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We investigate the dynamics of three-strategy (rock-paper-scissors) replicator equations in which the fitness of each strategy is a function of the population frequencies delayed by a time interval $$T$$ . Taking $$T$$ as a bifurcation parameter, we demonstrate the existence of (non-degenerate) Hopf bifurcations in these systems and present an analysis of the resulting limit cycles using Lindstedt's method. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new cortex-pallidum model to study the origin mechanism of Parkinson's oscillations in the cortex. In contrast to many previous models, the globus pallidus internal (GPi) and externa (GPe) both exert direct inhibitory feedback to the cortex. Using Hopf bifurcation analysis, two new critical conditions for oscillations, which can include the self-feedback projection of GPe, are obtained. In this paper, we find that the average discharge rate (ADR) is an important marker of oscillations, which can divide Hopf bifurcations into two types that can uniformly be used to explain the oscillation mechanism. Interestingly, the ADR of the cortex first increases and then decreases with increasing coupling weights that are projected to the GPe. Regarding the Hopf bifurcation critical conditions, the quantitative relationship between the inhibitory projection and excitatory projection to the GPe is monotonically increasing; in contrast, the relationship between different coupling weights in the cortex is monotonically decreasing. In general, the oscillation amplitude is the lowest near the bifurcation points and reaches the maximum value with the evolution of oscillations. The GPe is an effective target for deep brain stimulation to alleviate oscillations in the cortex. [ABSTRACT FROM AUTHOR]

The dynamics of integer-order Cohen-Grossberg neural networks with time delays has lately drawn tremendous attention. It reveals that fractional calculus plays a crucial role on influencing the dynamical behaviors of neural networks (NNs). This paper deals with the problem of the stability and bifurcation of fractional-order Cohen-Grossberg neural networks (FOCGNNs) with two different leakage delay and communication delay. The bifurcation results with regard to leakage delay are firstly gained. Then, communication delay is viewed as a bifurcation parameter to detect the critical values of bifurcations for the addressed FOCGNN, and the communication delay induced-bifurcation conditions are procured. We further discover that fractional orders can enlarge (reduce) stability regions of the addressed FOCGNN. Furthermore, we discover that, for the same system parameters, the convergence time to the equilibrium point of FONN is shorter (longer) than that of integer-order NNs. In this paper, the present methodology to handle the characteristic equation with triple transcendental terms in delayed FOCGNNs is concise, neoteric and flexible in contrast with the prior mechanisms owing to skillfully keeping away from the intricate classified discussions. Eventually, the developed analytic results are nicely showcased by the simulation examples. [ABSTRACT FROM AUTHOR]

This paper outlines a methodology for the construction of vector fields that can enable a multi-robot system moving on the plane to generate multiple dynamical behaviors by adjusting a single scalar parameter. This parameter essentially triggers a Hopf bifurcation in an underlying time-varying dynamical system that steers a robotic swarm. This way, the swarm can exhibit a variety of behaviors that arise from the same set of continuous differential equations. Other approaches to bifurcation-based swarm coordination rely on agent interaction which cannot be realized if the swarm members cannot sense or communicate with one another. The contribution of this paper is to offer an alternative method for steering minimally instrumented multi-robot collectives with a control strategy that can realize a multitude of dynamical behaviors without switching their constituent equations. Through this approach, analytical solutions for the bifurcation parameter are provided, even for more complex cases that are described in the literature, along with the process to apply this theory in a multi-agent setup. The theoretical predictions are confirmed via simulation and experimental results with the latter also demonstrating real-world applicability. [ABSTRACT FROM AUTHOR]

Understand the dynamics of cancer stem cells (CSCs), prevent the non-recurrence of cancers and develop therapeutic strategies to destroy both cancer cells and CSCs remain a challenge topic. In this paper, we study both analytically and numerically the dynamics of CSCs under radiotherapy effects. The dynamical model takes into account the diffusion of cells, the de-differentiation (or plasticity) mechanism of differentiated cancer cells (DCs) and the time delay on the interaction between microRNAs molecules (microRNAs) with DCs. The stability of the model system is studied by using a Hopf bifurcation analysis. We mainly investigate on the critical time delay τ c , that represents the time for DCs to transform into CSCs after the interaction of microRNAs with DCs. Using the system parameters, we calculate the value of τ c for prostate, lung and breast cancers. To confirm the analytical predictions, the numerical simulations are performed and show the formation of spatiotemporal circular patterns. Such patterns have been found as promising diagnostic and therapeutic value in management of cancer and various diseases. The radiotherapy is applied in the particular case of prostate model. We calculate the optimum dose of radiation and determine the probability of avoiding local cancer recurrence after radiotherapy treatment. We find numerically a complete eradication of patterns when the radiotherapy is applied before a time t < τ c . This scenario induces microRNAs to act as suppressors as experimentally observed in prostate cancer. The results obtained in this paper will provide a better concept for the clinicians and oncologists to understand the complex dynamics of CSCs and to design more efficacious therapeutic strategies to prevent the non-recurrence of cancers. [ABSTRACT FROM AUTHOR]

