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• A new car-following model is proposed to describe the dynamic behavior of autonomous vehicle. • The bifurcation characteristic of traffic flow composed of autonomous vehicle is analyzed. • A controller that considers multi-step memory and multi-step prediction effect is designed. • A hybrid control strategy is proposed, which combines multi-step prediction and memory mechanism with PID. This paper is committed to capturing the dynamic behaviors of homogenous flow of autonomous vehicles (AVs), and exploring the control strategies to improve traffic conditions, which can alleviate traffic congestion and improve traffic efficiency. Firstly, a car-following model of AVs considering real-time driving state is established. Secondly, based on bifurcation theory and stability theory, bifurcation analysis is carried out and the relationship between bifurcation and stability is revealed. In order to suppress the bifurcation and improve the stability, a controller considering multi-step prediction and memory mechanism (MPM) is designed, and the root trajectories for eigenvalues and stable time length of the model controlled by MPM controller are calculated. In response to the limitations of the MPM controller, a hybrid controller including the MPM controller and PID controller is further proposed, and it is found that the model controlled by hybrid controller has greater range of stable bifurcation parameter and stable time length, which means better ability of bifurcation suppression. Finally, the capabilities of the controller proposed in this paper are effectively demonstrated by numerical experiments in MATLAB and simulation experiments in the ROS-Gazebo environment. [ABSTRACT FROM AUTHOR]

We are concerned in this paper with dynamic behaviors and an optimal control problem of a new delayed epidemic model. There are three major ingredients. The first one is the dynamic behaviors of the state system. The locally asymptotic stability of the disease-free equilibrium and the endemic equilibrium are investigated and the effect of time delay on stability is also discussed. It is also found that the Hopf bifurcation appears at a specific time delay. The second, which is the main new ingredient of this paper, is an optimal control problem. Applying vaccine strategy in the system, an optimal control problem is proposed to minimize the total number of infected individuals as much as possible, maximize the total number of the uninfected individuals, and reduce the total control cost. In view of Pontryagin's maximum principle, the specific characteristics of the optimal control policy are given. The third ingredient is the numerical simulations of the theoretical results. • The model considers time delay, competition and spread of disease. • It studies dynamics and an optimal control problem of a delayed model. • Examples and numerical simulations are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the control and dynamic behavior of the inverted pendulum system with time delay under fractional Gaussian noise excitation. For H=1/2 and H ∈ (1 / 2 , 1) , we analyze the stochastic dynamic characteristics of the system under Hopf bifurcation, utilizing time delay and noise intensity as bifurcation parameters, and validate the theoretical conclusions through numerical simulations. We also defined the engineering application range of angle and angular velocity under both asymptotically stable and periodic oscillation dynamic states. Furthermore, using the stochastic I t o ˆ equation, we determined the values of time delay and noise intensity that satisfy the maximum engineering application range of angle and angular velocity, and verified their accuracy against the original equation. Additionally, we observed stochastic D-bifurcation and P-bifurcation arising from the combined effects of time delay and noise. Our results exhibit remarkable consistency between analytical and numerical findings, affirming the robustness of our approach and shedding light on the intricate dynamics of the system. • We investigate systems with time delays under fractional Gaussian noise excitation, thus paving the way for a deeper exploration of nonlinear phenomena in stochastic delay differential equations. • In the case of H = 1/2 and H ∈ (1/2, 1), we use different stochastic average methods to derive the Itô stochastic differential equation of the system, and verify the effectiveness and accuracy of the reduction through numerical simulations, and give the maximum time delay and noise intensity values for the engineering application range satisfying the angular and angular velocity under the excitation of Gaussian noise and fractional-order Gaussian noise, respectively. • Under H = 1 / 2 and H ∈ (1 / 2 , 1) , we utilize numerical simulations to unveil the complex dynamics of the system under different conditions. Simultaneously, we provide theoretical support for the control range of angle and angular velocity in engineering applications from another perspective, complementing the control section of the inverted pendulum. [ABSTRACT FROM AUTHOR]

In this paper, we use a cortex-basal ganglia resonance network to explore the possible Hopf bifurcation mechanism of beta oscillation. Different from traditional viewpoints, this model contains a direct inhibitory projection from the subthalamic nucleus (STN) to the cortex excitatory nuclei (EXN). First, we obtain the oscillation critical conditions of the delay by Hopf bifurcation analysis. Then, we explore the effects of some key parameters on Hopf bifurcation points. Delay is an indispensable factor in this model, oscillations will occur only when delay increases to a certain extent. Interestingly, the amplitude of oscillation increases with an increase in delay, but the trend of frequency is opposite; and heterogeneous delays have different effects on amplitude and frequency. The inhibitory coupling weight in the pathway " S T N → E X N " can reduce the firing activation level of cortical circuit, and the increase of coupling weights in the closed loop of the STN and globus pallidus external (GPe) can disinhibition this effect. On the stable boundary curve, the quantitative relationship of coupling weights between the direct and indirect projection from the basal ganglia to cortex is monotonously increasing. An increase in some critical coupling weights in the model, the bidirectional Hopf bifurcations (supercritical and subcritical) occur alternately, which can uniformly explain the origin mechanism of oscillations. In the inhibition-excitation closed loops, the excitatory and inhibitory connection weights have different effects on oscillations, the quantitative relationship of which is monotonically decreasing under Hopf bifurcation critical conditions. In the oscillation region, we observe that the amplitude is the smallest near bifurcation points and gradually increases with the evolution of oscillations. Finally, we find that the direct inhibitory projection from the STN to the EXN is a key pathway for external stimuli to control cortical oscillations. All theoretical analysis results compares well to numerical simulations, we hope that theoretical mechanisms obtained in this paper can inspire experimental research in Parkinson's disease. • The projection from the STN to the EXN can inhibit cortical circuit. • The quantitative relationship of various projections from BG to cortex is increasing. • The supercritical and subcritical Hopf bifurcations can occur alternately. • The quantitative relationship of different weights in closed loops is decreasing. • The oscillation amplitude is the smallest near bifurcation points. [ABSTRACT FROM AUTHOR]

