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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]

This paper researches the stability and multi-frequency dynamic characteristics of nonlinear grid-connected pumped storage-wind power interconnection system (PS-WPIS). Firstly, a nonlinear model of grid-connected PS-WPIS is established. Then, the system stability and multi-frequency characteristics are revealed through stable domain and dynamic response analysis. Furthermore, the coupling mechanism of grid-connected PS-WPIS is explained, and the effect of capacity ratio on system stability is studied. Finally, the effect of hydraulic, mechanical and electrical parameters on grid-connected PS-WPIS is revealed. The results show that the stable domain of grid-connected PS-WPIS consists of two horizontal bifurcation lines and one curving bifurcation line. The former is related to wind power subsystem, and the latter is related to pumped storage subsystem. The grid-connected PS-WPIS contains the phenomenon of multi-frequency oscillations. The multi-frequency oscillations are generated by the coupling effect of pumped storage subsystem and wind power subsystem. The capacity increase of pumped storage or wind power worsens the stability and dynamic response of grid-connected PS-WPIS. The regulation performance of grid-connected PS-WPIS can be significantly improved by selecting smaller values of flow inertia time constant of penstock and time constant of wind turbine shafting. [ABSTRACT FROM AUTHOR]

The paper contributes to unveil how drivers—either human or not—may lose control of road vehicles after a disturbance. First, a simple vehicle-and-driver model is considered: Its motion is characterized by the existence of limit cycles whose amplitude depend on vehicle forward velocity (both oversteering and understeering vehicles may exibit this property). Such limit cycles are originated by a Hopf bifurcation occurring at a relatively high vehicle forward velocity. A mathematical proof of the existence of Hopf bifurcations is given. The existence of Hopf bifurcations and saddle limit cycles is confirmed by experimental tests performed by a dynamic driving simulator with a complex vehicle model and human in the loop. By a Zubov method, a Lyapunov function is derived to compute the region of asymptotic stability for the simple vehicle-and-driver model. A necessary and sufficient condition is derived for global asymptotic stability. Such a condition refers to the variation of the kinetic energy which must vanish at the end of the disturbed motion. This occurrence has been detected at the driving simulator too. Just a single stable equilibrium has been found inside the domain of attraction in all of the examined cases. [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]

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 have proposed a prey-predator model for the study of dynamical behaviors of three species such as toxin-producing Phytoplankton, Zooplankton and Fish in a fishery system. The stability condition, existence condition of equilibrium and bifurcation have also been established. In this paper, Holling type II functional response function has been considered to analysis of the proposed model. All equilibriums of the proposed system are determined, and the behavior of the system is also investigated near the positive equilibrium point. At the end, local stability of the system is analyzed by numerical illustrations. [ABSTRACT FROM AUTHOR]

It is usually essential to reveal the relationship between continuous-time systems and discrete-time ones. First, a discrete-time recurrent neural network is presented by the Euler scheme in this paper. Then, the time step size is set to a bifurcation parameter and frequency domain approach is adopted for Hopf bifurcation analysis. Moreover, the periodic solutions are obtained by the harmonic balance method; then the stability conditions are presented. The critical step size is determined with which the discrete-time recurrent neural network can inherit the stable state of the continuous-time one. Finally, one numerical example of the discrete-time recurrent neural network is given to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper focuses on the analysis of multiple types of time delays and fractional-order proportional-derivative (PD α ) controller on the dynamics of fractional-order p53 model. The multiple types of time delay schemes are dexterously designed during the bifurcation control for the proposed system, and the comparative study is elaborately performed on bifurcation control theoretically. By analyzing the corresponding characteristic equation, some explicit conditions for the local asymptotical stability of the trivial equilibrium can be given for the fractional-order controlled system. Analysis reveals that the negative feedback delay restrains the delay threshold, while the positive feedback delay promotes the delay threshold and makes the system more robust. System dynamics is affected by fractional order, which is more conducive to the generation of system oscillations. We have adapted control system techniques and designed a PD α controller, which is used to activate p53 protein. Numerical results display that the addition of PD α controller can increase the richness of the dynamic system. It is worth noting that the differential gain parameter k d of PD α controller has a significant impact on the system period, and the proportional gain parameter k p can amplify the system amplitude. In addition, k p can significantly advance the time threshold of system compared to k d . The PD controller is a good strategy to control the oscillation behaviors of the system. [ABSTRACT FROM AUTHOR]

