Stability Of Hopf Bifurcations Research Articles

Bifurcation analysis on a river population model with varying boundary conditions

A delayed reaction-diffusion-advection equation subject to constant-flux and free-flow boundary conditions is considered, which models single population dynamics in a river. At first, we show the existence of a nonconstant steady state induced by the change of constant flux value. Then by analyzing the distribution of eigenvalues, the stability of the constant and nonconstant steady states and the existence of Hopf bifurcations are obtained. And an algorithm for determining the direction and stability of Hopf bifurcations is derived by applying the center manifold theory and normal form method for PFDEs. Finally, the effects of advection and downstream boundary condition on periodic oscillations are discussed theoretically and numerically.

Emergence of hidden dynamics in different neuronal network architecture with injected electromagnetic induction

The diverse firing responses in a single neuron model as well as in a neuronal network play a major role in understanding the collective neuronal dynamics. In this paper, we consider an excitable slow-fast memristive model and study its various intrinsic dynamics by allowing a periodic external stimulus. The single model exhibits various types of spiking, bursting, mixed-mode oscillations (MMOs), and mixed-mode bursting oscillations (MMBOs) depending on the amplitude and frequency of the periodic injected current. Corresponding bifurcation analysis reveals the existence of supercritical and subcritical Hopf bifurcations in the system depending on the major predominant parameters that establish the scenarios of the neuronal responses. We have verified analytically the existence and stability of Hopf bifurcations. The memristive system can produce cascade of period doubling bifurcations for particular parameter regimes. Next, we investigate the emergence of multi-arm antispiral waves in the diffusively coupled system with proper analytical justification. We have also computed the group and phase velocities to discern the spiral and antispiral waves near the Hopf instability. A transition from target wave to multi-arm antispiral wave is observed in the diffusive system. Moreover, we use a random network architecture different from the diffusive network to study the dynamics of the coupled network for certain firing activities. It is observed that the network of heterogeneous desynchronized neurons with higher electrical couplings, can generate MMOs, MMBOs, and bursting for different external stimuli. However, the uncoupled systems cannot reveal such typical dynamics for particular parameters. Further, we introduce a reduced-order model to summarize the complete dynamics of the larger random network and report the findings.

Bifurcation analysis of an extended macro model considering time delay and anticipation effect

In this paper, a novel macro traffic flow model is proposed integrating time delay and anticipation effect on the headway and velocity, which are indispensable factors in the course of driving. The stability condition of the novel model is derived by linear stability analysis method. The corresponding stability diagrams show that time delay deteriorates the stability of traffic flow, while anticipation effect is conducive to traffic flow stability. The existence and stability of Hopf bifurcations are discussed through bifurcation theory. Numerical results demonstrate that the improved continuum model presented can well describe the local cluster effect, fuel consumption and exhaust emissions. Considering the factor of anticipation and time delay has positive and negative impact on dynamic characteristic of macroscopic flow, respectively. In Hopf bifurcation simulation, partial data from Portable Emissions Measurement System (PeMS) is adopted to calibrate the parameters of the novel model. Meanwhile, the simulation results verify the existence of Hopf bifurcation. The occurrence of bifurcations brings a significant impact on the stability of traffic flow, in which the density wave of traffic flow arises equal-amplitude oscillations.

Effect of hunting cooperation on the dynamic behavior for a diffusive Holling type II predator-prey model

In this paper, the dynamics of a diffusive Holling type II predator-prey model with hunting cooperation are studied. For the local model, the stability of nonnegative equilibria, the detailed behavior of Hopf bifurcation, saddle-node bifurcation, and Bogdanov–Takens bifurcation are investigated. It is shown that under suitable conditions the model exhibits a supercritical and orbitally asymptotically stable limit cycle when the predator cooperation in hunting rate pass through a threshold value. For the reaction-diffusion model, we analyze the diffusion-driven Turing instability of both the positive equilibria and bifurcating periodic solutions, the existence, direction and stability of Hopf bifurcations. Our result shows that the hunting cooperation plays a crucial role in determining the stability and bifurcation behavior to the model, that is, it is beneficial to the coexistence of predator and prey, it can have a destabilizing effect, it can induce the bistability phenomenon, it can undergo not only the saddle-node bifurcation but also Bogdanov–Takens bifurcation, and it can induce the rise of Turing instability, which is a strong contrast to the case without hunting cooperation. Moreover, some numerical simulations are performed to visualize the complex dynamic behavior.