Based on the study of scholars at home and abroad, this paper improves the three-dimensional IS-LM model in macroeconomics, analyzes the equilibrium point of the system and stability conditions, focuses on the parameters and complex dynamic characteristics when Hopf bifurcation occurs in the three-dimensional IS-LM macroeconomics system. In order to analyze the stability of limit cycles when Hopf bifurcation occurs, this paper further introduces the first Lyapunov coefficient to judge the limit cycles, i.e. from a practical view of the business cycle. Numerical simulation results show that within the range of most of the parameters, the limit cycle of 3D IS-LM macroeconomics is stable, that is, the business cycle is stable; with the increase of the parameters, limit cycles becomes unstable, and the value range of the parameters in this situation is small. The research results of this paper have good guide significance for the analysis of macroeconomics system. [ABSTRACT FROM AUTHOR]

In this paper, we present a complete parametric analysis on a nonlinear predator–prey system with the generalized Holling type III functional response and a strong Allee effect. We apply the hierarchical parametric analysis to derive explicit conditions for the existence and stability of equilibrium solutions in a 5-dimensional parameter space. Specifically, a detailed study on Hopf bifurcation is given to show bifurcation of multiple limit cycles, and complex dynamics of multistable states including bistable, tristable and tetrastable phenomena. We also conduct simulations and provide a biological interpretation of the multistable states under various conditions. • A predator–prey generalized Holling type III functional response and a strong Allee effect is investigated. • The hierarchical parametric analysis is applied to study the existence and stability of equilibrium solutions in a 5-dimensional parameter space. • A detailed study on Hopf bifurcation is given to show bifurcation of multiple limit cycles. • Complex dynamics of multistable states including bistable, tristable and tetrastable phenomena are explored. [ABSTRACT FROM AUTHOR]

Nonlinear dynamics and control of the galloping of elastic structures is an important research topic that has been addressed in the present paper. Some recent research has used high-frequency excitation in controlling galloping oscillations under steady wind. However, galloping under unsteady wind is potentially more dangerous because of the presence of an additional parametric instability on top of self-excited oscillation. No scientific study is thus far available in the literature concerning the control of galloping under unsteady wind. The present paper investigates the efficacy of high-frequency excitation, both parametric and direct, for controlling unsteady galloping. Analytical and numerical methods are employed to understand the dynamics of the system with and without high-frequency excitation. A detailed nonlinear analysis of the problem reveals rich dynamic characteristics of the system under different types of possible aerodynamic damping profiles. Overall, it is shown that both direct and parametric high-frequency excitation can effectively control the galloping oscillation. Direct excitation can completely eliminate the periodic as well as quasiperiodic galloping arising due to unsteady wind. However, parametric excitation can only limit the oscillation amplitude. • Effect of high-frequency excitations (HFE) on periodic and quasiperiodic galloping under unsteady wind flow is studied. • Method of direct partition of motion (MDPM) and method of multiple scales (MMS) are used. • Direct HFE can mitigate the periodic and quasiperiodic oscillation. • Parametric HFE can reduce the amplitude of the galloping response by shifting the resonance region. [ABSTRACT FROM AUTHOR]

• We introduce a simple two-parameter quadratic three-dimensional system. • It only has six terms, two nonlinearities and one equilibrium. • A homoclinic flip bifurcation of case inward twist C in is present. • Chaotic behavior with Smale horseshoes and strange attractors is warranted. • It is the first example of a 3D vector field exhibiting the case C in. In this paper we consider a two-parameter quadratic three-dimensional system with only six terms and two nonlinearities. First we analyze the Hopf bifurcation of its only equilibrium detecting several degeneracies. With this information we numerically obtain various bifurcation diagrams of periodic orbits in which saddle-node and period-doubling bifurcations as well as homoclinic connections appear. A careful study of the homoclinic orbits, in a region of the two-parameter plane where the equilibrium is a real saddle, shows the presence of a homoclinic flip bifurcation of case C. Here this orbit changes from orientable to non-orientable, being the lowest codimension for a homoclinic bifurcation to a real saddle equilibrium that results in chaotic behavior. More specifically, we determine that it corresponds to the case inward twist C in , in such a way that, as far as we know, it is the first example of a 3D vector field exhibiting this case. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the impact of delayed tax revenues on the outcomes of fiscal policy. The analytical framework is the Goodwin growth cycle model, which is founded on the Volterra predator-prey equations. We study the dynamic behavior of the system analytically proving the existence of Hopf bifurcations, which may be supercritical and subcritical. In the numerical simulations, which follows the qualitative analysis, it is shown that, given the degree of competition in the markets, fiscal policy purposes may become consistent with their real outcomes only if policy makers are able to control the delay in the structure of the tax system. Nevertheless, there are in the system elements out of the control of the policy makers. These elements imply the possibility to make partially ineffectual the stabilization policy, because of the risk to overcome the minimum public expenditure able to stabilize the system. [ABSTRACT FROM AUTHOR]