A simplified wheelset model with a nonlinear smooth equivalent conicity function is taken into account. The goal is to investigate the influence of the different nonlinear equivalent conicity functions on the bifurcation characteristics. The equivalent conicity functions of different types are fitted smoothly by the measured data, especially when the lateral displacement of the wheelset is less than 3 mm. In addition to qualitative analysis of the stability and Hopf bifurcation of the equilibrium, the mechanism that the type of the nonlinear equivalent conicity affects the Hopf bifurcation characteristics is explained using the normal form theory. Analytical studies reveal that if the second-order derivative of the equivalent conicity function with respect to the lateral displacement of the wheelset at the equilibrium is positive (negative), the Hopf bifurcation of the wheelset system is subcritical (supercritical). The limit cycle motion caused by the Hopf bifurcation is analyzed, such as the fold bifurcation, the period-doubling bifurcation, the cusp bifurcation, and the fold-flip bifurcation. It is noted that the concept of equivalent conicity applied in this paper is a purely heuristic engineering concept, which has been applied by many railway engineers, while it is still short of a strict mathematical basis. [ABSTRACT FROM AUTHOR]

In the present paper, a spatially discretized diffusive Brusselator model with zero-flux boundary conditions is considered. Firstly, the global existence and uniqueness of the positive solution are proved. Then the local stability of the unique spatially homogeneous steady state is considered by analyzing the relevant eigenvalue problem with the aid of decoupling method. Hence, the occurrence conditions of Turing bifurcation and Hopf bifurcation for the model at this steady state are obtained. Meanwhile, the comparative simulations on the stability regions of the steady state between the spatially discretized diffusive Brusselator model and its counterpart in continuous space are given. Furthermore, the approximate expressions of the bifurcating periodic solutions are derived according to Hopf bifurcation theorem. The bifurcating spatially nonhomogeneous periodic solutions show the formation of a special kind of periodic structures for this model. Finally, numerical simulations are given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the forced vibration and dynamic stability of a simply supported axially moving beam coupled to a nonlinear energy sink with the aim of passive vibration control. The equations of motion of the coupled system are solved using harmonic balance and pseudo-arc-length continuation methods. The impacts of the beam velocity, excitation frequency, and magnitude of the external force on the behavior of the system are studied. It is observed that the absorber should be designed in a way that the strongly modulated response (SMR) and weakly modulated response occur in the response of the system when the moving beam is excited near its primary resonances. The results show that the saddle node and Hopf bifurcations boundaries, as well as the absorber performance, would be reduced by increasing the beam velocity. It is also realized that the excitation frequency in which the response of the system enters the SMR region is decreased by enhancing the beam velocity, as well as the magnitude of the external force. Using the results of this paper, one can readily simulate the vibration control of simply supported moving beams utilizing a nonlinear energy sink. [ABSTRACT FROM AUTHOR]

The anti-control of Hopf bifurcation is a common means used in the applications of chaos. Hopf bifurcation phenomenon can be advanced or delayed by the anti-control of Hopf bifurcation to meet the practical needs of engineering. At present, there are few research achievements on anti-control of Hopf bifurcation of low-dimensional chaotic systems, and the control process is complex, and the control effect is not outstanding. This paper takes the high-dimensional chaotic system with coexisting attractors as an example and uses dynamic state feedback control method to implement anti-control of Hopf bifurcation. By introducing adjusted parameters, Hopf bifurcation can be generated at any predetermined location of high-dimensional chaotic system to achieve the desired purpose of anti-control of Hopf bifurcation. The method for anti-control of Hopf bifurcation avoids the complicated calculation process and has remarkable control effect. According to the results of simulation, the anti-control of Hopf bifurcation for high-dimensional chaotic system is verified. [ABSTRACT FROM AUTHOR]