This paper lucubrates the Hopf bifurcation of fractional-order Hopfield neural network (FOHNN) with three nonidentical delays. The type of delays in the model include leakage delay, self-connection delay and communication delay. Differentiating from traditional bifurcation exploration of delayed fractional-order system, this paper presents a succinct and systematic approach as much as possible to settle the bifurcation problem when all three delays fluctuate and aren't convertible. In addition, this paper furnishes a humble opinion for solving bifurcation cases caused by arbitrary unequal delays. At length, we address three simulation examples to corroborate the correctness of key fruits. [ABSTRACT FROM AUTHOR]

As is well known that time delays are inevitable in practice due to the finite switching speed of amplifiers and information transmission between neurons. So the study on the Hopf bifurcation of delayed neural networks has aroused extensive attention in recent years. However, it's worth mentioning that only the communication delays between neurons were generally considered in most existing relevant literatures. Actually, it has been proven that a kind of so-called leakage delays cannot be ignored because the self-decay process of a neuron's action potential is not instantaneous in hardware implementation of neural networks. Though leakage delays have been taken into account in a few more recent works concerning the Hopf bifurcation of fractional-order bidirectional associative memory neural networks, the addressed neural networks were low-dimension or the involved time delays were single. In this paper, we propose a five-neuron fractional-order bidirectional associative memory neural network model, which includes leakage delays and discrete communication delays to meet the characteristics of real neural networks better. Then we use the stability theory of fractional differential equations and Hopf bifurcation theory to investigate its dynamic behavior of Hopf bifurcation. The Hopf bifurcation of the proposed model are studied by taking the involved two different leakage delays as the bifurcation parameter respectively, and two kinds of sufficient conditions for Hopf bifurcation are obtained. A numerical example as well as its simulation plots and phase portraits are given at last. Our results indicate that a Hopf bifurcation rises near the zero equilibrium point when the leakage delay reaches its critical value which is given by an explicit formula. Particularly, the results of numerical simulations show that the leakage delay would narrow the stability region of the proposed system and make the Hopf bifurcation occur earlier. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the algal blooms model of plankton interactions initially proposed by Das and Sarkar (DCDIS-A, 14(3):401–414, 2007), where the oscillatory mode in the interaction between phytoplankton and zooplankton is observed. We provide a detailed analysis of the dependence of the equilibria and their stability on various parameters in the model. The bifurcation behaviors around equilibrium (e.g., Hopf bifurcation, Bogdanov–Takens bifurcation) are further found. Meanwhile, numerical simulations verify and illustrate the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we address the effects of two fiscal policy delays on the dynamical analysis of macroeconomics. First, a time gap between the accrual of taxes and their payment is considered. Second, the time spent between the purchasing decisions and the actual expenditure is also taken into consideration. Since both these delays are significant in controlling macroeconomic conditions, this paper incorporates aforementioned delays into the IS–LM model. At first, a mathematical model is developed using delayed differential equations. Then a unique steady state solution is obtained. Around the equilibrium point, linear stability analysis is done. Also, the occurance of Hopf bifurcation is observed when delay crosses a critical point and switches in stability are also detected. Properties of Hopf bifurcation using center manifold theorem are discussed. Lastly, numerical simulations are run to verify our analysis. In this work, we considered a case study to perform simulation wherein GDP of India for last ten years is recorded for estimating some parameters. In different investment scenarios, numerical simulations corroborate the analytical findings of the model. Furthermore, rigorous analysis shows that adding the right mix of delays can help in maintaining/ regaining the stability after periods of instability, or even gaining stability in the long run. [ABSTRACT FROM AUTHOR]

In this paper, a two-neuron reaction–diffusion neural network with discrete and distributed delays is proposed, and the state feedback control strategy is adopted to achieve control of its spatiotemporal dynamical behaviours. Adding two virtual neurons, the original system is transformed into a neural network only containing the discrete delay. The conditions under which Hopf bifurcation and Turing instability arise are determined through analysis of the characteristic equation. Additionally, the amplitude equations are derived with the aid of weakly nonlinear analysis, and the selection of the Turing patterns is determined. The simulation results demonstrate that the state feedback controller can delay the onset of Hopf bifurcation and suppress the generation of Turing patterns. [ABSTRACT FROM AUTHOR]

A general time-delay reaction–diffusion–advection system with the Dirichlet boundary condition and spatial heterogeneity is investigated in this paper. By using the implicit function theorem, we obtain the existence and asymptotic expression of the spatially non-homogeneous positive steady-state solution. This is the steady-state bifurcation from zero equilibrium. Via analyzing the corresponding characteristic equation, the stability of the spatially non-homogeneous positive steady-state solution and the occurrence of Hopf bifurcation at the positive steady-state solution are obtained, and the spatially non-homogeneous periodic solution is derived from Hopf bifurcation, this is the secondary bifurcation behavior of the system. Utilizing the normal form method and center manifold theory, we prove that the direction of Hopf bifurcation is supercritical and the bifurcating spatially non-homogeneous periodic solution is stable. Furthermore, We show that there exist two sequences Hopf bifurcation values and the orders of two sequences Hopf bifurcation values are given. Moreover, theoretical and numerical results are applied to competition and cooperation systems, respectively. Finally, the effect of the advection rate and spatial heterogeneity are discussed. [ABSTRACT FROM AUTHOR]

