large enough, strategies' coexistence becomes the dominant defect. [ABSTRACT FROM AUTHOR]

Synthetic drug addiction represents an escalating worldwide concern. In this paper, we introduce a comprehensive synthetic drug addiction model that considers both psychosocial and environmental factors. Additionally, our model encompasses aspects related to relapse and the drug supply chain. Through a dynamic analysis of this model, we uncover a multitude of intricate dynamical phenomena, including saddle–node bifurcation, Hopf bifurcation. Meanwhile, we have also obtained an elliptic-type nilpotent singularity. Furthermore, we provide bifurcation diagrams and corresponding phase diagrams to offer epidemiological insights into these intricate dynamic phenomena. [ABSTRACT FROM AUTHOR]

• A diffusive plankton model with defensive and offensive effects is considered. • Hopf bifurcations of plankton model without and with diffusion are studied. • Turing instability of diffusive model is analyzed. • The impacts of defense and offense on the coexistence of plankton are discussed. This paper investigates the dynamic behavior of a diffusive plankton model with defensive and offensive effects in two cases. For the single compartment model, we first derive the sufficient conditions for the stability and Hopf bifurcation of coexisting equilibrium, which implies that the changes of defense and offense can cause oscillation of planktonic population. Then the properties of Hopf bifurcation are discussed by center manifold theorem. For the spatially extended model, we obtain the sufficient conditions for Turing instability and Hopf bifurcation. It is observed that spatial patterns put in place, under the interaction of diffusion, defense and offense. Finally, some numerical simulations are carried out to support the analytical results. [ABSTRACT FROM AUTHOR]

• Four mathematical models for derivation of CSA of surge chamber are constructed. • Stability of hydropower station with surge chamber is analyzed. • Formulas for CSA of surge chamber are derived. • Verification and comparison of different CSAs of surge chamber are conducted. • Correctness and rationality of formulas for CSA of surge chamber are explained. This paper aims to study the critical sectional area (CSA) of surge chamber considering the nonlinearity of head loss of diversion tunnel and steady output of turbine. Firstly, three basic equations for hydropower station with surge chamber are established. Four mathematical models for the derivation of CSA of surge chamber are constructed. Then, the stability of hydropower station with surge chamber is analyzed by Hopf bifurcation. Based on the critical stable state of hydropower station, the formulas for the CSA of surge chamber are derived. Finally, the verification and comparison of different CSAs are conducted. The correctness and rationality of obtained formulas are explained. The results indicate that, under load increase operation condition, both the nonlinearity of head loss of diversion tunnel and nonlinearity of steady output of turbine can reduce the value of CSA of surge chamber and are favorable for the stability of hydropower station. Under load decrease operation condition, the rules are opposite. Under both load increase and load decrease operation conditions, the effect of the nonlinearity of head loss of diversion tunnel on CSA of surge chamber is much more significant than that of the nonlinearity of steady output of turbine. The formula for CSA of surge chamber considering both the nonlinearity of head loss of diversion tunnel and nonlinearity of steady output of turbine can be expressed as an amplification coefficient times of the Thoma formula. That formula has a higher precision than Thoma formula. [ABSTRACT FROM AUTHOR]

In this paper, we consider a predator–prey model with fear response delay, gestation delay, fear effect, prey refuge, harvesting and Crowley–Martin type functional response. First, we discuss the model without delays and corroborate the positivity and boundedness of the solution. Then, we give some sufficient conditions for the existence and stability of three equilibriums. For the model with delays, we not only analyze the local stability of the positive equilibrium and the occurrence of Hopf bifurcation, but also obtain the crossing curves to study the stability switches of the positive equilibrium on the delays plane. Furthermore, we calculate the normal form of Hopf bifurcation and hence get the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. At last, we support our findings by numerical simulations. • Considering the cost of fear into both reproduction and death term of prey. • Fear that influence birth and death rate of prey have similar impact on populations. • The stability crossing curve method is creatively applied to the model. • The model can exhibit stability switches on the delays plane. [ABSTRACT FROM AUTHOR]

Although there are many factors that influence population fluctuations, the impact of delay on population dynamics is one of the most significant ones. In this paper, we explore how the Allee effect in combination with a double delay (competition and cooperation) affects predator–prey population dynamics. One can determine the requirements for the equilibrium's asymptotic stability and the presence of Hopf bifurcations by investigating the roots of a characteristic equation. The rigorous mathematical proofs of the stability and direction of the Hopf bifurcating periodic solutions are derived with the help of the normal form theory and the center manifold theorem. Numerical simulation shows that different delay mechanisms can generate different behaviors, such as the competitive delay that produces regular and irregular oscillations, as well as the chaotic, while the cooperative delay results in the stability switch. These theoretical and numerical results are illustrated that depending on the chosen delayed mechanism, delay may not necessarily lead to instability, and also can help for understanding the biological implications corresponding to the interaction dynamics of predator and prey. [ABSTRACT FROM AUTHOR]