Stability and direction of Hopf bifurcations of a cyclical growth model with two-time delays and one-delay dependent coefficients

• We study the impact of two discrete-time delays on the basic Goodwin growth cycle model. • We study a delayed differential equation system with discrete-time delays and delay-dependent coefficients depending only on one of the time delay. • Having chosen the time delays as bifurcation parameters, we study the stability-switching properties of the transcendental characteristic equation. • We analyze the direction and stability of Hopf bifurcation by means of the first Lyapunov coefficient. This paper deals with the impact of two discrete-time delays on the basic Goodwin growth cycle model. The former concerns the existence of a finite time delay for building capital goods as suggested by Kalecki. The latter pertains to the wage lag hypothesis. This is because, taking the current change of the employment rate into account, workers and capitalists bargain new wage periodically. There are no examples in the literature on the Goodwin model of the use of both those lags in order to explore the GDP dynamics. From the analytical point-of-view, what we obtain is a delayed differential equation system with discrete-time delays and delay-dependent coefficients depending only on one of the time delays. Having chosen the time delays as bifurcation parameters, we study the stability-switching properties of the transcendental characteristic equation resulting from the stability analysis and the direction of the Hopf bifurcations. Although the system with no lag displays a stable focus, the introduction of the two lags preserves the stable solution only for particular combinations of parameters and length of the lags. In any other case, instability prevails and regular cycles or chaotic fluctuations emerge. Finally, we provide the analytical results with the necessary economic interpretations.

Bifurcation analysis in a nonlinear electro-optical oscillator with delayed bandpass feedback

The dynamics of a nonlinear electro-optical oscillator with delayed bandpass feedback is investigated. By analyzing the distribution of the eigenvalues, the existence and stability of Hopf bifurcations are obtained. Particularly, the stability switches are found as the delay varies, where the time delay is regarded as bifurcation parameter. And then, by applying the normal form method and center manifold theory of functional differential equations, an algorithm for determining the sense of the Hopf bifurcations and stability of the bifurcating periodic solutions is derived. For illustrating the theoretical results, some numerical simulations are performed.

Diffusive induced global dynamics and bifurcation in a predator-prey system

In this paper, a diffusive Leslie-type predator-prey model is investigated. The existence of a global positive solution, persistence, stability of the equilibria and Hopf bifurcation are studied respectively. By calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. Finally, our theoretical results are illustrated by a model with homogeneous kernels and one-dimensional spatial domain.

Dynamics of a ratio‐dependent stage‐structured predator‐prey model with delay

In this paper, we investigate the dynamics of a time‐delay ratio‐dependent predator‐prey model with stage structure for the predator. This predator‐prey system conforms to the realistically biological environment. The existence and stability of the positive equilibrium are thoroughly analyzed, and the sufficient and necessary conditions for the stability and instability of the positive equilibrium are obtained for the case without delay. Then, the influence of delay on the dynamics of the system is investigated using the geometric criterion developed by Beretta and Kuang. We show that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions. The formulas determining the direction and the stability of Hopf bifurcations are explicitly derived by using the center manifold reduction and normal form theory. Finally, some numerical simulations are performed to illustrate and expand our theoretical results.

Bistability and delay-induced stability switches in a cancer network with the regulation of microRNA

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.

