Analysis of the stability and Hopf bifurcation of money supply delay in complex macroeconomic models

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.     

Bifurcations in a predator-prey model with discrete and distributed time delay

In this paper, a class of predator-prey model with discrete and distributed time delay is considered. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. By using the normal form theory and center manifold theory, we derive some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, main conclusions are included.     

Dynamics analysis of a non-smooth Filippov pest-natural enemy system with time delay

In this paper, we propose a non-smooth Filippov system that describes the interaction of the pest and natural enemy with considering time delay, which represents the change in the growth rate of natural enemies before it is released to prey on pests. When the number of the pest is below the threshold, no control is applied; otherwise, control measures will be adopted. We discuss the stability of the equilibria and the existence of Hopf bifurcation. The results show that the Hopf bifurcation occurs when the time delay passes through some critical values. By applying the Filippov convex method, we obtain the dynamics of the sliding mode. The solutions of the system eventually tend toward the regular equilibrium, the pseudo-equilibrium or a standard periodic solution. Numerical simulations show that time delay plays an important role in local and global sliding bifurcations. We can obtain boundary focus bifurcations from boundary node bifurcations by varying time delay. Furthermore, touching, buckling and crossing bifurcations can be obtained frequently by increasing time delay. The results can provide some insights in pest control.

     

Dynamics of an inertial two-neuron system with time delay

In this paper, we considered a delayed differential equation modeling two-neuron system with both inertial terms and time delay. By analyzing the distribution of the eigenvalues of the corresponding transcendental characteristic equation of its linearized equation, local stability criteria are derived for various model parameters and time delay. By choosing time delay as a bifurcation parameter, the model is found to undergo a sequence of Hopf bifurcation. Furthermore, the direction and the stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Also, resonant codimension-two bifurcation is found to occur in this model. Some numerical examples are finally given for justifying the theoretical results. Chaotic behavior of this inertial two-neuron system with time delay is found also through numerical simulation, in which some phase plots, waveform plots, power spectra and Lyapunov exponent are computed and presented.     

Stability and Hopf bifurcation of a two-dimensional supersonic airfoil with a time-delayed feedback control surface

The time-delayed feedback control for a supersonic airfoil results in interesting aeroelastic behaviors. The effect of time delay on the aeroelastic dynamics of a two-dimensional supersonic airfoil with a feedback control surface is investigated. Specifically, the case of a 3-dof system is considered in detail, where the structural nonlinearity is introduced in the mathematical model. The stability analysis is conducted for the linearized system. It is shown that there is a small parameter region for delay-independently stability of the system. Once the controlled system with time delay is not delay-independently stable, the system may undergo the stability switches with the variation of the time delay. The nonlinear aeroelastic system undergoes a sequence of Hopf bifurcations if the time delay passes the critical values. Using the normal form method and center manifold theory, the direction of the Hopf bifurcation and stability of Hopf-bifurcating periodic solutions are determined. Numerical simulations are performed to illustrate the obtained results.     

Stability and global Hopf bifurcation in toxic phytoplankton–zooplankton model with delay and selective harvesting

Consider that some zooplankton can be harvested for food and some phytoplankton can liberate toxin; a toxin producing phytoplankton–zooplankton model with delay and selective harvesting is proposed and investigated. We discuss the stability of equilibria and perform the analysis of Hopf bifurcation. More precisely, the global asymptotical stability of equilibria is investigated by the Lyapunov method and Dulac theorem. In addition, the computing formulas of stability and direction of the Hopf bifurcating periodic solutions are also given. Furthermore, we prove that there exists at least one positive periodic solution as a time delay varies in some regions by using the global Hopf-bifurcation result of Wu (Trans. Am. Math. Soc. 350:4799–4838, 1998) for functional differential equations. Finally, the impact of harvesting is discussed along with numerical results to provide some support to the analytical findings.     

Hopf Bifurcation on a Two-Neuron System with Distributed Delays: A Frequency Domain Approach

In this paper, a more general two-neuron model with distributed delays and weak kernel is investigated. By applying the frequency domain approach and analyzing the associated characteristic equation, the existence of bifurcation parameter point is determined. Furthermore, we found that if the mean delay is used as a bifurcation parameter, Hopf bifurcation occurs for the weak kernel. This means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value. The direction and stability of the bifurcating periodic solutions are determine by the Nyquist criterion and the graphical Hopf bifurcation theorem. Some numerical simulations for justifying the theoretical analysis are also given.     