This paper is to investigate the motion and stability of a four-wheel-steering (4WS) vehicle with the nonlinearity of the lateral tyre forces, and the effects of the steering mechanism are considered as well. A new nonlinear model is developed for the steady state motion of the vehicle. The stability of a typical steady state motion generated by the model is also presented with application of bifurcation method. It is demonstrated in the paper that Hopf bifurcation of the steady state motion of the 4WS vehicle is evident. [Copyright &y& Elsevier]

In this paper, a mathematical model describing the interaction of malignant glioma cells, macrophages and glioma specific CD8+T cells is discussed. The biologically feasible equilibria and corresponding local stability are deduced. The bifurcations and related dynamical behaviors of this model are further studied thoroughly. The existence of transcritical bifurcation and saddle–node bifurcation is derived based on Sotomayor's theorem and Hopf bifurcation is well discussed. The codimension 2 bifurcation such as Bogdanov–Takens bifurcation is investigated using the normal form theory and center manifold theorem in more detail. Finally, numerical simulations are obtained to validate our analytical findings by varying the parameters. • The dynamics of a mathematical model describing the interaction of malignantglioma cells, macrophages and glioma specific CD8+T cells are discussed. • The co-existence of equilibria of the model are obtained by numerical simulation. • The existence of codimension 1 and codimension 2 bifurcations is investigated. • Extensive numerical simulations are obtained to validate our analytical findings. [ABSTRACT FROM AUTHOR]

This article considers a Cournot duopoly model in a continuous-time framework and analyze its dynamic behavior when the competitors are heterogeneous in determining their output decision. Specifically the model is expressed in the form of differential equations with discrete delays. The stability conditions of the unique Nash equilibrium of the system are determined and the emergence of Hopf bifurcations is shown. Applying some recent mathematical techniques (stability switching curves) and performing numerical simulations, the paper confirms how different time delays affect the stability of the economy. [ABSTRACT FROM AUTHOR]

This paper is comprehensively concerned with the dynamics of a class of high-dimension fractional ring-structured neural networks with multiple time delays. Based on the associated characteristic equation, the sum of time delays is regarded as the bifurcation parameter, and some explicit conditions for describing delay-dependent stability and emergence of Hopf bifurcation of such networks are derived. It reveals that the stability and bifurcation heavily relies on the sum of time delays for the proposed networks, and the stability performance of such networks can be markedly improved by selecting carefully the sum of time delays. Moreover, it is further displayed that both the order and the number of neurons can extremely influence the stability and bifurcation of such networks. The obtained criteria enormously generalize and improve the existing work. Finally, numerical examples are presented to verify the efficiency of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a discriminant for judging the stability of differential dynamical systems is given. The discriminant is a rational expression in the coefficients of the original system. How to transform a differential dynamical system to the normal form is shown, which is the origin of the discriminant. This method is more simple and practical than the traditional Lyapunov method. Examples are presented to show the applicability of the methodology and the convenience of the discriminant. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a cancer network including a protein module and a corresponding microRNA cluster that inhibits the synthesis of proteins. The existence of multiple steady states and their stability depending on the parameters are firstly determined. Bistability and dependency on the parameters, Hopf bifurcations and the corresponding properties like direction and stability of Hopf bifurcations are determined by computing the normal form on the center manifold. Then, the role of the delay in the process of synthesis of the protein is investigated. We show that the delay can stabilize the unstable equilibrium and destabilize the stable equilibrium. Some simulations are carried out to numerically illustrate the obtained theoretical results. Finally, the biological interpretation of the theoretical results is discussed. [ABSTRACT FROM AUTHOR]

In this paper, a delayed-diffusive predator-prey model with a ratio-dependent functional response subject to Neumann boundary condition is studied. More precisely, Turing instability of positive equilibrium, instability and Hopf bifurcation induced by time delay are discussed. In addition, by the theory of normal form and center manifold, conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived. Numerical simulations are conducted to illustrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

Calcium ions are important in cell process, which control cell functions. Many models on calcium oscillation have been proposed. Most of existing literature analyzed calcium oscillations using numerical methods, and found rich dynamical behaviours. In this paper, we explore a further study on an established three-store model, which contains endoplasmic reticulum (ER), mitochondria and calcium binding proteins. We conduct bifurcation analysis to identify two Hopf bifurcations, and apply normal form theory to study their stability and show that one of them is supercritical while the other is subcritical. Further, we transform the model into a slow-fast system, and then apply the geometrical singular perturbation theory to investigate the mechanism of generating slow-fast motions. The study reveals that the mechanism of generating the slow-fast oscillating behaviour in the three-store calcium model for certain parameter values is due to the relative fast change in the free calcium in cytosol, and relative slow changes in the free calcium in mitochondria and in the bounded Ca 2 + binding sites on the cytosolic proteins. A further parametric study may provide some useful information for controlling harmful effect, by adjusting the amount of calcium in a human body. Numerical simulations are present to demonstrate the correct analytical predictions. [ABSTRACT FROM AUTHOR]