Hopf bifurcation and chaos in a delayed Nicholson’s blowflies equation with nonlinear density-dependent mortality rate

The stability and bifurcation of the Nicholson’s blowflies equation with nonlinear density-dependent mortality rate and delay feedback are investigated. First, we study the existence of positive equilibria and the stability of positive equilibria in the absence of delay. Then, how delay influences the stability of positive equilibria is studied. According to our analysis, as delay increases, there are a sequence of critical values where the system undergoes Hopf bifurcations. Using the theory of normal form and center manifold reduction, the direction and stability of Hopf bifurcations can be determined according to the parameters of the system. Finally, numerical simulations are applied to illustrate and expend our theoretical analysis, and delay-induced chaos has been found.

Bifurcation analysis and Turing instability in a diffusive predator-prey model with herd behavior and hyperbolic mortality

In this paper, we consider a predator-prey model with herd behavior and hyperbolic mortality subject to the homogeneous Neumann boundary condition. Firstly, we prove the existence and uniqueness of positive equilibrium for this model by analytical skills. Then we analyze the stability of the positive equilibrium, Turing instability, and the existence of Hopf, steady state bifurcations. Finally, by calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. Meanwhile, for the steady state bifurcation, the possibility of pitchfork bifurcation can be concluded by the normal form, which does also determine the stability of spatially inhomogeneous steady states. Furthermore, some numerical simulations to illustrate the theoretical analysis are also carried out and expand our theoretical results.

Bifurcation analysis of a diffusive predator–prey system with a herd behavior and quadratic mortality

In this paper, a diffusive predator–prey system, in which the prey species exhibits herd behavior and the predator species with quadratic mortality, has been studied. The stability of positive constant equilibrium, Hopf bifurcations, and diffusion‐driven Turing instability are investigated under the Neumann boundary condition. The explicit condition for the occurrence of the diffusion‐driven Turing instability is derived, which is determined by the relationship of the diffusion rates of two species. The formulas determining the direction and the stability of Hopf bifurcations depending on the parameters of the system are derived. Finally, numerical simulations are carried out to verify and extend the theoretical results and show the existence of spatially homogeneous periodic solutions and nonconstant steady states. Copyright © 2014 John Wiley & Sons, Ltd.

Persistence, Stability and Hopf Bifurcation in a Diffusive Ratio-Dependent Predator–Prey Model with Delay

In this paper, we study the persistence, stability and Hopf bifurcation in a ratio-dependent predator–prey model with diffusion and delay. Sufficient conditions independent of diffusion and delay are obtained for the persistence of the system and global stability of the boundary equilibrium. The local stability of the positive constant equilibrium and delay-induced Hopf bifurcation are investigated by analyzing the corresponding characteristic equation. We show that delay can destabilize the positive equilibrium and induce spatially homogeneous and inhomogeneous periodic solutions. By calculating the normal form on the center manifold, the formulae determining the direction and the stability of Hopf bifurcations are explicitly derived. The numerical simulations are carried out to illustrate and extend our theoretical results.

Bifurcation analysis of a diffusive ratio-dependent predator–prey model

In this paper, a ratio-dependent predator–prey model with diffusion is considered. The stability of the positive constant equilibrium, Turing instability, and the existence of Hopf and steady state bifurcations are studied. Necessary and sufficient conditions for the stability of the positive constant equilibrium are explicitly obtained. Spatially heterogeneous steady states with different spatial patterns are determined. By calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. For the steady state bifurcation, the normal form shows the possibility of pitchfork bifurcation and can be used to determine the stability of spatially inhomogeneous steady states. Some numerical simulations are carried out to illustrate and expand our theoretical results, in which, both spatially homogeneous and heterogeneous periodic solutions are observed. The numerical simulations also show the coexistence of two spatially inhomogeneous steady states, confirming the theoretical prediction.