The new result on delayed finance system

In this paper, a finance system with time delay is considered. By linearizing the system at the unique equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the unique equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.     

Stability, bifurcation and chaos of a delayed oscillator with negative damping and delayed feedback control

This paper presents a detailed analysis on the dynamics of a delayed oscillator with negative damping and delayed feedback control. Firstly, a linear stability analysis for the trivial equilibrium is given. Then, the direction of Hopf bifurcation and stability of periodic solutions bifurcating from trivial equilibrium are determined by using the normal form theory and center manifold theorem. It shows that with properly chosen delay and gain in the delayed feedback path, this controlled delayed system may have stable equilibrium, or periodic solutions, or quasi-periodic solutions, or coexisting stable solutions. In addition, the controlled system may exhibit period-doubling bifurcation which eventually leads to chaos. Finally, some new interesting phenomena, such as the coexistence of periodic orbits and chaotic attractors, have been observed. The results indicate that delayed feedback control can make systems with state delay produce more complicated dynamics.     

Rich dynamics in a non-local population model over three patches

We consider a system of nonlinear delay differential equations that describes the growth of the mature population of a species with age-structure living over three patches. We analyze existence of non-negative homogeneous equilibria and their stability and discuss possible Hopf bifurcation from these equilibria. More precisely, by employing both the standard Hopf bifurcation theory and the symmetric bifurcation theory for functional differential equations, we obtain very rich dynamics for the system, including bistable equilibria, transient oscillations, synchronous periodic solutions, phase-locked periodic solutions, mirror-reflecting waves and standing waves.     

A delayed predator–prey model with strong Allee effect in prey population growth

In this paper, we consider a delayed predator-prey system with intraspecific competition among predator and a strong Allee effect in prey population growth. Using the delay as bifurcation parameter, we investigate the stability of coexisting equilibrium point and show that Hopf-bifurcation can occur when the discrete delay crosses some critical magnitude. The direction of the Hopf-bifurcating periodic solution and its stability are determined by applying the normal form method and the centre manifold theory. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using the global Hopf-bifurcation result of Wu ({Trans. Am. Math. Soc.} 350:4799?C4838, 1998) for functional differential equations, we establish the global existence of periodic solutions. Numerical simulations are carried out to validate the analytical findings.     

Complex dynamics of a four neuron network model having a pair of short-cut connections with multiple delays

This paper deals with dynamic behaviors on Hopfield type of ring neural network of four neurons having a pair of short-cut connections with multiple time delays. By suitable transformation and under certain assumptions on multiple time delays, the model is reduced to four dimensional nonlinear delay differential equations with three delays. Regarding these time delays as parameters, delay independent sufficient conditions for no stability switches of the trivial equilibrium of the linearized system are derived. Conditions for stability switching with respect to one delay parameter which is not associated with short-cut connection are obtained. Hopf bifurcations with respect to two other delays which are associated with short-cut connection are also obtained. Using the normal form method and center manifold theory, the direction of the Hopf bifurcation, stability and the properties of Hopf-bifurcating periodic solutions are determined. Using numerical simulations of the nonlinear model, different rich dynamical behaviors such as quasiperiodicity, torus attractor and chaotic-bands are also observed for suitable range of three delay parameters. Lyapunov exponents are also calculated using the AnT 4.669 tool for verification of chaotic dynamics.     

Global stability and Hopf-bifurcation in a zooplankton–phytoplankton model

We deal with a predator–prey model, representing a resource (phytoplankton) and two predators (zooplankton) system with toxin-producing delay. The response function is assumed here to be concave in nature. Firstly, the stability criterion of the model is analyzed both from a local and a global point of view. Our results imply that the toxin’s intrinsic characteristics, such as toxic liberation rate and toxin-producing delay, will not change the stability of the system irreversibly. Secondly, Hopf bifurcation of both systems with delay and without delay can occur via system parameters pertaining to the toxin. Our results indicate that the toxin produced by phytoplankton may be used as a bio-control agent for the Harmful Algal Bloom problems. Furthermore, the explicit algorithm for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained using the normal form method and center manifold theorem. Finally, some numerical simulations are carried out to illustrate the results.     

