Stability and Hopf bifurcation in a class of nonlocal delay differential equation with the zero‐flux boundary condition

The aim of this paper is to study the stability and Hopf bifurcation in a general class of differential equation with nonlocal delayed feedback that models the population dynamics of a two age structured spices. The existence of Hopf bifurcation is firstly established after delicately analyzing the eigenvalue problem of the linearized nonlocal equation. The direction of the Hopf bifurcation and stability of the bifurcated periodic solutions are then investigated by means of center manifold reduction. Subsequently, we apply our main results to explore the spatial‐temporal patterns of the nonlocal Mackey‐Glass equation. We obtain both spatially homogeneous and inhomogeneous periodic solutions and numerically show that the former is stable while the latter is unstable. We also show that the inhomogeneous periodic solutions will eventually tend to homogeneous periodic solutions after transient oscillations and increasing of the immature mobility constant will shorten the transient oscillation time.     

Saddle-node-Hopf bifurcation in a modified Leslie–Gower predator-prey model with time-delay and prey harvesting

From the saddle-node-Hopf bifurcation point of view, this paper considers a modified Leslie–Gower predator-prey model with time delay and the Michaelis–Menten type prey harvesting. Firstly, we discuss the stability of the equilibria, obtain the critical conditions for the saddle-node-Hopf bifurcation, and give the completion bifurcation set by calculating the universal unfoldings near the saddle-node-Hopf bifurcation point by using the normal form theory and center manifold theorem. Then we derive the parameter conditions for the existence of monostable coexistence equilibrium and the parameter regions in which both the prey-extinction and the coexistence equilibrium (or coexistence periodic or quasi-periodic solutions) are simultaneously stabilized. We also investigate the heteroclinic bifurcation, and describe the phenomenon that the periodic behavior disappears as through the heteroclinic bifurcation. Finally, some numerical simulations are performed to support our analytic results.     

Hopf-transcritical bifurcation in toxic phytoplankton–zooplankton model with delay

In this paper, a phytoplankton–zooplankton model with toxic liberation delay is considered. Firstly, the critical values of Hopf bifurcation, transcritical bifurcation and Hopf-transcritical bifurcation are given, and to give more detailed information about the periodic oscillations, the direction and stability of Hopf bifurcation is studied by using the normal-form theory and center manifold theorem. Then, we give the detailed bifurcation set by calculating the universal unfoldings near the Hopf-transcritical bifurcation point. Finally, we show that the plankton system may exhibit quasi-periodic oscillations, which are verified both theoretically and numerically, and explain the experimental observed fluctuation phenomenon of plankton population.     

On the new solutions of the generalized Burgers–Huxley equation

In this paper, we investigate a class of generalized Burgers–Huxley equation by employing the bifurcation method of planar dynamical systems. Firstly, we reduce the equation to a planar system via the traveling wave solution ansatz; then by computing the singular point quantities, we obtain the conditions of integrability and determine the existence of one stable limit cycle from Hopf bifurcation in the corresponding planar system. From this, some new exact solutions and a special periodic traveling wave solution, which is isolated as a limit, are obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Hopf-pitchfork bifurcation and periodic phenomena in nonlinear financial system with delay

In this paper, we identify the critical point for a Hopf-pitchfork bifurcation in a nonlinear financial system with delay, and derive the normal form up to third order with their unfolding in original system parameters near the bifurcation point by normal form method and center manifold theory. Furthermore, we analyze its local dynamical behaviors, and show the coexistence of a pair of stable periodic solutions. We also show that there coexist a pair of stable small-amplitude periodic solutions and a pair of stable large-amplitude periodic solutions for different initial values. Finally, we give the bifurcation diagram with numerical illustration, showing that the pair of stable small-amplitude periodic solutions can also exist in a large region of unfolding parameters, and the financial system with delay can exhibit chaos via period-doubling bifurcations as the unfolding parameter values are far away from the critical point of the Hopf-pitchfork bifurcation.     

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.     

周期激励Stuart-Landau方程的某些动力学行为

研究了周期激励Stuart-Landau方程的锁频周期解．利用奇异性理论分别研究了这些解关于外部激励振幅和频率的分岔行为．结果表明:关于外部激励振幅的普适开折具有余维3,在某些条件下,得到了转迁集及分岔图．另外还证明:关于频率的分岔问题具有无穷余维,因此该情形下的动力学分岔行为非常复杂．发现了一些新的动力学现象,它们是孙亮等所获结果的补充．     

Numerical Computation of Connecting Orbits in Delay Differential Equations

We discuss the numerical computation of homoclinic and heteroclinic orbits in delay differential equations. Such connecting orbits are approximated using projection boundary conditions, which involve the stable and unstable manifolds of a steady state solution. The stable manifold of a steady state solution of a delay differential equation (DDE) is infinite-dimensional, a problem which we circumvent by reformulating the end conditions using a special bilinear form. The resulting boundary value problem is solved using a collocation method. We demonstrate results, showing homoclinic orbits in a model for neural activity and travelling wave solutions to the delayed Hodgkin–Huxley equation. Our numerical tests indicate convergence behaviour that corresponds to known theoretical results for ODEs and periodic boundary value problems for DDEs.     

Exponential trichotomy and homoclinic bifurcation with saddle-center equilibrium

In this paper the bifurcation of a homoclinic orbit is studied for an ordinary differential equation with periodic perturbation. Exponential trichotomy theory with the method of Lyapunov–Schmidt is used to obtain some sufficient conditions to guarantee the existence of homoclinic solutions and periodic solutions for this problem. Some known results are extended.     