The coronavirus disease 2019 (COVID-19) remains serious around the world and causes huge deaths and economic losses. Understanding the transmission dynamics of diseases and providing effective control strategies play important roles in the prevention of epidemic diseases. In this paper, to investigate the effect of delays on the transmission of COVID-19, we propose a delayed SEIR model to describe COVID-19 virus transmission, where two delays indicating the incubation and recovery periods are introduced. For this system, we prove its solutions are nonnegative and ultimately bounded with the nonnegative initial conditions. Furthermore, we calculate the disease-free and endemic equilibrium points and analyze the asymptotical stability and the existence of Hopf bifurcations at these equilibrium points. Then, by taking the weighted sum of the opposite number of recovered individuals at the terminal time, the number of exposed and infected individuals during the time horizon, and the system cost of control measures as the cost function, we present a delay optimal control problem, where two controls represent the social contact and the pharmaceutical intervention. Necessary optimality conditions of this optimal control problem are exploited to characterize the optimal control strategies. Finally, numerical simulations are performed to verify the theoretical analysis of the stability and Hopf bifurcations at the equilibrium points and to illustrate the effectiveness of the obtained optimal strategies in controlling the COVID-19 epidemic. [ABSTRACT FROM AUTHOR]

In this paper, we couple population dynamics and evolutionary game theory to establish a prey–predator system in which individuals in the predator population need to choose between group hunting strategies and isolated hunting strategies. This system includes two types of delay: fitness delay and hunting delay. In the absence of delays, we discuss the stability of boundary and interior equilibria. In addition, the condition that the non-delayed system undergoes transcritical bifurcation is obtained. For the delayed system, we explore the stability of the interior equilibrium and obtain the conditions for the existence of Hopf bifurcation. The conditions for determining the direction and stability of the Hopf bifurcation and the periodic variation in the periodic solution are introduced by using the normal form theory and center manifold theory. Finally, we simulate non-delayed and delayed systems. The results indicate that when the availability of prey is high, the isolated hunting strategy is the dominant strategy. When the availability of prey is low, mixed strategies appear and the proportion of the group hunting strategy increases as the availability of prey decreases. Furthermore, large delays lead to the disappearance of the mixed hunting strategy and its replacement by pure hunting strategies. [ABSTRACT FROM AUTHOR]

In this paper, considering that fractional-order calculus can more accurately describe memory and genetic properties, we introduce fractional integral operators into neural networks and discuss the stability and Hopf bifurcation of a fractional-order bidirectional associate memory (BAM) neural network with two delays. In addition, the hybrid controller is proposed to achieve Hopf bifurcation control of the system. By taking two time delays as the bifurcation parameters and analyzing of the corresponding characteristic equation, stability switching curves of the controllable system for two delays are obtained. The direction of the characteristic root crossing the imaginary axis in stability switching curves is determined. Sufficient criteria are sequentially given to judge the local stability and the existence of Hopf bifurcation of a fractional-order BAM neural network system. The numerical simulation results show that the hybrid controller can effectively control Hopf bifurcation of a fractional-order BAM neural network system with two delays. [ABSTRACT FROM AUTHOR]

In this paper, we propose a diffusive predator–prey model with two delays, i.e., a spatial memory delay and a toxic effect delay. Initially, we analyze the global existence of the solution of the system. We then analyze the equilibria’s local stability without delays and investigate the Hopf bifurcation induced by one delay. Subsequently, we establish an analytical framework for constructing the stability switching curve in the delay space. Finally, we present numerical simulations to validate the theoretical results and verify the emergence of various spatial patterns in the system. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a diffusive prey–predator system with strong Allee effect in prey, spatial memory, and gestation delay of predator. Turing instability and Hopf bifurcation are investigated extensively. The system more easily creates Turing patterns, when the predator owns sluggish memory-induced diffusion. If the predator owns rapid memory-induced diffusion, then the system may present abundant dynamics. Predator with long memory displays a spatial nonhomogeneous periodic spread once the system only contains spatial memory delay. Populations are in a homogeneous or nonhomogeneous periodic spread with gestation period. However, under the combined influences of memory delay and gestation delay, the stability switches appear. Strong Allee effect has obvious influences on the system. The positive equilibrium is linearly stable if there is no strong Allee effect; once strong Allee effect is reasonable, it is unstable. But if the Allee effect is huge enough, then two populations can die out. Finally, we investigate the results by numerical simulations, which demonstrate that spatial memory, gestation delay, and strong Allee effect are conductive to the dynamics of the system. [ABSTRACT FROM AUTHOR]

