Hopf-pitchfork bifurcation in van der Pol's oscillator with nonlinear delayed feedback

First, we identify the critical values for Hopf-pitchfork bifurcation. Second, we derive the normal forms up to third order and their unfolding with original parameters in the system near the bifurcation point, by the normal form method and center manifold theory. Then we give a complete bifurcation diagram for original parameters of the system and obtain complete classifications of dynamics for the system. Furthermore, we find some interesting phenomena, such as the coexistence of two asymptotically stable states, two stable periodic orbits, and two attractive quasi-periodic motions, which are verified both theoretically and numerically.     

On the coexistence of periodic or quasi-periodic oscillations near a Hopf-pitchfork bifurcation in NFDE

Normal form method is first employed to study the Hopf-pitchfork bifurcation in neutral functional differential equation (NFDE), and is proved to be an efficient approach to show the rich dynamics (periodic and quasi-periodic oscillations) around the bifurcation point. We give an algorithm for calculating the third-order normal form in NFDE models, which naturally arise in the method of extended time delay autosynchronization (ETDAS). The existence of Hopf-pitchfork bifurcation in a van der Pol’s equation with extended delay feedback is given and the unfoldings near this critical point is obtained by applying our algorithm. Some interesting phenomena, such as the coexistence of several stable periodic oscillations (or quasi-periodic oscillations) and the existence of saddle connection bifurcation on a torus, are found by analyzing the bifurcation diagram and are illustrated by numerical method.     

Hopf-transcritical bifurcation in retarded functional differential equations

Firstly, we analyze a codimension-two unfolding for the Hopf-transcritical bifurcation, and give complete bifurcation diagrams and phase portraits. In particular, we express explicitly the heteroclinic bifurcation curve, and obtain conditions under which the secondary bifurcation periodic solutions and the heteroclinic orbit are stable. Secondly, we show how to reduce general retarded functional differential equation, with perturbation parameters near the critical point of the Hopf-transcritical bifurcation, to a 3-dimensional ordinary differential equation which is restricted on the center manifold up to the third order with unfolding parameters, and further reduce it to a 2-dimensional amplitude system, where these unfolding parameters can be expressed by those original perturbation parameters. Finally, we apply the general results to the van der Pol’s equation with delayed feedback, and obtain the existence of stable or unstable equilibria, periodic solutions and quasi-periodic solutions.     

On the stability and Hopf bifurcation of a prey-generalist predator system with independent age-selective harvesting

Age-selective harvesting where harvesting of species after a certain age is a scientific strategy with respect to biological and economical point of views. By this method we can overcome the unexpected extinction risk of any harvested population due to random harvesting below its maturation (age, body size or weight). The objective of this paper is to study dynamic behavior of preypredator system with alternative form of time delay in harvesting. Arino et al. [2] have given alternative expression for a delayed logistic equation. Using this expression of time delay, a preypredator system with Holling type III functional response and independent age-selective harvesting is proposed and analyzed. We find out the critical values of delay parameters under different dynamical situations and observe that system is stable and unstable when the delay parameters are bellow and above the critical values respectively and there is Hopf bifurcation when delay parameters cross the critical values. System shows these interesting dynamical features under different critical parametric restrictions. Using the normal form theory and the center manifold theorem, we determine the stability and direction of the bifurcating periodic solutions. Numerical simulations illustrate the analytical results.     

Turing-Hopf bifurcation in the reaction-diffusion system with delay and application to a diffusive predator-prey model

The interactions of diffusion-driven Turing instability and delay-induced Hopf bifurcation always give rise to rich spatiotemporal dynamics. In this paper, we first derive the algorithm for the normal forms associated with the Turing-Hopf bifurcation in the reaction-diffusion system with delay, which can be used to investigate the spatiotemporal dynamical classification near the Turing-Hopf bifurcation point in the parameter plane. Then, we consider a diffusive predator-prey model with weak Allee effect and delay. Through investigating the dynamics of the corresponding normal form of Turing-Hopf bifurcation induced by diffusion and delay, the spatiotemporal dynamics near this bifurcation point can be divided into six categories. Especially, stable spatially homogeneous/inhomogeneous periodic solutions and steady states, coexistence of two stable spatially inhomogeneous periodic solutions, coexistence of two stable spaially inhomogeneous steady states and the transition from one kind of spatiotemporal patterns to another are found.     

