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 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-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.     

Bifurcation analysis and chaos control of the modified Chua’s circuit system

From the view of bifurcation and chaos control, the dynamics of modified Chua’s circuit system are investigated by a delayed feedback method. Firstly, the local stability of the equilibria is discussed by analyzing the distribution of the roots of associated characteristic equation. The regions of linear stability of equilibria are given. It is found that there exist Hopf bifurcation and Hopf-zero bifurcation when the delay passes though a sequence of critical values. By using the normal form method and the center manifold theory, we derive the explicit formulas for determining the direction and stability of Hopf bifurcation. Finally, chaotic oscillation is converted into a stable equilibrium or a stable periodic orbit by designing appropriate feedback strength and delay. Some numerical simulations are carried out to support the analytic results.     

Global Analysis of Predator–Prey System with Hawk and Dove Tactics

Functional response of the Holling type II is incorporated into a predator–prey model with predators using hawk‐dove tactics to consider combination effects of nonlinear functional response and individual tactics. By mathematical analysis, it is shown that the model undergoes a sequence of bifurcations including saddle‐node bifurcation, supercritical Hopf bifurcation and homoclinic bifurcation. New phenomena are found that include the bistable coexistence of prey and predators in the form of a stable limit cycle and a stable positive equilibrium, the bistable coexistence of prey and predators in a large stable limit cycle that encloses three positive equilibria and a stable positive equilibrium within the cycle, and the bistable coexistence of two stable limit cycles.     

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.     

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.     

具有时滞和毒素系统的动力学分析

In this paper, a toxin producing phytoplankton-zooplankton model with inhibitory substrate and time delay is investigated. A discrete time delay is induced to both of the consume response function and distribution of toxic substance term. Moreover, Tissiet type function is used for zooplankton grazing to account for the effect of toxication by the TPP population. The conditions to guarantee the coexistence of two species and stability of coexistence equilibrium are given. In particular, we show that there exist critical values of the delay parameters below which the coexistence equilibrium is stable and above which it is unstable. Hopf bifurcation occurs when the delay parameters cross their critical values. Some numerical simulations are executed to validate the analytical findings.     

New results on the parametric stability of nonlinear systems

In this paper, we derive some new results on the parametric stability of nonlinear systems. Explicitly, we derive a necessary and sufficient condition for a nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. Next, we derive some new results on the parametric stability of discrete-time nonlinear systems. As in the continuous case, we derive a necessary and sufficient condition for a discrete-time nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the discrete-time nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. We illustrate our results with some classical examples from the bifurcation theory.     

Dynamics in a diffusive predator–prey system with a constant prey refuge and delay

In this paper, a diffusive predator–prey system with a constant prey refuge and time delay subject to Neumann boundary condition is considered. Local stability and Turing instability of the positive equilibrium are studied. The effect of time delay on the model is also obtained, including locally asymptotical stability and existence of Hopf bifurcation at the positive equilibrium. And the properties of Hopf bifurcation are determined by center manifold theorem and normal form theorem of partial functional differential equations. Some numerical simulations are carried out.     

Dynamics of Leslie–Gower type generalist predator in a tri-trophic food web system

In this paper, the dynamics of a tri-trophic food web system consists of Leslie–Gower type generalist predator has been explored. The system is bounded under certain conditions. The Hopf-bifurcation has been established in the phase planes. The bifurcation diagrams exhibit coexistence of all three species in the form of periodic/chaotic solutions. The “snail-shell” chaotic attractor has very high Lyapunov exponents. The coexistence in the form of stable equilibrium is also possible for lower values of parameters. The two-parameter bifurcation diagrams are drawn for critical parameters.     

多重时滞富营养化生态模型的稳定性与分支分析

本文研究了多重时滞富营养化生态模型的稳定性与分支问题.利用特征值方法,分别研究了具有单时滞和双时滞模型的线性稳定性.发现当模型中的时滞经过一系列临界值时,模型在平衡点附近经历了Hopf分支和Hopf-zero分支,并给出Hopf分支和Hopf-zero分支存在的充分条件.最后数值模拟验证了理论结果.     