This paper proposes a fractional-order reserve biological prey-predator system with a practical specimen biological system incorporating a mixed functional response and fear effect on prey populace because of predation. The fractional-order system describes using the Holling type I and III functional responses. The existence, non-negativity, uniqueness, and boundedness of the solutions along with the local asymptotic stability of the equilibrium points, are investigated from a mathematical point of view. For the reserve biological fractional-order prey-predator system, we have investigated how the fear effect interacts with fractional influencing the local stability and Hopf bifurcation of the system. In this instance, the prerequisites for the fractional-order model’s stability and the occurrence of Hopf bifurcation are illustrated. Numerical examples are exhibited to approve the hypothetical outcomes of the fractional-order prey-predator system. [ABSTRACT FROM AUTHOR]

In this paper, we have considered a Lorenz-like model with slight changes in the nonlinear terms. Here we have studied the system dynamics for different range of values of parameters σ , r . The Hopf bifurcation analysis of the system has been done using center manifold theorem for σ = - 1 , r > 1 . Phase portraits of solutions of the system are plotted for various system parameters to substantiate the change in dynamics. The bifurcation diagram and the Lyapunov exponent evaluation plots also help to explain the behaviour of the system. Using Fishing principle, we have shown the existence of homoclinic orbit and consequently, observed the existence of homoclinic as well as heteroclinic orbits in the numerical simulation for σ > 0 , r > 1 . [ABSTRACT FROM AUTHOR]

This paper is concerned with a delayed reaction–diffusion equation on a unit disk. By means of the singularity theory and Lyapunov–Schmidt reduction, we not only derive universal conclusions about the existence of inhomogeneous steady-state solutions and the equivariant Hopf bifurcation theorems, but also obtain some more extraordinary properties of bifurcating solutions, which are produced by the radial symmetry through abstract methods based on the Lie group representation theory. Meanwhile, we illustrate our results by an application to a population model with a time delay. Furthermore, the methods established in this paper are applicable to specific delayed reaction–diffusion models with other symmetries. [ABSTRACT FROM AUTHOR]

At present, there are few studies on the delayed kernel function of hyper-strong kernel. This paper attempts to analyze the stability and bifurcation of neural networks with distributed delayed hyper-strong kernels. Firstly, considering the average delay as a bifurcation parameter, the study discusses the characteristic equations of delayed kernels with weak kernel, strong kernel and hyper-strong kernel to provide sufficient conditions for the stability and generation of Hopf bifurcation. Secondly, it applies the normal theory and the center manifold theory to derive the formulas for determining the stability and direction of the bifurcating periodic solution. Finally, it verifies the correctness of the calculation results by numerical simulation with an example. [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]

The dynamics of prey–predator system, when one or both the species are harvested non-linearly, has become a topic of intense study because of its wide applications in biological control and species conservation. In this paper we have discuss different bifurcation analysis of a two dimensional prey–predator model with Beddington–DeAngelis type functional response in the presence of prey refuge and non-linear harvesting of both species. We have studied the positivity and boundedness of the model system. All the biologically feasible equilibrium points are investigated and their local stability is analyzed in terms of model parameters. The global stability of coexistence equilibrium point has been discussed. Depending on the prey harvesting effort ( E 1 ) and degree of competition among the boats, fishermen and other technology ( l 1 ) used for prey harvesting, the number of axial and interior equilibrium points may change. The system experiences different type of co-dimension one bifurcations such as transcritical, Hopf, saddle-node bifurcation and co-dimension two Bogdanov–Takens bifurcation. The parameter values at the Bogdanov–Takens bifurcation point are highly sensitive in the sense that the nature of coexistence equilibrium point changes dramatically in the neighbourhood of this point. The feasible region of the bifurcation diagram in the l 1 - E 1 parametric plane divides into nine distinct sub-regions depending on the number and nature of equilibrium points. We carried out some numerical simulations using the Maple and MATLAB software to justify our theoretical findings and finally some conclusions are given. [ABSTRACT FROM AUTHOR]

In this paper, a fractional model of a special structure of bidirectional associative memory (BAM) neural networks called tri-diagonal BAM neural networks (TdBAMNNs) is considered. The Hopf bifurcation analysis is made for the proposed fractional system in the presence of leakage and communication delays. The feasibility of the obtained theoretical results is verified by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the general Langford system under homogeneous Neumann boundary conditions. The stabilities of constant solutions are discussed for the general Langford ODE and PDE systems, respectively. Based on the stability results, for the Langford ODE system, the existence, bifurcation direction and stability of periodic solutions are established. Then for the Langford PDE system, by the center manifold theory and the normal form method, the direction of Hopf bifurcation and the stability of spatially homogeneous periodic solutions are investigated. Finally, numerical simulations are shown to support and supplement the results of theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we propose a predator–prey model with prey refuge and fear effect under non-diffusion and diffusion. For the non-diffusion ODE model, we first analyze the existence and stability of equilibria. Then, the existence of transcritical bifurcation, Hopf bifurcation and limit cycle is discussed, respectively. We find that when the cost of minimum fear η is taken as the bifurcation parameter, it not only influence the occurrence of Hopf bifurcation but also alters its direction. For diffusion predator–prey model under homogeneous Neumann boundary conditions, we observe that the Turing instability does not occur, but the Hopf bifurcation will manifest near the interior equilibrium. By considering η as the bifurcation parameter, the direction and stability of spatially homogeneous periodic orbits are established. At last, the validity of the theoretical analysis are verified by a series of numerical simulations. The results indicate that prey refuge and fear effect play an key role in the stability of populations. [ABSTRACT FROM AUTHOR]