Malware spreading, characterized by differential equations, in networked systems has become a common topic of study in the scientific literature. In this paper, we focus primarily on the propagation dynamics of malware in cyber–physical systems (CPSs). Although some work prior has been done to understand how the malware dynamics evolves in complex networks, these studies have restricted their attention to stability, bifurcation and oscillation based on ordinary differential equation models. Here, we address the question "how does one detect tipping phenomena which result in a total collapse of CPSs if the spread of malware is related to both time and space?" To this end, we use a reaction–diffusion equation to model spatio-temporal dynamic evolutions of malware in CPSs and accurately predict the emergence of tipping points due to the Turing instability and Hopf bifurcation for the first time. In order to further reveal the mechanism of tipping, the specific conditions of the Turing instability and Hopf bifurcation are given. It is confirmed that the introduction of space factor and transmission delay causes CPSs to change from stable to unstable, and then the tipping occurs. Moreover, the delay-dependent stability is discussed and the stability and direction of Hopf bifurcation are explored. These studies provide a strategic guidance for the prediction and control of malicious virus in CPSs. Finally, some numerical simulation examples are provided to substantiate the theoretical results. • A novel reaction-diffusion malware spreading model with multiple delays is proposed. • The basic reproductive ratio is calculated and the Turing instability induced by tipping is studied. • The tipping point of time delay is given and the direction of Hopf bifurcation is determined. • The spatial-temporal evolution of infected nodes in CPSs can be predicted quantitatively. [ABSTRACT FROM AUTHOR]

In this paper, the influence of the fear effect and Leslie–Gower function on the dynamic behavior of the predator–prey model is considered. First, the well-posedness analysis of this model is demonstrated, and the existence and local stability of the equilibrium point are given. Then by using the bifurcation theory and selecting the appropriate bifurcation parameters, many types of bifurcation phenomena in the system are discovered, including transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Meanwhile, the numerical simulations on the basis of theoretical analysis are further carried out to intuitively show the influence of fear effect on population. As the degree of fear increases, the system will undergo multiple dynamic behaviors switching until the final extinction of prey population, while the predator population will survive due to the presence of substitute prey. In addition, the initial density of predator and prey can determine whether the solution of system tends to a coexisting steady state or a periodic oscillation. • A novel prey–predator model with the impact of fear is introduced. • The system can experience many types of bifurcation phenomena. • Numerical simulations and sensitivity analysis are performed. • Our results show that fear effect plays an important role on deriving the complex dynamics. [ABSTRACT FROM AUTHOR]

In this paper two coupled (linear and nonlinear) equations are studied. These equations describe dynamics of surface plasmon polaritons at dielectric-metal-dielectric surfaces. Previously constructed bifurcation diagram of an analytical spatially homogeneous solution is refined and verified by direct simulations. Further research is conducted using numerical scheme based on spectral method. The 2/3 de-aliasing rule is used to cope with nonlinearity along with pseudo-spectral approach. An Andronov–Hopf bifurcation of a stable limit cycle and a period-doubling bifurcation of a stable limit torus were found. Floquet theory was applied to verify bifurcations. [ABSTRACT FROM AUTHOR]

In this paper we investigate stability conditions for small-amplitude periodic solutions emerging near symmetry-preserving Hopf bifurcations in a time-delayed fully-connected N-node PLL network. The study of this type of systems which includes the time delay between connections has attracted much attention among researchers mainly because the delayed coupling between nodes emerges almost naturally in mathematical modeling in many areas of science such as neurobiology, population dynamics, physiology and engineering. In a previous work it has been shown that symmetry breaking and symmetry preserving Hopf bifurcations can emerge in the parameter space. We analyze the stability along branches of periodic solutions near fully-synchronized Hopf bifurcations in the fixed-point space, based on the reduction of the infinite-dimensional space onto a two-dimensional center manifold in normal form. Numerical results are also presented in order to confirm our analytical results. [ABSTRACT FROM AUTHOR]