Stability and bifurcation in a ratio-dependent Holling-III system with diffusion and delay

A diffusive ratio-dependent predator-prey system with Holling-III functional response and delay effects is considered. Global stability of the boundary equilibrium and the stability of the unique positive steady state and the existence of spatially homogeneous and inhomogeneous periodic solutions are investigated in detail, by the maximum principle and the characteristic equations. Ratio-dependent functional response exhibits rich spatiotemporal patterns. It is found that, the system without delay is dissipative and uniformly permanent under certain conditions, the delay can destabilize the positive constant equilibrium and spatial Hopf bifurcations occur as the delay crosses through some critical values. Then, the direction and the stability of Hopf bifurcations are determined by applying the center manifold reduction and the normal form theory for partial functional differential equations. Some numerical simulations are carried out to illustrate the theoretical results.

Stability and Hopf Bifurcation Analysis of a Gene Expression Model with Diffusion and Time Delay

We consider a model for gene expression with one or two time delays and diffusion. The local stability and delay-induced Hopf bifurcation are investigated. We also derive the formulas determining the direction and the stability of Hopf bifurcations by calculating the normal form on the center manifold.

Stability and Bifurcation Analysis of a Delayed Leslie-Gower Predator-Prey System with Nonmonotonic Functional Response

A delayed Leslie-Gower predator-prey model with nonmonotonic functional response is studied. The existence and local stability of the positive equilibrium of the system with or without delay are completely determined in the parameter plane. Using the method of upper and lower solutions and monotone iterative scheme, a sufficient condition independent of delay for the global stability of the positive equilibrium is obtained. Hopf bifurcations induced by the ratio of the intrinsic growth rates of the predator and prey and by delay, respectively, are found. Employing the normal form theory, the direction and stability of Hopf bifurcations can be explicitly determined by the parameters of the system. Some numerical simulations are given to support and extend our theoretical results. Two limit cycles enclosing an equilibrium, one limit cycle enclosing three equilibria and different types of heteroclinic orbits such as connecting two equilibria and connecting a limit cycle and an equilibrium are also found by using analytic and numerical methods.

STEADY-STATE, HOPF AND STEADY-STATE-HOPF BIFURCATIONS IN DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS TO A DAMPED HARMONIC OSCILLATOR WITH DELAY FEEDBACK

In this paper, employing the normal form theory of delay differential equations due to Faria and Magalhães, we present explicit formulas of the coefficients of a normal form associated with the flow on a center manifold with the unfolding for general delay differential equations under the cases of steady-state, Hopf and steady-state-Hopf singularities. The explicit conditions determining the transcritical and pitchfork bifurcations for steady-state singularity, determining the direction and stability of Hopf bifurcations, and determining the coefficients of a normal form with universal unfolding for steady-state-Hopf singularity up to third order are obtained. Using the obtained results, we give a complete description of bifurcation scenario of the damped harmonic oscillator with delay feedback near the zero equilibrium. Finally, numerical simulations are given to illustrate our theoretical results and some numerical extensions are obtained as a supplement to our theoretical analysis.

STABILITY AND BIFURCATION ANALYSIS IN A DIFFUSIVE BRUSSELATOR SYSTEM WITH DELAYED FEEDBACK CONTROL

A diffusive Brusselator model with delayed feedback control subject to Dirichlet boundary condition is considered. The stability of the unique constant equilibrium and the existence of a family of inhomogeneous periodic solutions are investigated in detail, exhibiting rich spatiotemporal patterns. Moreover, it shows that Turing instability occurs without delay. And under certain conditions, the constant equilibrium switches finite times from stability to instability to stability, and becomes unstable eventually, as the delay crosses through some critical values. Then, the direction and the stability of Hopf bifurcations are determined by the normal form theory and the center manifold reduction for partial functional differential equations. Finally, some numerical simulations are carried out for illustrating the analysis results.

Stability and Hopf Bifurcation in a Diffusive Predator-Prey System with Beddington-DeAngelis Functional Response and Time Delay

This paper is concerned with a diffusive predator-prey system with Beddington-DeAngelis functional response and delay effect. By analyzing the distribution of the eigenvalues, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated. Also, it is shown that the small diffusion can affect the Hopf bifurcations. Finally, the direction and stability of Hopf bifurcations are determined by normal form theory and center manifold reduction for partial functional differential equations.