• We firstly introduce a fractional-order PD scheme to control the Hopf bifurcation caused embedded in congestion control system. • The condition of stability, the criterion of Hopf bifurcation and the range of periodic oscillations for the congestion control system are derived. • The initiation of the inherent Hopf bifurcation can be effectively delayed or advanced by setting appropriate proportional and differential gain parameters. In this paper, we consider the control of nonlinear dynamic in a congestion control system. A novel fractional-order proportional-derivative (PD) feedback law is firstly designed to control the Hopf bifurcation caused by the system. The proposed P D 1 n controller has the different order with the original congestion system. Meanwhile, the communication delay is elected as the bifurcation parameter to study the stability, bifurcations and periodic oscillations of the controlled congestion system. By the stability mechanism of fractional-order systems, we can obtain the criteria for satisfying the stability and Hopf bifurcation. It has been discovered that the Hopf bifurcation point can be delayed or advanced under the adjustment of appropriate control gain parameters and the order. Therefore, the congestion system becomes controllable and the desirable behaviors can be realized. Finally, numerical simulations are presented to demonstrate the validity of the main results and the efficiency of the control strategy in the designed fractional-order PD controller. [ABSTRACT FROM AUTHOR]

• The dynamics of predator-prey system with two time delays and Monod–Haldane response function are considered. • The local stability of the interior equilibrium is established and the conditions for existence of the Hopf bifurcation are obtained. • The direction of the Hopf bifurcation and the stability of the bifurcation period solutions are investigated. • The Hopf bifurcation of the original system is controlled by using parameter disturbance and state feedback control. • The numerical examples are given to carry out to support the obtained theoretical findings. In this paper, the dynamics of a stage-structured predator-prey system with two time delays and Monod–Haldane response function are considered. By taking the two time delays as the bifurcation parameter, the local stability of the interior equilibrium is established and the conditions for existence of the Hopf bifurcation are obtained. Furthermore, based on the normal form method and center manifold theorem, the direction of the Hopf bifurcation and the stability of the bifurcation period solutions are investigated. In addition, by using parameter disturbance and state feedback control to act on the system, we succeeded in controlling the Hopf bifurcation of the original system. Numerical examples are given to carry out to support the obtained theoretical findings. [ABSTRACT FROM AUTHOR]

Highlights • A diffusive predator-prey system with time-delay and competitive interference is considered. • Time delay induced instability and Hopf bifurcation at coexisting equilibrium are obtained. • The direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are determined. Abstract In this paper, we studied a diffusive predator-prey model with competitive interference and Crowley-Martin type functional response. The conditions for local stability of coexisting equilibrium are given by analyzing the eigenvalue spectrum. By using delay as bifurcation parameter, conditions for occurrence of Hopf bifurcation are also given. The property of bifurcating period solutions is investigated by calculating the normal form. Some numerical simulations are performed to support our theoretical result. Our conclusions show that diffusion and delay are two factors that should be considered in establishing the predator-prey model, since they can induce spatially bifurcating period solutions. [ABSTRACT FROM AUTHOR]

Highlights • We study the impact of herd shape on the the prey and predator equilibrium densities. • In the absence of the time delay we studied the local stability and bifurcation. • The effect of the time delay on the stability of the interior equilibrium where Hopf bifurcation is obtained. Abstract In this paper, a delayed predator-prey model with the presence of a social behavior for the prey population has been investigated. A new functional response is obtained. We studied the effect of the herd shape for the prey population on the prey and predator equilibrium densities. The analysis of the system has been also established where the boundedness, stability, Hopf bifurcation, stability of Hopf bifurcation are obtained. [ABSTRACT FROM AUTHOR]

Abstract The stability and bifurcation behavior of a kind of active magnetic bearing rotor are investigated in this paper. The analysis is carried out both analytically and numerically. It is found that a Hopf bifurcation occurs in the system by using center manifold and normal form. Numerical simulation is conducted to validate the theoretical predictions. More precisely, the dynamic characteristics of this system in two-dimensional parameter space are analyzed, the phase diagrams assist us to identify multi-attractor coexisting that makes the dynamical behaviors of the system become more enrich and complex. These results we represent can be useful in designing and selection of suitable operating parameters. As a result, the system can avoid the undesirable behavior. [ABSTRACT FROM AUTHOR]