In this paper, we study a delayed diffusive predator–prey model with nonlocal competition in prey and habitat complexity. The local stability of coexisting equilibrium is studied by analyzing the eigenvalue spectrum. Time delay inducing Hopf bifurcation is investigated by using time delay as bifurcation parameter. We give some conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution by utilizing the normal form method and center manifold theorem. Our results suggest that only nonlocal competition and diffusion together can induce stably spatial inhomogeneous bifurcating periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, based on an original mass transfer model, we propose a new volatile organic compound (VOC) emission model which contains a homogeneous linear parabolic partial differential equation with mixed boundary conditions and a nonlinear ordinary differential equation. First, we transfer above equations to nonlinear delay differential equations by using an integral mean value theorem. Second, we prove the global asymptotic stability of the unique positive equilibrium for ordinary differential equation. Next, we show linear stability analysis and obtain the existence of Hopf and Hopf-zero bifurcations associated with the improved nonlinear delayed VOC emission system. Moreover, we derive explicit formulae for determining the stability of Hopf-bifurcating periodic solutions and the direction of Hopf bifurcation. Finally, detailed numerical simulations, such as local asymptotically stable equilibrium, global asymptotically stable equilibrium, the effects of parameters on VOC emission, are presented to demonstrate the applications of the theoretical results, and the explanations and comparisons for numerical results are also presented. In particular, the interesting simulation results associated with the global asymptotic stability of equilibrium for delayed differential equation for some region of delay are also shown. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate a delayed predator–prey model with stage structure for prey and Ivlev-type functional response. Four assumptions about this model are made as follows: (1) there are a single predator and a single prey population in the model; (2) the prey is divided by the age into two stage structures, immature and mature; (3) the trophic transfer from prey to predator is incomplete; (4) there are two time delays due to the time of maturation of prey and the time of gestation of predator. Some properties of equilibria for the model without delays are provided. Further, the stability of equilibria and existence of Hopf bifurcation are studied by discussing the different cases of time delays for the model with delays. We observe that delays can cause a stable equilibrium to become unstable one, even occur Hopf bifurcation when delays pass through their corresponding critical values. Meanwhile, we derive explicit formulae to determine the properties of Hopf bifurcation such as the direction of Hopf bifurcation and the stability of periodic solutions. Numerical simulations of all theoretical analyses are given for verifying our theoretical results. In the present work, we demonstrate that the validity and universality of oscillations induced by the time delays both theoretically and numerically. These results of this paper may be helpful for us to further understand the role of the critical values of time delays in stabilizing the predator–prey model. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic competition model with time delay and harvesting is investigated. By means of the stochastic center manifold reduction principle and stochastic averaging method, the model is simplified to a one-dimensional Markov diffusing process. The singular boundary theory and invariant measure are applied in analyzing the stochastic stability and bifurcation. The T-S fuzzy model of the system is constructed, and the H ∞ state feedback controller is designed to eliminate the instability phenomenon by using a linear matrix inequality approach. Finally, numerical simulations are given to demonstrate our results. [ABSTRACT FROM AUTHOR]

Considering that time delay (delay) is a common phenomenon in biological systems, reaction-diffusion equations with delay are widely used to study the dynamic mechanism of those systems, in which delay can induce the loss of stability and degradation of performance. In this paper, taking into account the inhomogeneous distribution of species in space and this can be considered as a random network, the pattern dynamics of a prey-predator network system with diffusion and delay are investigated. The effect of delay and diffusion on the network system is obtained by linear stability analysis, including the stability and Hopf bifurcation as well as Turing pattern. Our results show that the stability of the system changes with the value of delay. Moreover, we obtain Turing pattern related to the network connection probability and diffusion. Finally, the numerical simulation verifies our results. [ABSTRACT FROM AUTHOR]

In this paper, the dynamic behaviors of coupled reaction-diffusion neural oscillator system with excitatory-to-inhibitory connection and time delay under the Neumann boundary conditions are investigated. By constructing a basis of phase space based on the eigenvectors of Laplace operator, the characteristic equation of this system is obtained. Then, the local stability of zero solution and the occurrence of Hopf bifurcation are established by regarding the time delay as the bifurcation parameter. In particular, by using the normal form theory and center manifold theorem of the partial differential equation, the normal forms are obtained, which determine the bifurcation direction and the stability of the periodic solutions. Finally, two examples are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents stability and bifurcations of two synaptically coupled identical Hindmarsh-Rose neurons with one time delay. The parameters we choose contribute to the single neuron exhibiting excitable behavior with a unique stable equilibrium. In the absence of time delay, we find coupling-induced oscillations, i.e., the excitable neurons can fire with different types of regular or irregular periodic spiking/bursting behaviors due to the synaptic coupling. With the help of stability and bifurcation theory, the asymptotic stability of equilibrium, fold and Hopf bifurcation are studied from the corresponding characteristic equation. In case of time delay, a detailed Hopf bifurcation analysis is given. And an explicit formula about the coupling strength and time delay for the occurrence of Hopf bifurcation is derived, based on which a series of periodic orbits generate when time delay passes through the critical value. Finally, numerical simulations are carried out for supporting our theoretical results; meanwhile, branches of Hopf bifurcation curves are plotted in the two-parameter plane. [ABSTRACT FROM AUTHOR]