In this paper we study a three-dimensional tumor–immune system interaction model consisted of tumor cells, activated T cells, and immune checkpoint inhibitor anti-PD-1. Based on the uncontrollable character of tumor cells in the absence of immune response and treatment, the growth of tumor cells is assumed to be exponential. We discuss the distribution of equilibria qualitatively and study the stability of all possible equilibria with and without anti-PD-1 drug. When no drug is applied, the model has a tumor-free equilibrium and at most one tumorous equilibrium. Biologically, there exists a threshold d T 1 for the death rate d T of T cells: when d T ≥ d T 1 tumor cells will keep growing; when d T < d T 1 tumor cells may be eradicated for some positive initial values and keep growing for some other positive initial values. For the case with anti-PD-1 treatment, the model has at most five tumor-free equilibria and two interior equilibria. Our analysis indicates that there exists a threshold γ A 1 for the intravenous continuous injection γ A : when γ A ≤ γ A 1 the fate of tumor cells is the same as the case with no drug applied; when γ A > γ A 1 the model may exhibit bistable phenomena and periodic orbits. Furthermore, we establish the existence of local Hopf bifurcation around the interior equilibrium and determine the stability of the bifurcating periodic orbits. Our simulations show that the model exhibits a stable periodic orbit which implies the long term coexistence and balance of the tumor and immune system. • Study a tumor–immune system interaction model with an immune checkpoint inhibitor. • Discuss the number and stability of all possible equilibria. • Establish the existence of Hopf bifurcation and the stability of the periodic orbit. • Carry out numerical simulations to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the equivalence of the multiple time scales (MTS) method and the center manifold reduction (CMR) method for deriving the normal form of Hopf bifurcation for delayed reaction–diffusion equations. First, we generalize the MTS method associated with functional differential equations to delayed reaction–diffusion equation for solving the normal form of Hopf bifurcation. This provides a user-friendly approach of analyzing the stability of time-periodic solutions, particularly, for the engineering researchers who are not familiar with CMR method. Second, we outline the CMR method associated with the computation of normal form of Hopf bifurcation for delayed reaction–diffusion equations. Third, we prove that the two methods can derive equivalent up to the third order normal form of Hopf bifurcation. Finally, different types of examples are presented, such as reaction–diffusion equations with Neumann boundary condition and Dirichlet boundary conditions, single delay and two delays, spatially homogeneous and spatially inhomogeneous periodic solutions. We obtain the equivalent Hopf bifurcation normal forms results derived by using the CMR and MTS methods for above examples, respectively. • We generalize MTS method to Hopf bifurcation of delayed reaction–diffusion equation. • We outline CMR method for Hopf bifurcation of delayed reaction–diffusion equations. • We prove the equivalence of the two methods up to the third order normal form. • Four typical examples are presented. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the species interaction model with the ratio-dependent Holling III functional response and strong Allee effect. To explore nonhomogeneous solutions of the model, we consider the existence and non-existence of non-constant steady states and temporal bifurcation. Then we show the boundedness of the global positive solutions for the parabolic system and present the upper and lower bounds of positive solutions for the associated elliptic system. The non-existence and existence of the non-constant steady states of the elliptic system with the homogeneous Neumann boundary conditions are obtained by using the priori estimates, maximum principle and index theory. Furthermore, the existence and the direction of Hopf bifurcation are investigated via the stability analysis, center manifold theory and normal form reduction. Numerical simulations are carried out to verify our theoretical analysis and to illustrate that the ratio-dependent Holling III functional response and strong Allee effect have strong impact on dynamical behaviors of the species interaction systems. • Boundedness of parabolic and elliptic models is analyzed. • Nonconstant steady states are presented. • Hopf bifurcation with strong Allee effect is described. [ABSTRACT FROM AUTHOR]

In this paper, we systematically study Parkinson oscillation origin mechanism, oscillation amplitude and frequency characteristics in an improved cortex-basal ganglia (EXN-INN-STN-GPe) resonance model. Firstly, Hopf bifurcation critical conditions T 1 0 and T 2 0 for heterogeneous delays are derived, which are greatly affected by coupling weights. The delay is a key factor to affect the frequency of oscillation, many frequency bands (such as gamma oscillation, beta oscillation, alpha oscillation, etc.) appear with an increase in delay. According to T 1 0 and T 2 0 , the state of STN-GPe loop can be divided into four types: stable state, induced oscillation, spontaneous oscillation and compound oscillation, which represent different origin mechanism of oscillation. They have different amplitude characteristics, which increases with an increase in delay in most cases. The amplitude of the induced oscillation is smaller than the amplitude of the spontaneous oscillation in many cases. In the compound oscillation region, the EXN-STN antiphase oscillation mode can inhibit the oscillation amplitude in the STN-GPe. Induced oscillation may indicate that the oscillation origins in the cortex, which spreads to the STN-GPe loop with the evolution of disease. Self-feedback connections of EXN (W E E) and INN (W I I) play a very important role in affecting the amplitude of induced oscillation. The induced oscillation amplitude of the GPe is increased with an increase in W E E , but the trend is the opposite when it is affected by W I I. As an increase in coupling weight between the EXN and INN, the minimum of W E E and W I I required to induce oscillation are both increased. The theoretical analysis is in good agreement with the numerical simulation, the mechanism obtained may inspire experimental and new theoretical research in Parkinson's disease. • Hopf bifurcation critical conditions T 1 0 and T 2 0 for heterogeneous delays are obtained. • The delay is a key factor to affect the frequency of oscillation. • According to T 1 0 and T 2 0 , the state of STN-GPe loop can be divided into four types. • The amplitude of induced oscillation is smaller than that of spontaneous oscillation. • EXN-STN antiphase oscillation mode can inhibit oscillation amplitude in the STN-GPe. [ABSTRACT FROM AUTHOR]

This paper considers a mutualism system of three species in which each species provides resource for the next one in a one-directional loop, while there exists spatial competition among them. The system is characterized by a lattice gas model and the cases of obligate mutualisms, obligate–facultative mutualisms and facultative mutualisms are considered. Using dynamical systems theory, it is shown that (i) the mutualisms can lead to coexistence of species; (ii) A weak mutualism or an extremely strong mutualism will result in extinction of species, while even the superior facultative species will be driven into extinction by its over-strong mutualism on the next one; (iii) Initial population density plays a role in the coexistence of species. It is also shown that when there exists weak mutualism, an obligate species can survive by providing more benefit to the next one, and the inferior facultative species will not be driven into extinction if it can strengthen its mutualism on the next species. Moreover, Hopf bifurcation, saddle-node bifurcation and bifurcation of heteroclinic cycles are shown in the system. Projection method is extended to exhibit bistability in the three-dimensional model: when saddle-node bifurcation occurs, stable manifold of the saddle-node point divides int R + 3 into two basins of attraction of two equilibria. Furthermore, Lyapunov method is applied to exhibit unstability of heteroclinic cycles. Numerical simulations confirm and extend our results. [ABSTRACT FROM AUTHOR]