Hopf bifurcation analysis on an Internet congestion control system of arbitrary dimension with communication delay

This paper carries out a Hopf bifurcation analysis on a model of Internet congestion control system for a network with arbitrary topology. The general form of the rate-based Kelly model for a multi-source multi-link network with a communication delay is considered. Assuming the communication delay as a bifurcation parameter, we find that when the delay parameter passes a critical value, a periodic solution bifurcates from the equilibrium point. The stability and direction of bifurcating periodic solutions are studied by using the center manifold theorem and the normal form theory. We simulate our model for a typical example to show the applicability of the approach.     

Hopf‐pitchfork bifurcation in a two‐neuron system with discrete and distributed delays

Both discrete and distributed delays are considered in a two‐neuron system. We analyze the influence of interaction coefficient and time delay on the Hopf‐pitchfork bifurcation. First, we obtain the codimension‐2 unfolding with original parameters for Hopf‐pitchfork bifurcation by using the center manifold reduction and the normal form method. Next, through analyzing the unfolding structure, we give complete bifurcation diagrams and phase portraits, in which multistability and other dynamical behaviors of the original system are found, such as a stable periodic orbit, the coexistence of two stable nontrivial equilibria, and the coexistence of a stable periodic orbit and two stable equilibria. In addition, the obtained theoretical results are verified by numerical simulations. Finally, we perform the comparisons of the obtained results of Hopf‐pitchfork bifurcation with other Hopf‐fold bifurcation results in some biological neural systems and give the obtained mathematical results corresponding to the physical states of neurons. Copyright © 2015 JohnWiley & Sons, Ltd.     

Hopf bifurcations in a predator–prey system with a discrete delay and a distributed delay

A delayed Lotka–Volterra two-species predator–prey system with discrete hunting delay and distributed maturation delay for the predator population described by an integral with a strong delay kernel is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that under suitable conditions on the parameters the positive equilibrium is asymptotically stable when the hunting delay is less than a certain critical value and unstable when the hunting delay is greater than this critical value. Meanwhile, according to the Hopf bifurcation theorem for functional differential equations (FDEs), we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the hunting delay crosses through a sequence of critical values. In particular, by applying the normal form theory and the center manifold reduction for FDEs, an explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions occurring through Hopf bifurcations is given. Finally, to verify our theoretical predictions, some numerical simulations are also included at the end of this paper.     

Stability and Hopf bifurcation in a delayed competition system

In this paper, Hopf bifurcation for two-species Lotka–Volterra competition systems with delay dependence is investigated. By choosing the delay as a bifurcation parameter, we prove that the system is stable over a range of the delay and beyond that it is unstable in the limit cycle form, i.e., there are periodic solutions bifurcating out from the positive equilibrium. Our results show that a stable competition system can be destabilized by the introduction of a maturation delay parameter. Further, an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the theory of normal forms and center manifolds, and numerical simulations supporting the theoretical analysis are also given.     

Double Hopf bifurcation and chaos in Liu system with delayed feedback

In this paper, we consider the stability of equilibria, Hopf and double Hopf bifurcation in Liu system with delay feedback. Firstly, we identify the critical values for stability switches and Hopf bifurcationusing the method of bifurcation analysis. When we choose appropriate feedback strength and delay, two symmetrical nontrivial equilibria of Liusystem can be controlled to be stable at the same time, and the stable bifurcating periodic solutions occur in the neighborhood of the two equilibria at the same time. Secondly, by applying the normal form method and center manifold theory,the normal form near the double Hopf bifurcation, as well as classifications of local dynamics are analyzed. Furthermore, we give the bifurcation diagram to illustrate numerically that a family of stable periodic solutions bifurcated from Hopf bifurcation occur in a large region of delay and the Liu system with delay can appear the phenomenon of ``chaos switchover''.     

Double Hopf bifurcation and quasi-periodic attractors in delay-coupled limit cycle oscillators

We investigate the double Hopf bifurcation at zero equilibrium point. Firstly, we give the critical values of Hopf and double Hopf bifurcations. Secondly, we implement the normal form method and the center manifold theory for delay-coupled limit cycle oscillators, and derive the universal unfolding and a complete bifurcation diagram of the system. Thirdly, many interesting phenomena, such as attractive periodic motion and three-dimensional invariant torus, are observed using numerical simulation. Finally, the normal forms of several strong resonant cases are listed.     