In recent research, we are interested in analyzing a double age dependence SIRS model. Our main interest is to seek the influence of the incubation period and the immunity period on the outbreak of contagious disease, these two periods can be considered as delays. These ones can generate different behaviors for the dynamical system, we focus in this paper on the Hopf bifurcation, which is discussed in detail. Further, the global behavior of solutions is studied using a suitable Lyapunov functional. We also expect that the incubation period duration may have an important impact on the value of the basic reproduction number R 0 , hence it can be considered as a control of the outbreak of the studied infectious disease. Numerically the obtained results are checked utilizing graphical representations. [ABSTRACT FROM AUTHOR]

In this paper, we show that the oscillations' amplitude increases and then saturates during the startup in the single-branch pulsating heat pipe (SBPHP) due to the interplay between an instability mechanism and a limiting mechanism produced by nonlinearities. The SBPHP is a small tube closed at one end (which is heated) and open at the other, in which sits a single vapor bubble followed by a liquid plug. Increasing the heater temperature above a critical value leads to oscillations of the liquid plug. The oscillations' amplitude initially increases over time and eventually saturates (the system reaches a steady state). To describe this dynamics, we start from a known theoretical SBPHP model. The model is derived from the momentum balance on the liquid plug which includes both the pressure difference between the vapor and the environment and the viscous friction between the liquid and the wall. Pressure in the vapor varies due to compression and expansion of the vapor bubble, but also due to evaporation and condensation as the liquid plug oscillates between the heat source and the heat sink. Both the pressure (described by perfect gas law) and the evaporation and condensation (limited in both the heat source and the heat sink) make the differential equations in the model nonlinear. By applying the center manifold reduction technique followed by the normal form reduction technique, we obtain an approximate analytical solution (describing the position and velocity of the liquid plug, and the mass of vapor, as a function of time). The analytical solution describes well the startup and the subsequent steady state. Using this solution, we show that the initial increase in oscillations' amplitude is due to the instability mechanism (produced by the interplay of phase change and viscous friction), while the following amplitude saturation is due to the pressure and phase-change nonlinearities, as the system reaches a steady state corresponding to a limit cycle. The limit cycle is created through a Poincaré–Andronov–Hopf bifurcation. We find that one can greatly increase the oscillations' amplitude in the steady state by increasing the equilibrium instability and by decreasing these nonlinearities. This can be done by increasing the phase change and by decreasing the friction. We conclude that the SBPHP dynamics should be viewed as a combination of an instability mechanism which tends to increase the oscillations' amplitude and a limiting mechanism produced by nonlinearities, which limits the amplitude. [ABSTRACT FROM AUTHOR]

In this paper, we delve into the Hopf bifurcation of a novel high-dimensional delayed fractional neural network model with bicyclic structure sharing a node. Under some mild assumptions, the developed model is equivalently transformed into a system with two delays representing, respectively, the sum of delays in each ring. By applying the formula of Coates's flow graph, we first derive the associated characteristic equation of the transformed system. Then, picking one delay as a control parameter and regarding the other one as a constant in its stable interval, we establish the respective critical value of each delay for the occurrence of Hopf bifurcation. It is shown that the stability behavior remains unchanged for the small control delay and will be lost once the delay passes through the critical value. Numerical simulations and sensitivity analysis are performed to validate the theoretical results and identify the most sensitive parameter to the Hopf bifurcation point. [ABSTRACT FROM AUTHOR]