A discrete-time predator–prey model with Allee effect is investigated in this paper. We consider the strong and the weak Allee effect (the population growth rate is negative and positive at low population density, respectively). From the stability analysis and the bifurcation diagrams, we get that the model with Allee effect (strong or weak) growth function and the model with logistic growth function have somewhat similar bifurcation structures. If the predator growth rate is smaller than its death rate, two species cannot coexist due to having no interior fixed points. When the predator growth rate is greater than its death rate and other parameters are fixed, the model can have two interior fixed points. One is always unstable, and the stability of the other is determined by the integral step size, which decides the species coexistence or not in some extent. If we increase the value of the integral step size, then the bifurcated period doubled orbits or invariant circle orbits may arise. So the numbers of the prey and the predator deviate from one stable state and then circulate along the period orbits or quasi-period orbits. When the integral step size is increased to a critical value, chaotic orbits may appear with many uncertain period-windows, which means that the numbers of prey and predator will be chaotic. In terms of bifurcation diagrams and phase portraits, we know that the complexity degree of the model with strong Allee effect decreases, which is related to the fact that the persistence of species can be determined by the initial species densities. [ABSTRACT FROM AUTHOR]

Understanding the effect of slowly varying control parameters in dynamical systems is important in many fields such as mechanics, biology, ecology and social sciences, where normally changes in parameters take place very slowly. When a control parameter becomes time varying, the system dynamics exhibits a delay in bifurcation, i.e., the system responds to the bifurcation scenario with a lag in real time. In this paper, we experimentally explore the delay associated with Hopf and pitchfork bifurcations in a parametrically driven nonlinear oscillator. For this study we choose a generic nonlinear oscillator, namely the parametrically driven Murali–Lakshmanan–Chua (PDMLC) oscillator. We identify and characterize the occurrence of delay in bifurcations in both the rising and falling edges of the external force and measure the delay associated with these bifurcations in both the edges. We show that the delay in Hopf and pitchfork bifurcations increase when the rate of change of control parameter decreases. We further show that the delay obeys a power law as a function of the external frequency. All the numerical simulation results are corroborated with the real-time electronic circuit experiment and we find a good qualitative agreement between the numerical and experimental results. [ABSTRACT FROM AUTHOR]

In the present paper a family of angiogenesis models that is a generalisation of the Hahnfeldt et al. model is proposed. Considered family of models consists of two differential equations with distributed time delays. The global existence and the uniqueness of the solutions are proved. Moreover, the stability of the unique positive steady state is examined in the case of Erlang and piecewise linear delay distributions. Theorems guaranteeing the existence of stability switches and occurrence of Hopf bifurcations are proved. Theoretical results are illustrated by numerical analysis performed for parameters estimated by Hahnfeldt et al. (1999) [47]. [ABSTRACT FROM AUTHOR]

In this paper, an existing method is modified for computing the focal values and period constants of switching systems associated with elementary singular points. In particular, a quadratic switching system is considered to illustrate the computational efficiency of this method. Further, with this method, a cubic switching system is constructed to show existence of 15 limit cycles, which is the best result so far obtained for cubic switching systems. [ABSTRACT FROM AUTHOR]

We call the intercept of the price function with the vertical axis the maximum price and the slope of the price function the marginal price . In this paper it is assumed that a monopolistic firm has full information about the marginal price and its own cost function but is uncertain on the maximum price. However, by repeated interaction with the market, the obtained price observations give a basis for an adaptive learning process of the maximum price. It is also assumed that the price observations have fixed delays, so the learning process can be described by a delayed differential equation. In the cases of one or two delays, the asymptotic behavior of the resulting dynamic process is examined, stability conditions are derived. Three main results are demonstrated in the two delay learning processes. First, it is possible to stabilize the equilibrium which is unstable in the one delay model. Second, complex dynamics involving chaos, which is impossible in the one delay model, can emerge. Third, alternations of stability and instability (i.e., stability switches) occur repeatedly. [ABSTRACT FROM AUTHOR]

Abstract: Inspired by a simulation specific to a delayed HIV model with stage-structure, some dynamic behaviors are studied in this paper, including global stability of disease-free equilibrium and local Hopf bifurcation when taking the delay as a parameter. The corresponding characteristic equation is a transcendental equation, with the parameters delay-dependent, thus we use the conventional analysis introduced by Beretta and Kuang to obtain sufficient conditions to the existence of Hopf bifurcation. Then some properties of Hopf bifurcation such as direction, stability and period are determined, and several examples illustrate our results. [Copyright &y& Elsevier]

Abstract: On the basis of the geometric singular perturbation theory and the theory of delayed Hopf bifurcation in slow–fast systems with delay, the stability of nonlinear systems with slowly and periodically varying delay is investigated in this paper. Sufficient conditions ensuring asymptotic stability of those systems are obtained. Especially, though a time-varying delay usually increases complexity in the analysis of system dynamics and it usually deteriorates system stability as well, the study indicates that under certain conditions, the stability of the systems with a time-invariant delay only can be improved by incorporating a slowly and periodically varying part into the constant delay. Two illustrative examples are given to validate the analytical results. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate Hopf bifurcation and center stability of a predator–prey biological economic model. By employing the local parameterization method, Hopf bifurcation theory and the formal series method, we obtain some testable results on these issues. The economic profit is chosen as a positive bifurcation parameter here. It shows that a phenomenon of Hopf bifurcation occurs as the economic profit increases beyond a certain threshold. Besides, we also find that the center of the biological economic model is always unstable. Finally, some numerical simulations are given to illustrate the effectiveness of our results. [Copyright &y& Elsevier]