This paper focuses on a cholera model with temporary immunity and recovery-age. We first investigate the existence of unique solution to the model by reformulating the system as an abstract Cauchy problem, and present the basic reproduction number. By analyzing the characteristic equation, we discuss the stability of the steady states and further verify the existence of Hopf bifurcation. The theoretical results show that the temporary immunity effect takes a crucial effect on the perturbation around the endemic equilibrium. Numerical simulations are performed to support our results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the high codimension bifurcations of a classical predator–prey system with Allee effects and generalized Holling type III functional response p (x) = m x 2 a x 2 + b x + 1 where b > - 2 a . We show that the maximal orders of nilpotent saddle, cusp singularity and weak focus are all three. The unfoldings of a cusp singularity and of a nilpotent saddle of order 3 with a fixed invariant line are developed. The dependence of codimension of degenerate Hopf bifurcation on b is thoroughly investigated. It is proven that there exist a homoclinic loop of order 2 and a heteroclinic loop of order 2 for - 2 a < b < 0 , and three limit cycles for b > 0 . Together with existing work for Holling type I, II and IV functional responses, our results complement the analysis of the classical predator–prey systems with Allee effects and four types of Holling functional responses. Furthermore, simple formulas are derived to characterize the order of nilpotent saddle, through which the existence and order of the heteroclinic loop can be easily obtained for a general class of predator–prey systems with any smooth functional response. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence, multiplicity, stability, and Hopf bifurcation of steady-state solutions of a diffusive Lotka–Volterra type model for two species with nonlocal delay effect and nonlinear boundary condition. It is found that there is no Hopf bifurcation when the interior reaction term is weaker than the boundary reaction term, and that the interior reaction delay determines the existence of Hopf bifurcation only when the interior reaction term is stronger than the boundary reaction term. This observation helps us to understand the nonlinear balance between the interior reaction and boundary flux in nonlinear boundary problems. Moreover, the general results are illustrated by applications to a model with homogeneous kernels. [ABSTRACT FROM AUTHOR]

The tea green leafhopper Empoasca onukii is one kind of insect pest threatening the tea production, and the mite Anystis baccarum has been used as an agent for pest control. In this paper, we introduce a generalist predator-prey model to study the dynamics for informing biological control. There have been some bifurcation studies of the generalist predator-prey model in the last few years. Except for the bifurcations include saddle-node bifurcation of codimension 1 and 2, Hopf bifurcations, and Bogdanov-Takens bifurcation of codimension 2 and 3, we also present the bifurcations of nilpotent singularities of elliptic and focus type of codimension 3. We find that the nilpotent singularities are associated with a cubic Liénard system, and the nilpotent bifurcations are three-parameter bifurcations of a codimension 4 nilpotent focus. Furthermore, we show that the nilpotent focus serves as an organizing center to connect all the codimension 3 bifurcations in the system. We also present the bifurcation diagrams to unfold the nilpotent singularities of codimension 3. One interesting observation is that we show numerically the existence of three limit cycles in the system. [ABSTRACT FROM AUTHOR]

In this paper, we consider a single species population model over patches with delay and nonlocal interactions, for which no symmetry for the dispersion (connection) matrix is assumed. We show that there exists a positive equilibrium when the dispersal rate is large. We also discuss the stability/instability of this positive equilibrium, establish the threshold dynamics and explore the associated Hopf bifurcation. Moreover, we demonstrate our theoretical results by a nonlocal logistic population model and by the Nicholson's blowflies model. [ABSTRACT FROM AUTHOR]

The bidirectional associative memory (BAM) neural network has the capability to store hetero-associative pattern pairs, which has high requirements for stability. This paper inquires into Hopf bifurcation of fractional-order bidirectional associative memory neural network (FOBAMNN) and implants three types of delays into the FOBAMNN. Namely leakage delay, self-connection delay, and communication delay, both of which are unequal. Drawing support from matrix eigenvalue theory, stability theory of fractional differential equations and bifurcation technology, The delay-dependent stability criterion and bifurcation point of the model are established by exploiting the characteristic polynomial. Afterwards, the self-connection delay or leakage delay is selected as the bifurcation parameter, and the unselected delay of the two is fixed in its stable interval, so as to obtain the condition of bifurcation. The results show that different types of delay affect the stability of the system. Simultaneously, once the delay outreaches the critical value of bifurcation, the stability of the model will be wrecked. Thereupon, in the application, we should adopt small delays to maintain the stability of the system. We illustrate that the leakage delay and self-connection delay can affect the stability of FOBAMNN. And the calculation method of the critical value of the delay will also be given. At length, the authenticity of the developed key fruits is elucidated via numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, a six-neuron incommensurate fractional order BAM neural network model with multi-delays is considered. We demonstrate that the equilibrium point of the system loses its stability and Hopf bifurcation emerges when the delay passes through a critical value. And the relationship between the critical delay of Hopf bifurcation and size of fractional orders is found. Finally, some numerical simulations are given to verify the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