Highlights • A network with wireless access links with a delay differential equation (DDE) model and input delay is considered. • Integral sliding mode control is assumed as the nonlinear robust active queue management in the network. • Hopf bifurcation analysis is investigated with the transmission delay considered as the Hopf bifurcation parameter. • The direction and the stability of the bifurcation periodic solutions are determined. • Simulation results are provided to illustrate the system behavior with different parameters. Abstract Transmission control protocol (TCP), which is usually implemented in the transport layer of a communication network, can detect the congestion after its occurrence. But congestion control can be more complicated in networks with wireless access links due to time-varying fading and packet error rate (PER). Thus, it is essential to use robust active queue management (RAQM) to prevent congestion with certain tolerance against serious disturbance of the wireless environment. As the stability of a TCP/RAQM is tremendously related to the transmission delay, this paper focuses on Hopf bifurcation analysis of a network governed by nonlinear delayed differential equations and an integral sliding mode control as a nonlinear RAQM. The transmission delay is assumed as the bifurcation parameter. It is shown that if the delay passes through a critical value, the closed loop system would not remain stable and the Hopf bifurcation occurs. Moreover, based on the normal form theory and the center manifold theorem, the direction and the stability of the bifurcation periodic solutions are determined. Finally, simulation results are given to confirm the analytical achievements and to show the effectiveness of the procedure compared to some recent protocols. [ABSTRACT FROM AUTHOR]

Highlights • Local and global stability analysis. • Numerical technique for fractional-order reaction–diffusion equation. • Modelling with the Caputo and Atangana–Baleanu fractional derivatives. • Numerical simulation results in one and two dimensions for different values of fractional power. Abstract In this paper, we numerically study a nonlinear time-fractional reaction-diffusion equation involving the Caputo and Atangana–Baleanu fractional derivatives of order α ∈ (0, 1). A novel algorithm known as the Laplace Adams–Bashforth method is formulated for the approximation of these derivatives. In the simulation framework, a tri-tropic food chain system is considered in which the classical time-derivatives are replaced with non-integer order derivatives. Mathematical analysis of the main system is examined for both stability and Hopf-bifurcations to occur. Numerical simulation results show the existence of chaotic behaviours and spatiotemporal oscillations as well as the emergence of some Turing patterns (such as, spots and stripes) in two-dimensional space. [ABSTRACT FROM AUTHOR]

Abstract Stability and Hopf bifurcation of a class of delayed complex-valued neural networks are investigated in this paper. First, using proper translations and coordinate transformations, we decompose the activation functions and connection weights into their real and imaginary parts, so as to construct an equivalent real-valued system. Then, the sufficient conditions for Hopf bifurcation and its directions are provided through normal form theory and central manifold theorem. In the end, some numerical simulations are given to illustrate the correctness of the results. [ABSTRACT FROM AUTHOR]

• We considered an opinion model of threshold-type delay differential equations. • We proved that the model with either a transcritical forward or a backward bifurcation, can exhibit Hopf bifurcation. • We showed that the obtained Hopf bifurcation appears only on upper bifurcated branches of the equilibria. • We performed numerical simulations to illustrate and support our theoretical results. In a recent paper [R. Qesmi, Dynamics of an opinion model with threshold-type delay, Chaos, Solitons & Fractals 98 (2020). (https://doi.org/10.1016/j.chaos.2020.110379)], we proposed a mathematical model of threshold-type delay differential equations describing the relationship between two subpopulations with opposite opinions and the opinion spread dynamics. The study there showed the possibility of a transcritical forward and backward bifurcations of positive equilibria. In the present paper, we show that the opinion model undergoes a Hopf bifurcation through which one of the bifurcation branches loses the stability and periodic solutions appear. One of the important consequences of the obtained dynamics is that the consensus of the both opinions could be lost by maintaining the balance between the time taken for an individual to become convinced of the outsider opinion, which need be short, and the number of individuals converted to the local opinion which need be low. Finally, we provide numerical simulations to illustrate and support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a cross-diffusion predator–prey system with Holling type functional response. We study the local stability, Turing instability, spatial pattern formation, Hopf and Turing–Hopf bifurcation of the equilibrium. Numerical simulation with zero-flux boundary conditions discloses that the system under consideration experiences the occurrence of cross-diffusion-driven instability. The dynamical system in Turing space emerges spots, stripe-spot mixtures and labyrinthine patterns, which reveals that the interaction of both self- and cross-diffusions play a significant role on the pattern formation of the present system in a way to enrich the pattern. We obtain the normal form of the Turing–Hopf bifurcation and observe that the system has stably spatially homogeneous periodic solutions, stable constant and nonconstant steady-state solutions, which indicates that the intrinsic growth rate coefficient and the environmental carrying capacity coefficient are two important factors for predator–prey system, and affect the stability of predator–prey system. • A cross-diffusion predator–prey system with habitat complexity and Holling type functional response is considered. • The Turing pattern is studied by the amplitude equation. • The Turing–Hopf bifurcation is investigated by using the method of the normal form and the center manifold theory. [ABSTRACT FROM AUTHOR]