一类在三次扰动下三次Hamilton系统的两参数开折

本文研究一类三次Ｈａｍｉｌｔｏｎ系统在三次扰动下的动力形态。利用向量场分支理论的方法讨论时该系统的两参数开折,并得到在参数平面原点领域的完整的分支图,进而对应分支图的每个区域给出相轨线图。     

Smoothness of bounded solutions of nonlinear evolution equations

It is shown that in many cases globally defined, bounded solutions of evolution equations are as smooth (in time) as the corresponding operator, even if a general solution of the initial-value problem is much less smooth; i.e., initial values for bounded solutions are selected in such a way that optimal smoothness is attained. In particular, solutions which bifurcate from certain steady states, such as periodic orbits, almost-periodic orbits and also homo- and heteroclinic orbits, have this property. As examples, a neutral functional differential equation, a slightly damped non-linear wave equation, and a heat equation are considered. In the latter case the space variable is included into the discussion of smoothness. Finally, generalized Hopf bifurcation in infinite dimensions is considered. Here smoothness of the bifurcation function is discussed and known results on the order of a focus are generalized.     

HOPF BIFURCATION AND OTHER DYNAMICAL BEHAVIORS FOR A FOURTH ORDER DIFFERENTIAL EQUATION IN MODELS OF INFECTIOUS DISEASE

ＨＯＰＦＢＩＦＵＲＣＡＴＩＯＮＡＮＤＯＴＨＥＲＤＹＮＡＭＩＣＡＬＢＥＨＡＶＩＯＲＳＦＯＲＡＦＯＵＲＴＨＯＲＤＥＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＩＮＭＯＤＥＬＳＯＦＩＮＦＥＣＴＩＯＵＳＤＩＳＥＡＳＥＪＩＮＧＺＨＵＪＵＮ（井竹君）（Ｉｎｓｔｉｔ...     

Bifurcations and exact solutions of nonlinear Schrodinger equation with an anti-cubic nonlinearity

In this paper, we consider the nonlinear Schr\"{o}dinger equation with an anti-cubic nonlinearity. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the corresponding planar dynamical system under different parameter conditions. Corresponding to different level curves defined by the Hamiltonian, we derive all exact explicit parametric representations of the bounded solutions (including periodic peakon solutions, periodic solutions, homoclinic solutions, heteroclinic solutions and compacton solutions).     

Travelling Wave Solutions and Conservation Laws of the (2+1)-dimensional Broer-Kaup-Kupershmidt Equatio

The travelling wave solutions and conservation laws of the (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK) equation are considered in this paper. Under the travelling wave frame, the BKK equation is transformed to a system of ordinary differential equations (ODEs) with two dependent variables. Therefore, it happens that one dependent variable $u$ can be decoupled into a second order ODE that corresponds to a Hamiltonian planar dynamical system involving three arbitrary constants. By using the bifurcation analysis, we obtain the bounded travelling wave solutions $u$, which include the kink, anti-kink and periodic wave solutions. Finally, the conservation laws of the BBK equation are derived by employing the multiplier approach.     

Versal unfoldings of predator-prey systems with ratio-dependent functional response

In this paper we study the versal unfolding of a predator-prey system with ratio-dependent functional response near a degenerate equilibrium in order to obtain all possible phase portraits for its perturbations. We first construct the unfolding and prove its versality and degeneracy of codimension 2. Then we discuss all its possible bifurcations, including transcritical bifurcation, Hopf bifurcation, and heteroclinic bifurcation, give conditions of parameters for the appearance of closed orbits and heteroclinic loops, and describe the bifurcation curves. Phase portraits for all possible cases are presented.     

具时滞的二维神经网络模型的分支

研究了一类具时滞的二维神经网络模型．通过对该模型的特征方程根的分布分析, 在适当的参数平面上给出了分支图．得到了pitchfork分支曲线是一条直线,进而研究了每个平衡点的稳定性和Hopf分支的存在性．最后,利用规范性方法和中心流形理论,得到了Hopf分支的分支方向和分支周期界的稳定性．     

Bifurcating time-periodic solutions of Navier-Stokes equations in infinite cylinders

Summary For the problem of hydrodynamical stability in an infinite cylindrical domain, we investigate all time-periodic solutions, not only spatially periodic ones, when a Hopf bifurcation occurs. When reflection symmetry is present, we show the existence of spatially quasiperiodic flows. We also show the existence of heteroclinic solutions connecting two symmetrically traveling waves that stay at each end of the cylinders (defect solutions). The technique we use rests on (i) a center manifold argument in a space of time-periodic vector fields, (ii) symmetry and normal form arguments for the reduced ordinary differential equation in two dimensions (without reflection symmetry) or in four dimensions (with reflection symmetry), and (iii) the integrability of the associated normal form. It then remains to prove a persistence result when we add the higher-order terms of the vector field.     

Periodic orbit of the pendulum with a small nonlinear damping

We study the pendulum with a small nonlinear damping, which can be expressed by a Hamiltonian system with a small perturbation. We prove that a unique periodic orbit exists for any initial position between the equilibrium point and the heteroclinic orbit of the unperturbed system, depending on the choice of the bifurcation parameter in the damping. The main tools are bifurcation theory and Abelian integral technique, as well as the Zhang''s uniqueness theorem on Li\''enard equations.     

Dynamics of a delayed two-coupled oscillator with excitatory-to-excitatory connection

In this paper, we consider a neural network model consisting of two coupled oscillators with delayed feedback and excitatory-to-excitatory connection. We study how the strength of the connections between the oscillators affects the dynamics of the neural network. We give a full classification of all equilibria in the parameter space and obtain its linear stability by analyzing the characteristic equation of the linearized system. We also investigate the spatio-temporal patterns of bifurcated periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. Moreover, the stability and bifurcation direction of the bifurcated periodic solutions are obtained by employing center manifold reduction and normal form theory. Some numerical simulations are provided to illustrate the theoretical results.