Purpose: In order to reduce the hazards caused by the nonlinear torsional vibration of the shaft system of the direct-drive permanent magnet wind turbine, the dynamic characteristics of the controlled shaft system torsional vibration were analyzed. Methods: First, the shaft system of the direct-drive permanent magnet wind turbine is equivalent to a two-mass model. Based on considering the electromechanical coupling and the nonlinear change of the main shaft damping, the torsional vibration dynamic model of the shaft system is established by the Lagrange–Maxwell equation. Second, with the main shaft damping as the bifurcation parameter, the Routh–Hurwitz criterion is used to calculate the Hopf bifurcation critical point and stability range of the shafting system. Finally, a nonlinear state feedback controller is designed, and the controlled shafting system paradigm is obtained by the direct method. Results: The torsional vibration characteristics of shafting and the influence of supercritical Hopf bifurcation on shafting are discussed. In order to suppress the unstable torsional vibration of the shafting system and analyze the influence of the controller's linear gain and nonlinear gain on the torsional vibration of the shafting system, the numerical simulation verifies the correctness of the theoretical analysis. Conclusions: This paper can provide a theoretical basis for the stable operation of the direct-drive permanent magnet wind turbine. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the stability and Hopf bifurcation of the fractional-order logistic equation with two different delays τ 1 , τ 2 > 0 : D α y (t) = ρ y (t - τ 1) 1 - y (t - τ 2) , t > 0 , ρ > 0 . We describe stability regions by using critical curves. We explore how the fractional order α , ρ , and time delays influence the stability and Hopf bifurcation of the model. Then, by choosing ρ , fractional order α , and time delays as bifurcation parameters, the existence of Hopf bifurcation is studied. An Adams-type predictor–corrector method is extended to solve fractional-order differential equations involving two different delays. Finally, numerical simulations are given to illustrate the effectiveness and feasibility of theoretical results. [ABSTRACT FROM AUTHOR]

Virotherapy treatment is a new and promising target therapy that selectively attacks cancer cells without harming normal cells. Mathematical models of oncolytic viruses have shown predator–prey like oscillatory patterns as result of an underlying Hopf bifurcation. In a spatial context, these oscillations can lead to different spatio-temporal phenomena such as hollow-ring patterns, target patterns, and dispersed patterns. In this paper we continue the systematic analysis of these spatial oscillations and discuss their relevance in the clinical context. We consider a bifurcation analysis of a spatially explicit reaction-diffusion model to find the above mentioned spatio-temporal virus infection patterns. The desired pattern for tumor eradication is the hollow ring pattern and we find exact conditions for its occurrence. Moreover, we derive the minimal speed of travelling invasion waves for the cancer and for the oncolytic virus. Our numerical simulations in 2-D reveal complex spatial interactions of the virus infection and a new phenomenon of a periodic peak splitting. An effect that we cannot explain with our current methods. [ABSTRACT FROM AUTHOR]

This paper is devoted to a charge transfer model with cross-diffusion under Neumann boundary conditions. We investigate how the cross-diffusion could destabilize the stable periodic solutions bifurcating from the unique positive equilibrium point. By the implicit function theorem and Floquet theory, we obtain some conditions on the self-diffusion and cross-diffusion coefficients under which the stable periodic solutions can become unstable. New irregular Turing patterns then generate by the destabilization of stable spatially homogeneous periodic solutions. We also present some numerical simulations to further support the results of theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper deals with Kalecki's 1935 business cycle model, where a finite time lag in the investment dynamics is assumed. The time lag is the gestation period elapsing between orders for capital goods and deliveries of finished industrial equipment. Including the actual mainstream theory, the economic literature agrees on the consequences that time lag has on the economic activity. It is a cause of persistent economic fluctuations. Following some recent research lines on this model, here we restate the Kalecki approach, assuming sigmoidal functions in addition to Kalecki's linear treatment and further considering a non-constant capital depreciation. Never made until now, this last assumption is such that to yield, in place of a delayed differential equation, a Volterra delayed integro-differential equation. Taken the time delay and the rate of capital depreciation as critical parameters, a qualitative study of that equation is carried out. We proved that with a small-time lag stable equilibria arise. But, when the delay increases, equilibria are destabilized through Hopf bifurcations and stability switches occur. Consequently, a variety of cyclical behaviors appear. [ABSTRACT FROM AUTHOR]