Two-parameter bifurcations in a network of two neurons with multiple delays

We consider a network of two coupled neurons with delayed feedback. We show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension 1 bifurcations (including a fold bifurcation and a Hopf bifurcation) and codimension 2 bifurcations (including fold-Hopf bifurcations and Hopf-Hopf bifurcations). We also give concrete formulae for the normal form coefficients derived via the center manifold reduction that give detailed information about the bifurcation and stability of various bifurcated solutions. In particular, we obtain stable or unstable equilibria, periodic solutions, quasi-periodic solutions, and sphere-like surfaces of solutions. We also show how to evaluate critical normal form coefficients from the original system of delay-differential equations without computing the corresponding center manifolds.     

HOPF BIFURCATION OF NONLINEAR ADVERTISING CAPITAL MODEL WITH DISCRETE AND CONTINUOUS TIME DELAYED FEEDBACK

In this paper,stability and Hopf bifurcation of a nonlinear advertising ca- pital model with time delayed are studied.By analyzing the change of delay, we obtain several sufficient conditions on stable and unstable properties.When delay passes a critical value,Hopf bifurcation may appear.Furthermore,the di- rection and stability of bifurcating periodic solutions are investigated by normal form and center manifold theory.Additionally,we also have some discussion about the model with continuous time delay.     

一类具有时滞的金融系统的稳定性

首先建立了一类具有时滞的金融模型,该模型以累计利润额为关键因素,接着以τ为参考元素研究了该模型的稳定性及Hopf分叉.发现当τ变化时,该系统的稳定性会发生变化,该模型会在某一确定值处出现Hopf分叉.最后用中心流形定理和规范型方法研究分叉周期解的稳定性.     

Bifurcation analysis of a modified Holling-Tanner predator-prey model with time delay

In this paper, a modified Holling-Tanner predator-prey model with time delay is considered. By regarding the delay as the bifurcation parameter, the local asymptotic stability of the positive equilibrium is investigated. Meanwhile, we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the delay crosses through a sequence of critical values. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions, an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included.     

Delay Induced Hopf Bifurcation of Small-World Networks

In this paper,the stability and the Hopf bifurcation of small-world networks with time delay are studied.By analyzing the change of delay,we obtain several sufficient conditions on stable and unstable properties.When the delay passes a critical value,a Hopf bifurcation may appear.Furthermore,the direction and the stability of bifurcating periodic solutions are investigated by the normal form theory and the center manifold reduction.At last,by numerical simulations,we further illustrate the effectiveness of theorems in this paper.     

Hopf-zero bifurcation of a delayed predator-prey model with dormancy of predators

In this paper, We investigate Hopf-zero bifurcation with codimension 2 in a delayed predator-prey model with dormancy of predators. First we prove the specific existence condition of the coexistence equilibrium. Then we take the mortality rate and time delay as two bifurcation parameters to find the occurrence condition of Hopf-zero bifurcation in this model. Furthermore, using the Faria and Magalhases normal form method and the center manifold theory, we obtain the third order degenerate normal form with two original parameters. Finally, through theoretical analysis and numerical simulations, we give a bifurcation set and a phase diagram to show the specific relations between the normal form and the original system, and explain the coexistence phenomena of several locally stable states, such as the coexistence of multi-periodic orbits, as well as the coexistence of a locally stable equilibrium and a locally stable periodic orbit.     

Hopf bifurcation and stability analysis on discrete-time Hopfield neural network with delay

In this paper, a discrete-time Hopfield neural network with delay is considered. We give some sufficient conditions ensuring the local stability of the equilibrium point for this model. By choosing the delay as a bifurcation parameter, we demonstrated that Neimark–Sacker bifurcation (or Hopf bifurcation for map) would occur when the delay exceeds a critical value. A formula for determining the direction bifurcation and stability of bifurcation periodic solutions is given by applying the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical results are also provided.     

Stability and Hopf bifurcation for a regulated logistic growth model with discrete and distributed delays

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 τ as a bifurcation parameter, we prove that the system is locally asymptotically stable in a range of the delay and Hopf bifurcation occurs as τ 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.     

Global Hopf bifurcation in the Leslie-Gower predator-prey model with two delays

In this paper, the Leslie-Gower predator-prey system with two delays is investigated. By choosing the delay as a bifurcation parameter, we show that Hopf bifurcations can occur as the delay crosses some critical values. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of periodic solutions.