In today's society, the Internet has become an important tool of our life due to its potential applications in various areas such as economics, industry, agriculture, medical and health care, and information processing. To understand and grasp the law of the Internet, many competitive web site models of the Internet and some phenomena related to World Wide Web have been investigated systematically. However, many scholars only study the integer-order competitive web site models of the Internet. Up to now, there are few papers that focus on the dynamics of fractional-order competitive web site models of Internet, which possess memory property. In this paper, we are concerned with the stability and the existence of Hopf bifurcation of a fractional-order competitive web site model of Internet. By choosing the time delay as parameter and applying the Routh–Hurwitz criteria, we will establish a new sufficient condition guaranteeing the stability and the existence of Hopf bifurcation for fractional-order competitive web site model of Internet. The research reveals that fractional order and the delay play a key role in describing the stability and Hopf bifurcation of the considered system. Computer simulations are implemented to support the analytic results. Finally, a simple conclusion is presented. The theoretical findings of this article have a great significance in handling the competition dynamics among different web sites. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the complex dynamics of a predator–prey model with disease in the prey, which is characterized by the reduction of prey growth rate due to the anti-predator behavior. The value of this study lies in two aspects: Mathematically, it provides the existence and the stability of the equilibria and gives the existence of Hopf bifurcation. And epidemiologically, we find that the influence of the fear factor is complex: (i) The level of the population density decreases with the increasing of the fear factor; (ii) the cost of fear can destabilize the stability and benefit the emergency of the periodic behavior; and (iii) the high level of fear can induce the extinction of the predator. These results may enrich the dynamics of the eco-epidemiological systems. [ABSTRACT FROM AUTHOR]

In this paper, we study the global dynamics of a nonsmooth Rayleigh–Duffing equation x ¨ + a x ˙ + b x ˙ | x ˙ | + c x + d x 3 = 0 for the case d > 0 , i.e., the focus case. The global dynamics of this nonsmooth Rayleigh–Duffing oscillator for the case d < 0 , i.e., the saddle case, has been studied in the companion volume (Wang and Chen in Int J Non-Linear Mech 129: 103657, 2021). The research for the focus case is more complex than the saddle case, such as the appearance of five limit cycles and the gluing bifurcation which means that two double limit cycle bifurcation curves and one homoclinic bifurcation curve are very adjacent. We present bifurcation diagram, including one pitchfork bifurcation curve, two Hopf bifurcation curves, two double limit cycle bifurcation curves and one homoclinic bifurcation curve. Finally, numerical phase portraits illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a 4D autonomous system with complex hyper-chaotic dynamics is introduced. The Lyapunov exponent spectrum, bifurcation diagram and phase portrait are provided. Basic dynamical properties are also analyzed. In order to clarify the evolution of the complex dynamic behaviors of the system, the local bifurcation is studied and a Hopf bifurcation is proved to occur when the appropriate bifurcation parameter passes the critical value. All the conditions of Hopf bifurcation are derived by applying center manifold theorem and Poincaré-Andronov-Hopf bifurcation theorem. Numerical simulation results are consistent with the theoretical analysis. Besides, we present a rigorous study on the hyper-chaotic system by combining the topological horseshoe theory with a computer-assisted approach of Poincaré maps. Utilizing the algorithm for finding horseshoes in 3D hyper-chaotic maps, a horseshoe with two-directional expansion in the 4D hyper-chaotic system has been found, which rigorously proves the existence of hyper-chaos in theory. [ABSTRACT FROM AUTHOR]

In this paper, the lateral mathematical model of a railway wheel-set with two degrees of freedom has been built. To study the effect of yaw damper on the stability of high-speed wheel-set, the constrain between the wheel-set and the bogie is assumed as rigid. In this lateral model, only the nonlinear wheel/rail contact relationship has been taken into consideration in the lateral direction, due to both the nonlinear relationship between lateral displacement and contact angle of the wheel and the rail, and the nonlinear relationship between lateral displacement and the equivalent radius of the wheel at the contact point. And the nonlinear parameters are attained by using polynomial interpolation. By using Center Manifold Theorem, the method of Normal Form and Poincaré method, the model is reduced to a planar dynamical system, and the symbolic expression of the first-order fine focus is given, which can be used to determine which kind of bifurcation will occur at the critical speed. The simulation is established by using the parameters of the wheel-set of Chinese high-speed railway vehicle CRH3, and the result is consistent with the determination of the first-order fine focus. At last, the influence of different parameters on the stability of CRH3 wheel-set is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we have pointed out a time-delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis type functional responses. We discuss the structure of nonnegative equilibria and their stability analysis. Also, the invariance and boundedness of the model system are investigated. It is observed that delay destabilizes the system when the value of delay crosses its critical value. When the delay parameter is taken equals to its critical value, Hopf bifurcation occurs. Conditions for the system to become globally asymptotically stable at nonzero equilibria are also obtained. By applying the normal form theory and center manifold theorem, stability, direction, and period of bifurcating solutions of the model system are calculated. It is observed that supercritical Hopf bifurcation occurs for a set of parameter values, and the bifurcating periodic solutions are stable with decreasing period. Following the global Hopf bifurcation result of Wu (Trans Am Math Soc 35:4799-4838, ) for functional differential equations, we have shown that the local Hopf bifurcation implies the global Hopf bifurcation. The theoretical results are justified by presenting some numerical simulations. [ABSTRACT FROM AUTHOR]