Abstract: Recently, the coupling time delay has been considered as the source of the occurrence of the phase-flip bifurcation in time-delay coupled system. But the analytical results of how the coupling time delay affects this phenomenon is still lacking. In this paper, we consider a pair of identical tri-neuron network coupled with time delay. By using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups, we investigate the spatio-temporal patterns of Hopf bifurcating periodic oscillations induced by the coupling time delay. The explicit intervals of delay and the regions in the plane of the coupling strength and the gain of the inherent response function for the existence of synchronized in-phase or anti-phase oscillation are obtained. Our study show that the coupling time delay does not affect the spatio-temporal patterns of the individual neural loop but it has the significant impact on the spatio-temporal patterns between the two loops. These analytic results are then verified by numerical simulations. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence of both Hopf bifurcation and topological horseshoe for a novel finance chaotic system. First, through rigorous mathematical analysis, we show that a Hopf bifurcation occurs at systems’ three equilibriums S 0,1,2 and Hopf bifurcation at equilibrium S 0 is non-degenerate and supercritical. Second, the computer-assisted verifications for horseshoe chaos in the system are given. Simulation results are presented to support the analysis. [Copyright &y& Elsevier]

Abstract: In this paper, a three-species predator–prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we first show that Hopf bifurcation at the positive equilibrium of the system can occur as τ crosses some critical values. Second, we obtain the formulae determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included. [Copyright &y& Elsevier]

Abstract: Using the system parameter instead of the delay as the bifurcation parameter, linear stability and Hopf bifurcation analysis including its direction and stability of bidirectional ring network model are investigated in this paper. The main tools to obtain our results are the normal form method and center manifold theory. Numerical simulations show that the theoretically predicted values are in excellent agreement with the numerically observed behavior. [Copyright &y& Elsevier]

Abstract: This paper presents the use of Thyristor-Controlled Series Capacitor (TCSC) to control bifurcations of subsynchronous resonance (SSR) in multimachine power system. The modified second system of the IEEE second benchmark model of subsynchronous resonance is considered. The dynamics of the damper windings, automatic voltage regulator (AVR), and power system stabilizer (PSS) on SSR in power system are included. In the case of neglecting TCSC, the results show that the operating point of the system loses stability via subcritical Hopf bifurcation. When we add TCSC to the system, all bifurcations are eliminated. Therefore, the Thyristor-Controlled compensated system never loses stability at any realistic firing angle. [ABSTRACT FROM AUTHOR]

Abstract: This paper reports a four-dimension (4D) fast-slow hyperchaotic system with the structure of two time scales by adding a slow state variable w into a three-dimension (3D) chaotic dynamical system, studies the stability and Hopf bifurcation of origin point. Furthermore, based on the fast-slow dynamical bifurcation analysis and the phase planes analysis, different bursting phenomena, symmetric fold/fold bursting, symmetric sub-Hopf/sub-Hopf bursting and chaotic bursting, as well as chaotic and periodic spiking, are observed in the fast-slow hyperchaotic system. Numerical simulations are presented to show these results. [ABSTRACT FROM AUTHOR]

Abstract: Recently applications of mathematical modelings for gene expressions have received much attention. In this paper, we study the following system of gene expressions with delayswhich originated from the pattern mechanism of somites involving oscillating gene expression for zebrafish. The delays on mRNA and protein are due to the time needed for the gene to make the mRNA molecule and for the ribosome to translate mRNA into the protein molecule. The total delay τ = T m + T p is used as a bifurcation parameter to show that this system can exhibit Hopf bifurcations at certain critical values of τ. For T m ≠ T p and T m = T p , the normal form theory for general DDEs developed by Faria and Magalhães is used to perform center manifold reduction and determine the stability and direction of periodic solutions generated by Hopf bifurcation. The global existence of periodic solutions when T m = T p and T p =0 is attained by using a result from Wu (1998) . Examples are given to confirm the theoretical results. [Copyright &y& Elsevier]

Abstract: We investigate the stability of the in-phase mode in a system of two delay-coupled bubble oscillators. The bubble oscillator model is based on a 1956 paper by Keller and Kolodner. Delay coupling is due to the time it takes for a signal to travel from one bubble to another through the liquid medium that surrounds them. Using techniques from the theory of differential-delay equations as well as perturbation theory, we show that the equilibrium of the in-phase mode can be made unstable if the delay is long enough and if the coupling strength is large enough, resulting in a Hopf bifurcation. We then employ Lindstedt’s method to compute the amplitude of the limit cycle as a function of the time delay. This work is motivated by medical applications involving noninvasive localized drug delivery via microbubbles. [Copyright &y& Elsevier]

Abstract: This paper presents an investigation of stability and Hopf bifurcation of the synaptically coupled nonidentical FHN model with two time delays. We first consider the existence of local Hopf bifurcations, by regarding the sum of the two delays as a parameter, then derive explicit formulas for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions, using the normal form method and center manifold theory. Finally, numerical simulations are carried out for supporting the theoretical analysis. [Copyright &y& Elsevier]