Understanding the connection between spatial patterns in population densities and ecological heterogeneity is significant to the understanding of population dynamics and the governance of species in a given domain. In this paper, the spatiotemporal complexity of prey–predator dynamics with fractional Laplacian derivative and migration is investigated. To provide good guidelines on the choice of parameters, the linear stability analysis of the models was investigated. The fractional Laplacian operator was defined in terms of the left- and right-handed Riemann–Liouville derivative which in turn were approximated by using the fourth-order compact difference scheme, and the resulting system of ODEs was advanced in time using the fourth-order exponential time-differencing Runge–Kutta method. In the simulation experiments, different Turing dynamics such as spots, stripes, and other chaotic patterns are observed. Overall, pattern formation in predator–prey models is useful for understanding the dynamics of ecological systems, predicting the long-term behavior of the system, and studying the impact of environmental factors on the dynamics of the system. • Modification of predator–prey models with different functional responses. • The adoption of viable and reliable numerical techniques. • Local stability analysis. • Pattern formation scenarios in superdiffusive processes. • Numerical experiments in high-dimensional spaces. [ABSTRACT FROM AUTHOR]

This paper is concerned with a modified Leslie–Gower predator–prey diffusive dynamics system with Smith growth subject to homogeneous Neumann boundary condition, in which the functional response is Beddington–DeAngelis (Denote it by B–D) functional response term. Firstly, by applying the theory of stability and the Hopf bifurcation, we discuss the local stability and the existence of the Hopf bifurcation at the positive constant equilibrium solution of the ODE model, which the model undergoes the Hopf bifurcation when bifurcation parameter δ crosses the bifurcation critical value b 0. Moreover, stability of the bifurcation periodic solution is analyzed. Secondly, the Turing instability and the direction of Hopf bifurcation of the corresponding to PDE system are investigated by using Normal form theory and Centre manifold theory. Finally, we study the numerical simulations of this system to illustrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

The dynamic control over time evolution of ordinary differential systems has developed rapidly in recent years, but how to control the spatiotemporal evolution dynamics of partial differential systems is still an open question. Turing pattern is a major spatiotemporal evolution behavior in mussel-algal ecosystems and its control can pull the ecosystem back from the borderline of collapse and make it more stable. However, there has been relatively little research on the optimal control of Turing pattern in the mussel-algal system. In this paper, we first attempt to propose a proportional-derivative (PD) control strategy for the reaction–diffusion mussel–algae model under the influence of time delays. Turing instability driven by diffusion occurs in the diffusive mussel–algae model without control. The introduction of PD control can effectively suppress the occurrence of Turing instability and keep the ecological model stable in both time and space. When the time delay exceeds a critical value, the diffusive mussel–algae model will lose stability and undergo a Hopf bifurcation. PD controller can make Hopf bifurcation occur in advance or in hysteresis by changing the threshold point of Hopf bifurcation. The intrinsic mechanism of Hopf bifurcation is analyzed by means of the formal theory and the central manifold theorem. The numerical simulations demonstrate the efficiency and feasibility of the PD control scheme. [ABSTRACT FROM AUTHOR]

This paper focuses on the dynamic behavior of phytoplankton–zooplankton system with cell size and time delay. Remarkably, the existence of cell size and time delay make the dynamic behavior of the system more close to the real-world situation, essentially different from those in the existing related literature. To analyze the dynamic behavior of the system, the positiveness and boundedness of the solution are first derived. Then, the asymptotic stability of the coexistent equilibrium and Hopf bifurcation are studied by analyzing the associated characteristic equation. Once more, the direction of Hopf bifurcation and the stability of bifurcated periodic solution are determined. Finally, the theoretical results are illustrated by a numerical example. [ABSTRACT FROM AUTHOR]

The present paper deals with an eco-epidemiological prey–predator model with delay. It is assumed that infection floats in predator species only. Both the susceptible and infected predator species are subjected to harvesting at different harvesting rates. Differential predation rates for susceptible and infected predators are considered. It is shown that the time delay can even destabilize the otherwise globally stable non-zero equilibrium state. It is observed that coexistence of all the three species is possible through periodic solutions due to Hopf bifurcation. With the help of normal form theory and central manifold arguments, stability of bifurcating periodic orbits is determined. Numerical simulations have been carried out to justify the theoretical results obtained. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical analysis of a delayed reaction-diffusion virus infection model with logistic growth and chemotaxis for the uninfected target cells and humoral immune impairment is studied. By analyzing corresponding characteristic equations, the local stability of the infection-free equilibrium is established. The stability properties and Turing instability of the antibody-free equilibrium and antibody-present infection equilibrium have been extensively discussed. The existence of Hopf bifurcation with antibody response delay as a bifurcation parameter at the antibody-present infection equilibrium is established. The numerical simulations are carried out in order to illustrate the dynamical behavior of the model. [ABSTRACT FROM AUTHOR]