Considering the influence of environmental capacity and forgetting on rumor spreading, we improve the traditional SIR (susceptible–infected–removed) rumor propagation model and give two dynamic models of rumor propagation in heterogeneous environment and homogeneous environment, respectively. The main purpose of this paper is to make a dynamic analysis of rumor propagation models. In the spatial heterogeneous environment, we have analyzed the uniform persistence of the rumor propagation model and the asymptotic behavior of the positive equilibrium point when the diffusion rate of rumor–susceptible tends to zero. In the spatial homogeneous environment, we discuss the stability of rumor propagation model. Further, optimal control and the necessary optimality conditions are obtained by using the maximum principle. Finally, we study the Hopf bifurcation phenomenon through inducing time delay in the reaction–diffusion model. In addition, the existence of Hopf bifurcation is verified and the influence of diffusion coefficients is studied by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, a class of complex-valued neural networks with two time delays is considered. By considering that the activation function can be expressed by separating into its real and imaginary part and regarding the sum of time delays as a bifurcating parameter, the dynamical behaviors that include local asymptotical stability and local Hopf bifurcation are investigated. By analyzing the associated characteristic equation, the Hopf bifurcation occurs when the sum of time delays passes through a sequence of critical value. The linearized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results. [ABSTRACT FROM AUTHOR]

Descriptor systems, which are also called differential-algebraic systems, singular systems, degenerate systems, constrained systems and so on, have been one of the major research topics in engineering field. However, problems related to the local stability and the Hopf bifurcation of nonlinear descriptor systems with time delay have not been thoroughly investigated. In this paper, we consider the dynamical behavior of two-dimensional nonlinear descriptor systems with time delay. First, local stability of the system is analyzed using the location of the roots of the characteristic equation of the corresponding linearized system. It is well known that the equilibrium of the linearized system is locally asymptotically stable if all roots of the corresponding characteristic equation locate in the left half of the complex plane, and they are uniformly bounded away from the imaginary axis. Otherwise, the equilibrium is unstable. The conditions for the existence of Hopf bifurcation are investigated in detail by using the time delay as a bifurcation parameter. Furthermore, based on the descriptor and the 'neutral-type' model transformation, some special neutral differential equations can be equivalently transformed into the nonlinear descriptor systems. Finally, the correctness and the effectiveness of the theoretical analysis are justified by numerical examples. [ABSTRACT FROM AUTHOR]

This paper describes the dynamics of an infectious disease transmission modified Leslie-Gower type eco-epidemiological system in both deterministic and stochastic fluctuating environments with harvesting. The dynamics of the deterministic system is extensively investigated around coexistence equilibria. Sufficient conditions are derived for local and global stability of the system. The existence of Hopf bifurcation phenomenon is examined around interior equilibria of the system. Subsequently, we use normal form method and center manifold theorem to examine the nature of the Hopf bifurcation. The obtained results are useful to extract the criteria for disease extinction and control perspective. Later, a white noise term is incorporated to the system to describe the dynamics of the system in stochastic fluctuating environment. Sufficient conditions are derived for the mean square stability (MSS) of the system which can be used to evaluate necessary conditions for the asymptotic MSS and a threshold condition between asymptotic MSS and unstable system. Finally, some numerical simulations are carried out, and graphical illustrations are given in support of the analytical results obtained in both deterministic and stochastic systems. [ABSTRACT FROM AUTHOR]