This paper studies the mode coupling behavior and complex nonlinear dynamics of a piezoelectric driven L-shaped beam considering the added mass. To realize natural frequency adjustment and energy exchange among different modes, the added mass is delicately designed. The nonlinear governing equations, representing the first and second modes, are obtained by the Hamilton principle and Galerkin method. Perturbation and bifurcation analyses show that mode coupled vibration can lead to complex dynamic phenomena such as amplitude jump, amplitude saturation, double Hopf bifurcation and amplitude persistence. The physical conditions of amplitude jump, the critical voltage of amplitude saturation and the discriminant formula of double Hopf bifurcation are deduced theoretically and verified numerically. To be more convincing, an experimental test system is set up to observe the nonlinear dynamic behaviors. It is found that when the driving frequency is less than 26.18 Hz or more than 26.52 Hz, the first-order mode vibration jumps under the bifurcation driving voltage, which is qualitatively consistent with the theoretical results. Through mechanism investigation on subcritical Hopf bifurcation induced structure jumping phenomenon, the linear relationship between bifurcation voltage and perturbation mass is deduced. Both theoretical and experimental results demonstrate that small disturbance of added mass can significantly affect the bifurcation voltage of modal coupled vibration, which can realize the detection of micro-mass. The research results of this paper provide theoretical basis and experimental support for the development of micro-resonance device. [ABSTRACT FROM AUTHOR]

Usually, juvenile predators are small in size, comparatively feeble and in the earlier phase of learning the tricky manoeuvre skills of predation. Due to small prey–predator size-ratio, juvenile predators and preys become two deadliest opponents, therefore, whenever both face off, the outcome may be utterly unpredictable. In response to the juvenile predators' attack, preys also come up with a countermeasure and try to defend themselves but a sharp role-reversal in the predator–prey ecology occurs when preys kill the juvenile predators during the counter-attacking strategies. In this paper, we propose and analyze a biologically motivated delayed predator–prey system to explore the joint impact of counterattacking strategies and multiple delays on the predator–prey interactions. We have introduced the transformation of immature to mature predators including delay τ 1 as an average time to reach maturity. Another time delay τ 2 owing to the gestation of the adult predators, is taken into account. After the brief discussion of the basic ecosystem properties for the model in absence of delay, we have performed the stability analysis of the delayed system comprehensively. In this context, existence of Hopf bifurcation at the interior equilibrium is established and its stability as well as direction of the Hopf bifurcation is also investigated. Numerical simulations are also performed to corroborate the theoretical findings. Through this work, it has been illustrated that counterattacking of preys suppresses the unstable oscillations in the population, on the other hand, large delays always destabilize the interior steady state. Hence, in the realistic predator–prey ecology, where counterattack and delays are showing opposite behavior in the same ecosystem, the results of this paper are important to disclose the actual crossover and synergies between both the ecological properties. [ABSTRACT FROM AUTHOR]

The dynamics of a general reaction–diffusion–advection two species model with nonlocal delay effect and Dirichlet boundary condition is investigated in this paper. The existence and stability of the positive spatially nonhomogeneous steady state solution are studied. Then by regarding the time delay τ as the bifurcation parameter, we show that Hopf bifurcation occurs near the steady state solution at the critical values τ n (n = 0 , 1 , 2 , ...) . Moreover, the Hopf bifurcation is forward and the bifurcated periodic solutions are stable on the center manifold. The general results are applied to a Lotka–Volterra competition–diffusion–advection model with nonlocal delay. [ABSTRACT FROM AUTHOR]

It is well known that diffusion could have a destabilizing effect on the otherwise stable solution (including the spatially homogenous periodic solutions), which is called Turing instability. This paper analyzes dynamic behavior of the reaction–diffusion Lauffenburger–Kennedy bacterial infection system, and the focus is on the Turing instability of constant equilibrium solutions and spatially homogeneous Hopf bifurcating periodic solutions. The specific conditions about the diffusion rates under which constant equilibrium solutions and spatially homogeneous periodic solutions occur Turing instability are obtained. Then, the results of theoretical analysis are verified by the examples of numerical simulation. [ABSTRACT FROM AUTHOR]

A hepatitis B or C virus (HBV or HCV) epidemic model with intra- and extra-hepatic coinfection, immune delay and saturation incidence, as well as antiviral therapy is proposed in this paper. The existence of equilibria (infection-free, immune-free and immune-activated), the basic reproduction numbers, i.e., R 0 , R 1 , are given respectively, by which the criteria on (local and global) stability of above equilibria are established. Furthermore, if the immune delay τ > τ 0 , both the existence of subcritical (supercritical) Hopf bifurcation on the immune-activated equilibrium E ∗ , and the stability of bifurcating periodic solutions are obtained. Finally, the theoretical results are demonstrated by numerical simulations. We derive that the immune delay and intra- and extra-hepatic coinfection have significant influence on the transmission of HBV/HCV, could cause more complicated dynamics at E ∗ from stability to unstablity untill bifurcation, which greatly increases the difficulty of disease control. While effective antiviral therapy could evidently decrease the spread of HBV/HCV. [ABSTRACT FROM AUTHOR]

This paper is concerned with the bifurcation problem of a two-delayed fractional-order neural network(FONN) with three neurons. To begin with, the bifurcation progresses with regard to some types of time delays are captured by employing the self-connecting delay as a bifurcation parameter. Afterwards, communication delay is viewed as a bifurcation parameter to cultivate bifurcation outcomes for the developed FONN, and the communication delay triggered-bifurcation conditions are captured. There is an evidence that FONN emanates remarkable stability as long as a smaller value of them is selected, and the stability performance deterioration and Hopf bifurcations take place once choosing a larger control time delay. Eventually, the correctness of the developed theoretical discoveries is appraised by employing numerical revelations. [ABSTRACT FROM AUTHOR]