In this paper, birhythmicity in an enzymatic-substrate reaction described by a fractional-order extended van der Pol equation is investigated. The fractional derivatives are introduced in the system equations in order to model the memory property of the biological system. The residue harmonic balance scheme is used to study the periodic motions of the considered fractional-order van der Pol equations. It is shown that depending on system parameters and the fractional derivative order, the bistability area strongly increased. This fractional oscillator is analytically mapped, onto an ordinary bistable systems with a two stable amplitude. The obtained results clearly show an interesting collapse and revival of birhythmicity with the variation of the fractional derivative order. The amplitude and frequency of the fractional order van der Pol oscillator are derived. The analysis of amplitude equation corroborates with the results obtained by numerical simulations of the fractional-order differential equations describing the system. [ABSTRACT FROM AUTHOR]

This paper considers a fractional order Rosenzweig-MacArthur (R-M) model incorporating a prey refuge. The model is constructed and analyzed in detail. The existence, uniqueness, non-negativity and boundedness of the solutions as well as the local and global asymptotic stability of the equilibrium points are studied. Sufficient conditions for the stability and the occurrence of Hopf bifurcation for the fractional order R-M model are demonstrated. The resolution of the paradox of enrichment is investigated. The impact of fractional order and the prey refuge effects on the stability of the system are also studied both theoretically and by using numerical simulations. [ABSTRACT FROM AUTHOR]

The dynamics of a diffusive predator-prey system with anti-predator behaviour and maturation delay subject to Neumann boundary condition is investigated in this paper. The global stability of boundary equilibrium is studied. For coexisting equilibrium, Turing instability induced by diffusion and Hopf bifurcation induced by time delay are studied. By the theory of normal form and center manifold method, the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived. [ABSTRACT FROM AUTHOR]

In this paper, we study the bifurcation and stability of a ratio-dependent predator-prey model with nonconstant predator harvesting rate. The analysis is carried out both analytically and numerically. We determine stability and dynamical behaviours of the equilibria of this system and characterize codimension 1 and codimension 2 bifurcations of the system analytically. Our bifurcation analysis indicates that the system exhibits numerous types of bifurcation phenomena, including Fold, Hopf, Cusp, and Bogdanov–Takens bifurcations. We use the numerical software MATCONT, to compute curves of equilibria and to compute several bifurcation curves. We especially approximate a family of limit cycles emanating from a Hopf point. Our results generalize and improve some known results and show that the model has more rich dynamics than the ratio-dependent predator-prey model without harvesting rate. [ABSTRACT FROM AUTHOR]

In this paper, a three species predator-prey interaction model among Phytoplankton, Zooplankton and Fish has been developed in the presence of toxicant. It is assumed that Phytoplankton grows logistically and its growth rate is affected by toxin release in the environment through different natural and human activities. It is considered that Zooplankton consumes only Phytoplankton. Fish consumes both Phytoplankton as well as Zooplankton. Here, it is also assume that Phytoplankton releases some toxin in the environment which makes some death on Zooplankton. Also, it is considered that toxicant in the environment are increased constantly for different natural and human behavior. Then, different equilibrium points have been determined and the stability of the proposed system has been analyzed around these equilibrium points. Hopf bifurcation analysis has been done with respect to the contact rate between Phytoplankton and Environmental toxin ( γ ), constant rate of increase of Environmental toxin ( A ) , depletion rate of environmental toxin ( d 1 ), harvesting effort ( E ) and rate of releasing toxin by Phytoplankton ( ρ ). From this study, it is seen that these parameters have a big role on the stability of this predator prey system. Finally, some numerical simulation results have been shown to verify our analytical findings. [ABSTRACT FROM AUTHOR]

In this paper, based on quasi steady-state assumption, a phenomenological model of viral infections with intracellular delay is proposed to explore viral infection process. The existence as well as stability of equilibria and bifurcation are studied. The results indicate that the basic reproduction number is a sharp threshold parameter to determine the outcomes of infection when the intracellular delay is omitted. Otherwise, Hopf bifurcation may occur under some conditions. Combining with numerical simulations, we observe that the explicit dynamics of free virus can affect the dynamics behavior, the fluctuation scenario may appears in the case of full logistic proliferation term even if there is no intracellular delay. In addition, there are different patterns of bifurcation under different drug effectiveness and patient’s age interacting with intracellular delay, which indicates that intracellular delay may induce complicated kinetics of virus infection. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the control of chaotic Burke-Shaw system using Pyragas method. This system is derived from Lorenz system which has several applications in physics and engineering (e.g. secure communications). The linear stability and the existence of Hopf bifurcation of this system are investigated. Based on the characteristic equation, a theorem is stated and proved. This theorem is used to calculate the interval values of the time delay τ at which this system is stable (unstable). By establishing appropriate time delay τ and feedback strength K ranges, one of the unstable equilibria of this system can be controlled to be stable. We, also, introduced the fractional version of this system which is not studied in the literature as far as we know. The advantage of the fractional order system is that, the system has extra parameter which enriches its dynamics. Increasing the number of parameters may be used to increase the security of the transmitted information. We apply the Pyragas method to control the chaotic behavior of fractional Burke–Shaw system. As we did for the integer order, we determine the values of τ and K which guarantee that the fractional version is stable. Finally, to support the analytical results, some numerical simulations are carried out which indicate that chaotic solution is turned to be stable if τ passes through certain intervals. The bifurcation diagrams are calculated. [ABSTRACT FROM AUTHOR]