The railway bogie, the most important running component, has direct association with the dynamic performance of the whole vehicle system. The bifurcation type of the bogie that is affected by vehicle parameters will decide the behavior of the vehicle hunting stability. This paper mainly analyzes the effect of the yaw damper and wheel tread shape on the stability and bifurcation type of the railway bogie. The center manifold theorem is adopted to reduce the dimension of the bogie dynamical model, and the symbolic expression for determining the bifurcation type at the critical speed is obtained by the method of normal form. As a result, the influence of yaw damper on the bifurcation type of the bogie is given qualitatively in contrast to typical wheel profiles with high and low wheel tread effective conicities. Besides, the discriminant of bifurcation type for the wheel tread parameter variation is given which depicts the variation tendency of dynamics characteristics. Finally, numerical analysis is given to exhibit corresponding bifurcation diagrams. [ABSTRACT FROM AUTHOR]

The problem of determining the bounds of applicability of perturbation expansions in terms both of the system parameters and the state-space variable amplitude is a key point in the perturbation analysis of nonlinear systems. In the present paper an analysis in a finite neighborhood of a Hopf bifurcation is presented in order to analyze the conditions under which a Normal Form zero-divisors-based approach fails to describe the local dynamics and, therefore, a small divisor approach is required. The condition of “smallness” referred to the divisors is analyzed from both a qualitative and a quantitative point of view. Finally, a simple but effective analytical and numerical example is introduced to illustrate the theoretical issues along with an interpretation within a codimension-two framework. [ABSTRACT FROM AUTHOR]

Understanding how species' diffusion and environmental heterogeneity affect natural population dynamics is crucial in wildlife management and biodiversity conservation. In this paper, we consider a predation model with Beddington–DeAngelis functional response, where the prey moves between source–sink patches. Using Lyapunov theory and graph method, we demonstrate equilibrium stability, supercritical/subcritical bifurcations and persistence of the system. It is shown that there may exist two limit cycles in the system, and varying a diffusion rate can lead to transition of species' interaction outcomes between coexistence in periodic oscillation, persistence at a steady state, extinction of the predator and extinction of both species in a smooth fashion. Rigorous analysis on the equilibria shows that total population abundance of a diffusing prey may exceed total population abundance of a non-diffusing prey, even when the predator persists. Asymmetry in diffusion is also shown to play a role in the increase. [ABSTRACT FROM AUTHOR]

In this paper, we have considered a deterministic epidemic model with logistic growth rate of the susceptible population, non-monotone incidence rate, nonlinear treatment function with impact of limited hospital beds and performed control strategies. The existence and stability of equilibria as well as persistence and extinction of the infection have been studied here. We have investigated different types of bifurcations, namely Transcritical bifurcation, Backward bifurcation, Saddle-node bifurcation and Hopf bifurcation, at different equilibrium points under some parametric restrictions. Numerical simulation for each of the above-defined bifurcations shows the complex dynamical phenomenon of the infectious disease. Furthermore, optimal control strategies are performed using Pontryagin's maximum principle and strategies of controls are studied for two infectious diseases. Lastly using efficiency analysis we have found the effective control strategies for both cases. [ABSTRACT FROM AUTHOR]

In this paper, we consider a delayed food-limited model with feedback control. By regarding the delay as the bifurcation parameter and analyzing the corresponding characteristic equations, the linear stability of the system is discussed, and Hopf bifurcations are demonstrated. By the normal form and the center manifold theory, the explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some examples are presented to verify our main results. [ABSTRACT FROM AUTHOR]

Considering the macroeconomic model of money supply, this paper carries out the corresponding extension of the complex dynamics to macroeconomic model with time delays. By setting the parameters, we discuss the effect of delay variation on system stability and Hopf bifurcation. Results of analysis show that the stability of time-delay systems has important significance with the length of time delay. When time delay is short, the stable point of the system is still in a stable region; when time delay is long, the equilibrium point of the system will go into chaos, and the Hopf bifurcation will appear in certain conditions. In this paper, using the normal form theory and center manifold theorem, the periodic solutions of the system are obtained, and the related numerical analysis are also given; this paper has important innovation-theoretical value and acts as important actual application in macroeconomic system. [ABSTRACT FROM AUTHOR]