In this paper, we study the global existence of a bifurcating periodic solution for a two zooplankton-phytoplankton model with two delays. First, we demonstrate that the bifurcating periodic solution exists when one delay increases and the other delay remains unchanged. Second, we give simulation to describe the bifurcating periodic solution when one delay increases. Our work answers the question in Sect. 5 (Shi and Yu in Chaos Solitons Fractals 100:62–73, 2017). [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of bifurcation control for a delayed neuron system. By introducing a new fractional-order Proportional-Derivative (PD) feedback controller, this paper aims to control the stability and Hopf bifurcation through adjusting the control gain parameters. The order chosen in PD controller is different with that of the integer-order neuron system. Sufficient conditions for guaranteeing the stability and generating Hopf bifurcation are constructed for the controlled neuron system. Finally, numerical simulation results are illustrated to verify our theoretical derivations and the relationships between the onset of the Hopf bifurcation and the gain parameters are obtained. [ABSTRACT FROM AUTHOR]

This paper proposes a novel fractional-order delayed Ross–Macdonald model for malaria transmission. This paper aims to systematically investigate the effect of both the incubation periods of Plasmodium and the order on the dynamic behavior of diseases. Utilizing inequality techniques, contraction mapping theory, fractional linear stability theorem, and bifurcation theory, several sufficient conditions for the existence and uniqueness of solutions, the local stability of the positive equilibrium point, and the existence of fractional-order Hopf bifurcation are obtained under different time delays cases. The results show that time delay can change the stability of system. System becomes unstable and generates a Hopf bifurcation when the delay increases to a certain value. Besides, the value of order influences the stability interval size. Thus, incubation periods and the order have a major effect on the dynamic behavior of the model. The effectiveness of the theoretical results is shown through numerical simulations. [ABSTRACT FROM AUTHOR]

The Hopf bifurcation, Turing instability and steady state bifurcation to a fresh-water tussock sedge model with nonlocal interaction under Neumman boundary condition are investigated in this paper. First, we analyze the existence of constant steady states and the effect of the nonlocal term on the its stability and the existence of Hopf bifurcation. Furthermore, the occurrence conditions of Turing instability to such system are studied. Second, we focus on steady state bifurcation to the reaction–diffusion system with nonlocal interaction via Lyapunov–Schmidt reduced method. Finally, numerical simulations have been illustrated to verify our theoretical analysis. [ABSTRACT FROM AUTHOR]

It detects that fractional calculus incorporating into neural networks can neatly reflect the memory properties of neurons. Therefore, we focus on the dynamic bifurcations of a fractional-order Cohen–Grossberg neural network with diverse delays in this paper. Firstly, we use Cramer's rule to analyze the characteristic equation containing fourth-order transcendental terms and study the stability and Hopf bifurcation of the system. The acquired results show that communication and leakage delays significantly affect the time-delayed fractional-order neural network stability. Secondly, we numerically affirm that different fractional orders affect Hopf bifurcation eminently. The last two examples exhibit the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, an age-structured predator–prey system with Beddington–DeAngelis (B–D) type functional response, prey refuge and harvesting is investigated, where the predator fertility function f(a) and the maturation function β (a) are assumed to be piecewise functions related to their maturation period τ . Firstly, we rewrite the original system as a non-densely defined abstract Cauchy problem and show the existence of solutions. In particular, we discuss the existence and uniqueness of a positive equilibrium of the system. Secondly, we consider the maturation period τ as a bifurcation parameter and show the existence of Hopf bifurcation at the positive equilibrium by applying the integrated semigroup theory and Hopf bifurcation theorem. Moreover, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are studied by applying the center manifold theorem and normal form theory. Finally, some numerical simulations are given to illustrate of the theoretical results and a brief discussion is presented. [ABSTRACT FROM AUTHOR]

In the Antarctic, the whale population had been reduced dramatically due to the unregulated whaling. It was expected that Antarctic krill, the main prey of whales, would grow significantly as a consequence and exploratory krill fishing was practiced in some areas. However, it was found that there has been a substantial decline in abundance of krill since the end of whaling, which is the phenomenon of krill paradox. In this paper, to study the krill–whale interaction we revisit a harvested predator–prey model with Holling I functional response. We find that the model admits at most two positive equilibria. When the two positive equilibria are located in the region { (N , P) | 0 ≤ N < 2 N c , P ≥ 0 } , the model exhibits degenerate Bogdanov–Takens bifurcation with codimension up to 3 and Hopf bifurcation with codimension up to 2 by rigorous bifurcation analysis. When the two positive equilibria are located in the region { (N , P) | N > 2 N c , P ≥ 0 } , the model has no complex bifurcation phenomenon. When there is one positive equilibrium on each side of N = 2 N c , the model undergoes Hopf bifurcation with codimension up to 2. Moreover, numerical simulation reveals that the model not only can exhibit the krill paradox phenomenon but also has three limit cycles, with the outmost one crosses the line N = 2 N c under some specific parameter conditions. [ABSTRACT FROM AUTHOR]