This paper explores the classification of parameter spaces for reaction-diffusion systems of two chemical species on stationary rectangular domains. The dynamics of the system are explored both in the absence and presence of diffusion. The parameter space is fully classified in terms of the types and stability of the uniform steady state. In the absence of diffusion the results on the classification of parameter space are supported by simulations of the corresponding vector-field and some trajectories of the phase-plane around the uniform steady state. In the presence of diffusion, the main findings are the quantitative analysis relating the domain-size with the reaction and diffusion rates and their corresponding influence on the dynamics of the reaction-diffusion system when perturbed in the neighbourhood of the uniform steady state. Theoretical predictions are supported by numerical simulations both in the presence as well as in the absence of diffusion. Conditions on the domain size with respect to the diffusion and reaction rates are related to the types of diffusion-driven instabilities namely Turing, Hopf and Transcritical types of bifurcations. The first condition is a lower bound on the area of a rectangular domain in terms of the diffusion and reaction rates, which is necessary for Hopf and Transcritical bifurcation to occur. The second condition is an upper bound on the area of domain in terms of reaction-diffusion rates that restricts the diffusion-driven instability to Turing type behaviour, whilst forbidding the existence of Hopf and Transcritical bifurcation. Theoretical findings are verified by the finite element solution of the coupled system on a two dimensional rectangular domain. [ABSTRACT FROM AUTHOR]

In this paper, a diffusive predator–prey model with Allee effect and functional response of generalized Holling type IV is established. The complex pattern dynamics of and bifurcation phenomena of system are analyzed by using linear stability analyses and bifurcation theory, etc. Corresponding to spiral pattern, spot pattern or spot–stripe mix pattern and chaotic pattern, respectively, the conditions that arise Hopf instability, Turing instability and Hopf–Turing instability of constant positive steady solutions of reaction–diffusion system are presented. The impacts of Allee effect and cross-diffusion on pattern formations of the system are further discussed. Particularly, provided that cross-diffusion is absent, there will not appear the formation of Turing pattern, that is, Turing pattern is only driven by cross-diffusions. Addressed to the different pattern formations involving in cross-diffusion and Allee effect, overall numerical simulations are provided when the spatial dimension is two and the theoretical results are demonstrated. It is revealed that the larger Allee effect constant A and the cross-diffusion coefficient d 4 are advantageous for the appearing of formation of Turing pattern, while the cross-diffusion coefficient d 3 suppresses the formation of Turing pattern with its increasing. • Gave that the theoretical analysis on Turing instability, Hopf instability and Turing–Hopf instability of system (1.5) and revealed that the corresponding pattern dynamics of system (1.5). • Showed that the influences of the strong Allee effect and the weak Allee effect on the pattern formations of system (1.5) in a unified manner. It is the first time that the pattern problems are considered under the situation which do not distinguish the strong Allee effect and the weak Allee effect. • A series of complete numerical simulations including Turing pattern, spiral pattern and chaotic pattern are exhibited when the space form is taken as a bounded domain in two dimensional space. Based on the forward Euler method, a complex algorithm and MATLAB programme are used to achieve the simulations. Provided that multi-dimensional comparisons of curves ω (θ 2) and z (θ 2) as the changes of the parameters A , d 3 , d 4 , there has a more clear exhibition for the impacts of the parameters A , d 3 , d 4 on pattern formations. • Verified that Turing pattern is driven by cross-diffusions, the larger Allee effect constant A and the cross-diffusion coefficient d 4 are advantageous for the appearing of formation of Turing pattern, while the cross-diffusion coefficient d 3 suppresses the formation of Turing pattern with its increasing. [ABSTRACT FROM AUTHOR]