This paper investigates mixed-mode oscillations (MMOs) with a three-dimensional conductance-based cardiac action potential model, which makes the heart beat in a nonrenewable way. The 3D model was entailed by utilizing voltage-dependent timescales to describe the mechanism in which MMOs are generated. As expected, motivated by geometric singular perturbation theory, our analysis explains in detail the geometric mechanisms that there is a range of parameters under which the cardiac model highlights that the presence of MMOs is induced by the intrinsical canard phenomenon. Much is currently known about the geometric mechanisms, for a folded saddle, the two singular canards perturb to maximal canards. Characteristics of the stimulus current such as frequency and duration determine which early afterdepolarizations (EADs), as a special case of MMOs bears, as well as the article compares the detailed manifold structures of original and dimensionless systems with square wave pulses by setting the pacing cycle length. An exceedingly vital technique of the analysis is the slow–fast dynamics analysis by which the system governs multiple timescale structures analytically. A more novel and successful multiple-timescale approach divides the system so that there is only one fast variable and demonstrates that the MMOs arise from canard dynamics, such as using a three-variable model in which two variables are treated as "slow" and one treated as "fast", which the layer problem and the reduced problem are considered to explain the trajectory on the critical manifold. Meanwhile, if one variable was regarded as the single slow variable, substantial bifurcation properties are discovered for slow–fast system, as well as general one-parameter bifurcation type is discussed for the whole system similarly. By focusing on the first Lyapunov coefficient of the Hopf bifurcation, which decides whether the bifurcation is supercritical or subcritical, it was shown that an unstable limit cycle can arise via a delayed subcritical Hopf bifurcation for the original system. Meanwhile, the dynamical studies of cardiac model have major implications for further elaborating the complex dynamic behaviors, such as EADs, which can lead to tissue-level arrhythmias. Ultimately, it has turned these researches into a considerable player in the signal and information transmission for underlying nervous systems. [ABSTRACT FROM AUTHOR]

The Hopf bifurcation behavior is an important issue for the nonlinear dynamic analysis of gas foil bearing (GFB)-rotor systems. However, there is a lack of detailed study on different types of Hopf bifurcation and their corresponding characteristics for GFB-rotor systems. This paper is intended to provide a clear and systematic insight into the nonlinear dynamic characteristics of GFB-rotor systems with a supercritical or a subcritical Hopf bifurcation. The onset speed (OS) of instability (i.e., the bifurcation point) for the system is calculated by the linear stability analysis. The periodic solution of the system before or after the bifurcation point is obtained by the shooting method, and its stability is assessed by the Floquet multipliers. The shock stability and the unbalanced response characteristics of the GFB-rotor system with a supercritical or a subcritical Hopf bifurcation are presented. A GFB-rotor system with a supercritical Hopf bifurcation shows better dynamic characteristics than a system with a subcritical Hopf bifurcation. The parameter analysis reveals that the aspect ratio and the foil stiffness of the GFBs have obvious effects on the Hopf bifurcation type, while the loss factor has a relatively small effect. It is remarkable that although a lower foil stiffness increases the OS of instability, the actual speed limit would probably decrease as the Hopf bifurcation changes from a supercritical to a subcritical type. This can contribute to an understanding of the necessity of studies on actual available operating speed based on nonlinear analysis rather than conventional linear analysis for the bearing design. [ABSTRACT FROM AUTHOR]

The Hopf–Hopf bifurcations of an aircraft nose landing gear model are investigated in this paper based on two pairs of continuation parameters. It is shown that due to the effect of higher-order terms, there exists a deviation between the bifurcation curve of the original system and the bifurcation curve of its truncated amplitude system, which results in a contraction in the bistable region compared with the previous work. Two codimension 3 strong resonant points can be detected on the Neimark–Sacker bifurcation curve. Period-3 and period-4 limit cycles with torsional-lateral shimmy can emerge when the vertical force and the taxiing velocity are chosen near the resonant points. The occurrence of 1:3 and 1:4 resonant Hopf–Hopf bifurcations can induce tire-ground vibrations in torsional and lateral direction, which can greatly degrade the performance and safety and possibly destroy the airstrip. So it is of significance to investigate the resonate behaviors of the aircraft nose landing gear to ensure the safe landing. [ABSTRACT FROM AUTHOR]

In this paper, dynamic behaviors of a turbidostat model with distributed delay are concerned. Hopf bifurcations arise when the value of bifurcation parameter, the time delay of translation for the nutrient, crosses some critical values. Firstly, the type and stability of bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Moreover, the destabilization of periodic solutions is also discussed. Finally, numerical simulation results are given to support the theoretical conclusions. [ABSTRACT FROM AUTHOR]