Abstract: The complex dynamical behaviors of the hexagonal governor system with a spring are studied in this paper. We go deeper investigating the stability of the equilibrium points in the hexagonal governor system with a spring. These systems have a rich variety of nonlinear behaviors, which are investigated here by numerically integrating the Lagrangian equations of motion. A tiny change in parameters can lead to an enormous difference in the long-term behavior of the system. Hyperchaotic behavior is also observed in cases where two of the Lyapunov exponents are positive, one is zero, and one is negative. The routes to chaos are analyzed using Poincaré maps, which are found to be more complicated than those of nonlinear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincaré sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. By studying numerical simulations, it is possible to provide reliable theory and effective numerical method for other systems. [Copyright &y& Elsevier]

Abstract: In this paper, a four-neuron BAM neural network with distributed delays is considered, where kernels are chosen as weak kernels. Its dynamics is studied in terms of local stability analysis and Hopf bifurcation analysis. By choosing the average delay as a bifurcation parameter and analyzing the associated characteristic equation, Hopf bifurcation occurs when the bifurcation parameter passes through some exceptive values. The stability of bifurcating periodic solutions and a formula for determining the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, numerical simulation results are given to validate the theorem obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the stability and Hopf bifurcation of a new regulated logistic growth with discrete and distributed delays. By choosing the discrete delay as a bifurcation parameter, we prove that the system is locally asymptotically stable in a range of the delay and Hopf bifurcation occurs as crosses a critical value. Furthermore, explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Finally, an illustrative example is also given to support the theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, a dynamical systems analysis is presented for characterizing the motion of a group of unicycles in leader–follower formation. The equilibrium formations are characterized along with their local stability analysis. It is demonstrated that with the variation in control gain, the collective dynamics might undergo Andronov–Hopf and Fold–Hopf bifurcations. The vigor of quasi-periodicity in the regime of Andronov–Hopf bifurcation and heteroclinic bursts between quasi-periodic and chaotic behavior in the regime of Fold–Hopf bifurcation increases with the number of unicycles. Numerical simulations also suggest the occurrence of global bifurcations involving the destruction of heteroclinic orbit. [Copyright &y& Elsevier]

Abstract: We use center manifold theory to analyze a model of gene transcription and protein synthesis which consists of an ordinary differential equation (ODE) coupled to a delay differential equation (DDE). The analysis involves reformulating the problem as an operator differential equation which acts on function space, with the result that an infinite dimensional system is reduced to one of two dimensions. This work extends a previous CNSNS paper in which this problem was treated by Lindstedt’s method. The present work is shown to provide approximations of general motions, including the approach to a periodic motion, in contrast to Lindstedt’s method, which approximates only the periodic motion itself. In particular we show that the origin is asymptotically stable for the critical (bifurcation) value of the delay parameter. [Copyright &y& Elsevier]

• A new reaction-diffusion HFMD model with nonsmooth saturation treatment function is introduced. • The stability of equilibria, and the existence of Turing instability, Hopf bifurcation for our model are analyzed. • Simulations demonstrate the validity of theoretical results. • The model is used to fit the accumulated cases of HFMD in Guangxi Province, China from January, 2016 to May, 2019. In this paper, we propose a new reaction-diffusion model to investigate the spatial spread of hand, foot and mouth disease (HFMD). To reflect the actual factors which are limited medical resources for treatment and intensive treatment strategies which depend on a certain threshold, we also consider a continuous and nonsmooth treatment function in our model. First, existences of equilibria are determined by discussing the relation of parameters. Second, for the local stability of equilibria, the effect of the spatial diffusion including the variation of the size of the space domain and the diffusion coefficients of susceptible and infectious individuals on the stability is studied and Turing instability is demonstrated. Third, spatially homogeneous Hopf bifurcation and spatially homogeneous Hopf bifurcation are also studied by analyzing the corresponding characteristic equation. Finally, we use the model to fit the reported cases of HFMD in Guangxi Province, China from January, 2016 to May, 2019. Our results containing theoretical analysis and numerical simulations show that reducing the threshold I c with respect to intensive treatment and reducing the range of activities of infectious individuals can effectively control the spread of HFMD. [ABSTRACT FROM AUTHOR]

• A simple direct method to determine a base of the complementary spaces for the Lie transform is given. • The normal forms for vector field with double purely imaginary eigenvalues with geometric multiplicity one are considered. • Explicit formulas for the normal forms coefficients with three unfolding parameter is given. • Using this method, the normal form for the non-semisimple 1:1 resonance Hopf bifurcation can be given easily. In this paper, a new and efficient mechanism to compute the normal forms for 1:1 resonant Hopf bifurcation is developed. For a vector field given by ordinary differential equations, by assuming that eigenvalues at an equilibrium point are purely imaginary, double, and non-semisimple, the mechanism provides a direct method to calculate the coefficients for the normal forms. In particular, we present the following results: (1) a simple direct method to determine a basis of the complementary spaces for the Lie transform; (2) a simple direct method to determine the projection of any vector in H 4 3 to the complementary spaces; and (3) the normal forms for vector fields with double purely imaginary eigenvalues with geometric multiplicity one. In addition, explicit formulas for coefficients with three unfolding parameters are obtained and the normal forms for the non-semisimple 1:1 resonant Hopf bifurcation are obtained. [ABSTRACT FROM AUTHOR]