In this paper, a cytokine-enhanced viral infection model with time delays and CTL immune response is proposed and analyzed. Multiple time delays, namely intracellular delay τ 1 , virus replication delay τ 2 and immune response delay τ 3 are included. Firstly, two key parameters with strong biological significance, namely the virus reproductive number R 0 and the CTL immune reproductive number R 1 are derived. And then the stability of equilibria of the model is investigated by constructing suitable Lyapunov functionals and using LaSalle's invariance principle. We prove that if R 0 < 1 , then the infection-free equilibrium E 0 is globally asymptotically stable for any τ 1 , τ 2 , τ 3 ≥ 0 ; if R 0 > 1 , R 1 < 1 , then the immunity-inactivated equilibrium E 1 is globally asymptotically stable for any τ 1 , τ 2 , τ 3 ≥ 0 ; and if R 1 > 1 , then the immunity-activated equilibrium E ∗ is globally asymptotically stable for any τ 1 , τ 2 ≥ 0 and τ 3 = 0. For τ 1 , τ 2 ≥ 0 , τ 3 > 0 , theoretical analysis and numerical simulations show that with the increase of τ 3 , the equilibrium E ∗ loses its stability and the system generates Hopf bifurcation, which means the system produces periodic oscillation under certain conditions. [ABSTRACT FROM AUTHOR]

This paper first investigates the dynamical behavior of a class of reversible quadratic systems, providing all possible phase portraits on the plane. Then, we use generalized Melnikov function method to study the Hopf bifurcation of reversible quadratic systems under the perturbation of piecewise quadratic systems, finding 4 more limit cycles than the smooth case. [ABSTRACT FROM AUTHOR]

The continuous acceleration of information communication makes the spread of rumors constantly develop and change. Analysis of the dynamics of rumor propagation allows for its better control. In this paper, we study a rumor transmission model with a nonlinear transmission rate and a non-smooth threshold transmission function, based on the classical S I infectious disease model. In this model, we first prove the existence of the model solutions, and then obtain the number of non-negative equilibrium points. By dividing the model into spatially homogeneous and inhomogeneous parts with continuous and discontinuous, we discuss the characteristic equations of the equilibrium points and obtain the existence conditions for Saddle–node bifurcation, Turing bifurcation and Hopf bifurcation under the corresponding conditions. Finally, the feasibility of the theoretical calculation results is further verified by using the simulation results. • Bifurcation is studied. • Bifurcation direction is discussed. • Stability is proved. [ABSTRACT FROM AUTHOR]

Recently, the dynamics of the fractional delayed neural networks has been considerably concerned. It is illustrated that time delay has a remarkable influence on the dynamical behaviors of the fractional neural networks. Nevertheless, the results of the fractional neural network with leakage delay are extremely few. It is the first time that the stability and bifurcation of fractional BAM neural networks with time delay in leakage terms is examined in the current paper. The stability criterion and the conditions of bifurcation are obtained for the proposed systems with or without leakage delay by selecting time delay as the bifurcation parameter. It is amazing that the leakage delay has a destabilizing influence on the stability performance of such system and they cannot be ignored. Moreover, the relation between the bifurcation point and the order is fully discussed by careful calculation. Finally, numerical examples are addressed to verify the feasibility of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

Network with interacting loops and time delays are common in physiological systems. In the past few years, the dynamic behaviors of coupled interacting loops neural networks have been widely studied due to their extensive applications in classification of pattern recognition, signal processing, image processing, engineering optimization and animal locomotion, and other areas, see the references therein. In a large amount of applications, complex signals often occur and the complex-valued recurrent neural networks are preferable. In this paper, we study a complex value Hopfield-type network that consists of a pair of one-way rings each with four neurons and two-way coupling between each ring. We discuss the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. The existence of multiple branches of bifurcating periodic solution is obtained. We also found that the spatio-temporal patterns of bifurcating periodic oscillations alternate according to the change of the propagation time delay in the coupling, i.e., different ranges of delays correspond to different patterns of neural network oscillators. The oscillations of corresponding neurons in the two loops can be in phase or anti-phase depending on the parameters and delay. Some numerical simulations support our analysis results. [ABSTRACT FROM AUTHOR]

Fractional order dynamical systems admit chaotic solutions and the chaos disappears when the fractional order is reduced below a threshold value [1]. Thus the order of the dynamical system acts as a chaos controlling parameter. Hence it is important to study the fractional order dynamical systems and chaos. Study of fractional order dynamical systems is still in its infancy and many aspects are yet to be explored. In pursuance to this in the present paper we prove the existence of fractional Hopf bifurcation in case of fractional version of a chaotic system introduced by Bhalekar and Daftardar-Gejji [2]. We numerically explore the ( A, B, α ) parameter space and identify the regions in which the system is chaotic. Further we find (global) threshold value of fractional order α below which the chaos in the system disappears regardless of values of system parameters A and B . [ABSTRACT FROM AUTHOR]

A nonlocal delayed reaction-diffusive two-species model with Dirichlet boundary condition and general functional response is investigated in this paper. Based on the Lyapunov–Schmidt reduction, the existence, bifurcation direction and stability of Hopf bifurcating periodic orbits near the positive spatially nonhomogeneous steady-state solution are obtained, where the time delay is taken as the bifurcation parameter. Moreover, the general results are applied to a diffusive Lotka–Volterra type food-limited population model with nonlocal delay effect, and it is found that diffusion and nonlocal delay can also affect the other dynamic behavior of the system by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a diffusive predator–prey system with Crowley–Martin functional response and delay subject to Neumann boundary condition. More precisely, we study the stability and Turing instability of positive equilibrium for non-delay system, instability and Hopf bifurcation induced by time delay for delay system. In addition, by the theory of normal form and center manifold method, we derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution. [ABSTRACT FROM AUTHOR]