Hopf bifurcation and spatio-temporal patterns in?delay-coupled van der Pol oscillators

In this paper, the dynamics of a pair of van der Pol oscillators with delayed velocity coupling is studied by taking the time delay as a bifurcation parameter. We first investigate the stability of the zero equilibrium and the existence of Hopf bifurcations induced by delay, and then study the direction and stability of the Hopf bifurcations. Then 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. We find that there are different in-phase and anti-phase patterns as the coupling time delay is increased. The analytical theory is supported by numerical simulations, which show good agreement with the theory.     

Delayed feedback control and bifurcation analysis of Rossler chaotic system

In this paper, from the view of stability and chaos control, we investigate the Rossler chaotic system with delayed feedback. At first, we consider the stability of one of the fixed points, verifying that Hopf bifurcation occurs as delay crosses some critical values. Then, for determining the stability and direction of Hopf bifurcation we derive explicit formulae by using the normal-form theory and center manifold theorem. By designing appropriate feedback strength and delay, one of the unstable equilibria of the Rossler chaotic system can be controlled to be stable, or stable bifurcating periodic solutions occur at the neighborhood of the equilibrium. Finally, some numerical simulations are carried out to support the analytic results.     

Hopf bifurcation of an oscillator with quadratic and cubic nonlinearities and with delayed velocity feedback

This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms, and with linear delayed velocity feedback. The analysis indicates that for a sufficiently large velocity feedback gain, the equilibrium of the system may undergo a number of stability switches with an increase of time delay, and then becomes unstable forever. At each critical value of time delay for which the system changes its stability, a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay. The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability. It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions. The project supported by the National Natural Science Foundation of China (19972025)     

BIFURCATION IN A TWO-DIMENSIONAL NEURAL NETWORK MODEL WITH DELAY

IntroductionForunderstandingthedynamicsofneuralnetworks ,thepropertiesofstabilityandbifurcationinasimplifiednon_self_connectionneuralnetwork u1(t) =-μ1u1(t) aF(u2 (t-τ2 ) ) , u2 (t) =-μ2 u2 (t) bG(u1(t-τ1) ) ( 1 )hasbeenstudied .Forexample ,inRef.[1 ]ChenandWustudiedtheexistenceoftheslowlyoscillatingperiodicsolutionbyusingthemethodofdiscreteLiapunovfunction .InRef.[2 ]thesumoftimedelaysτ=τ1 τ2 beingregardedasabifurcationparameter,theexistenceoflocalHopfbifurcationandthepropertiesof…     

Stability and bifurcation analysis in tri-neuron model with time delay

A simple delayed neural network model with three neurons is considered. By constructing suitable Lyapunov functions, we obtain sufficient delay-dependent criteria to ensure global asymptotical stability of the equilibrium of a tri-neuron network with single time delay. Local stability of the model is investigated by analyzing the associated characteristic equation. It is found that Hopf bifurcation occurs when the time delay varies and passes a sequence of critical values. The stability and direction of bifurcating periodic solution are determined by applying the normal form theory and the center manifold theorem. If the associated characteristic equation of linearized system evaluated at a critical point involves a repeated pair of pure imaginary eigenvalues, then the double Hopf bifurcation is also found to occur in this model. Our main attention will be paid to the double Hopf bifurcation associated with resonance. Some Numerical examples are finally given for justifying the theoretical results.     

van der Pol-Duffing时滞系统的稳定性和Hopf分岔

研究了具有三次项的ｖａｎ ｄｅｒ Ｐｏｌ－Ｄｕｆｆｉｎｇ非线性时滞系统的稳定性和Ｈｏｐｆ分岔,分析了当线性化特征方程随两参数（时滞量和增益系数）变化时特征根的分布;证明了Ｈｏｐｆ分岔的存在性,通过构造中心流形并且使用范式方法给出的Ｈｏｐｆ分岔的方向以及周期解的稳定性,讨论时滞量对该系统的Ｈｏｐｆ分岔的影响。     

Hopf Bifurcation and Stability of Periodic Solutions for van der Pol Equation with Distributed Delay

The van der Pol equation with a distributed time delay is analyzed. Itslinear stability is investigated by employing the Routh–Hurwitzcriteria. Moreover, the local asymptotic stability conditions are alsoderived. By using the mean time delay as a bifurcation parameter, themodel is found to undergo a sequence of Hopf bifurcations. The directionand the stability criteria of the bifurcating periodic solutions areobtained by the normal form theory and the center manifold theorem. Somenumerical simulation examples for justifying the theoretical analysisare also given.