Complex multi-ring coupling structures are common in real-world networks. However, the current work is mainly limited to unidirectional multi-ring or those sharing a neuron. This paper is devoted to the stability and Hopf bifurcation of a class of double-ring neural network with two shared neurons and multiple time delays, where n and m neurons are distributed on the double-ring, respectively. First, we obtain the characteristic equation of the network at the trivial equilibrium point by using the Coates flow graph method. Then, based on this, some sufficient conditions for the stability and Hopf bifurcation of the double-ring network under two connection modes are given. Finally, we provide some numerical examples to illustrate the validness of the theoretical results, and the influences of fractional order, network size and the distance between two shared neurons on Hopf bifurcation are also discussed. [ABSTRACT FROM AUTHOR]

The paper is concerned with the nonlinear subharmonic resonance of an axially moving nanoscale beam with time-dependent velocity controlled by displacement and velocity time delay. To gain a thorough understanding of the system, the frequency response equation is first obtained using the multiple scales method. The stable intervals of the system's zero and non-zero solutions, as well as the necessary and sufficient conditions for Hopf bifurcation are specified. Then, the effects of nonlocal parameter, displacement feedback gain coefficient, velocity feedback gain coefficient, displacement and velocity time delay on the system's dynamic behaviors are discussed. The analysis demonstrates that the nonlocal parameters can not only reduce the amplitude of the system, but also change the stable regions of the non-zero solution. Furthermore, the time delay parameters can affect the number, amplitude, and stability of non-zero solutions. If the time delay parameters are chosen within the optimal value range, both positive and negative time delay feedback can effectively control system stability. These research results will serve as a theoretical reference for the design and optimization of axially moving nanostructures. • The nonlocal parameters can change the stable regions of the non-zero solution. • Time delay feedback can effectively control the system stability. [ABSTRACT FROM AUTHOR]

In this paper, a spatiotemporal predator-prey system with additional food supplied is investigated. By analyzing eigenvalues of the characteristic equation associated with delay parameter, the conditions of the existence of Hopf bifurcation in one dimension space are obtained. We analyze the properties of bifurcating period solutions based on the normal form theory and the center manifold theorem of partial functional differential equations (PFDs). Furthermore, numerical simulations confirm the theoretical results. The obtained results may provide some new insights on periodic oscillation in the densities of predator and prey. [ABSTRACT FROM AUTHOR]

In this paper, a new four-dimensional (4D) smooth quadratic autonomous system with complex hyper-chaotic dynamics is presented and analyzed. The Lyapunov exponent (LE) spectrum, bifurcation diagram and various phase portraits of the system are provided. The stability, Hopf bifurcation and pitchfork bifurcation of equilibrium point are discussed by using the center manifold theorem and bifurcation theory. Numerical simulation results are consistent with the theoretical analysis. Besides, by combining the topological horseshoe theory with a computer-assisted method of Poincaré maps and utilizing the algorithm for finding horseshoes in 3D hyper-chaotic maps, a horseshoe with two-directional expansions in the 4D hyper-chaotic system is successfully found, which rigorously proves the existence of hyper-chaos in theory. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel delay differential model with fractional-order for tumor immune system with external treatments. • Non-negativity of the solution of such model has been investigated. • We investigate the necessary and sufficient conditions for stability of the steady states and Hopf bifurcation with respect to two different time-delays τ 1 and τ 2. • The occurrence of Hopf bifurcation is captured when any of the time-delay passes through critical value τ 1 * , or τ 2 *. • Theoretical results are validated numerically by solving the governing system, using the modified Adams–Bashforth–Moulton predictor-corrector scheme. • Our findings show that the combination of fractional-order derivative and time-delay in the model improves the dynamics and increases complexity of the model. Time-delays and fractional-order play a vital role in modeling biological systems with memory. In this paper, we propose a novel delay differential model with fractional-order for tumor immune system with external treatments. Non-negativity of the solution of such model has been investigated. We investigate the necessary and sufficient conditions for stability of the steady states and Hopf bifurcation with respect to two differenttumor time-delays τ 1 and τ 2. The occurrence of Hopf bifurcation is captured when any of the time-delay passes through critical value τ 1 * , or τ 2 *. Theoretical results are validated numerically by solving the governing system, using the modified Adams–Bashforth–Moulton predictor-corrector scheme. Our findings show that the combination of fractional-order derivative and time-delay in the model improves the dynamics and increases complexity of the model. [ABSTRACT FROM AUTHOR]