一类不可压广义neo-Hookean球体的空穴分岔问题的定性研究

研究了一类不可压的广义neo-Hookean材料组成的球体的空穴分岔问题,该类材料可以看作是带有径向摄动的均匀各向同性不可压的neo-Hookean材料,得到了球体内部空穴生成的条件．与均匀各向同性的neo-Hookean球体的情况相比,证明了当摄动参数属于某些区域时,从平凡解局部向左分岔的空穴分岔解上存在一个二次转向分岔点,空穴生成时的临界载荷会比无摄动的材料的临界载荷小．用奇点理论证明了,空穴分岔方程在临界点附近等价于具有单边约束条件的正规形．用最小势能原理分别讨论了空穴分岔解的稳定性和实际稳定的平衡状态．     

Bogdanov-Takens singularity in the comprehensive national power model with time delays

In this paper, the comprehensive national power model with time delays is studied. The condition that there is only one trivial equilibrium in the model is given. Based on the analysis of the distribution of the eigenvalues at the trivial equilibrium, it is found that the trivial equilibrium is a Bogdanov-Takens singularity. Using the center manifold theory and the normal form method, the normal form with delay and ratio parameters of the model is obtained. Furthermore, the topological structures of the model near the bifurcation point with the variation of these two parameters are given. The associated development situations of the comprehensive national power for some topological structures are discussed. Finally, some numerical simulations are performed to support the analytic results.     

Bifurcation analysis in a recurrent neural network model with delays

In this paper, we study the dynamical behaviors of a three-node recurrent neural network model with four discrete time delays. We study several types of bifurcation, and use the method of multiple time scales to derive the normal forms associated with Hopf-zero bifurcation, non-resonant and resonant double Hopf bifurcations. Moreover, bifurcations are classified in two-dimensional parameter space near these critical points, and numerical simulations are presented to demonstrate the applicability of the theoretical results.     

Turing instability and Hopf bifurcation in a diffusive Leslie–Gower predator–prey model

In this paper, a reaction‐diffusion predator–prey system that incorporates the Holling‐type II and a modified Leslie‐Gower functional responses is considered. For ODE, the local stability of the positive equilibrium is investigated and the specific conditions are obtained. For partial differential equation, we consider the dissipation and persistence of solutions, the Turing instability of the equilibrium solutions, and the Hopf bifurcation. By calculating the normal form, we derive the formulae, which can determine the direction and the stability of Hopf bifurcation according to the original parameters of the system. We also use some numerical simulations to illustrate our theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Hopf Bifurcation of Delayed Predator-prey System with Reserve Area for Prey and in the Presence of Toxicity

A kind of three species delayed predator-prey system with reserve area for prey and in the presence of toxicity is proposed in this paper.Local stability of the coexistence equilibrium of the system and the existence of a Hopf bifurcation is established by choosing the time delay as the bifurcation parameter.Explicit formulas to determine the direction and stability of the Hopf bifurcation are obtained by means of the normal form theory and the center manifold theorem.Finally,we give a numerical example to illustrate the obtained results.     

Hopf bifurcation analysis of a density predator-prey model with crowley-martin functional response and two time delays

In this paper, a delayed density dependent predator-prey model with Crowley-Martin functional response and two time delays for the predator is considered. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of Hopf bifurcation at the coexistence equilibrium is established. With the help of normal form method and center manifold theorem, some explicit formulas determining the direction of Hopf bifurcation and the stability of bifurcating period solutions are derived. Finally, numerical simulations are given to illustrate the theoretical results.     

Analysis of the dynamics of Cournot team-game with heterogeneous players

In this paper, we study a dynamical system of a two-team Cournot game played by a team consisting of two firms with bounded rationality and a team consisting of one firm with naive expectation. The equilibrium solutions and the conditions of their locally asymptotic stability are studied. It is demonstrated that, as some parameters in the model are varied, the stability of the equilibrium will get lost and many such complex behaviors as the period bifurcation, chaotic phenomenon, periodic windows, strange attractor and unpredictable trajectories will occur. The great influence of the model parameters on the speed of convergence to the equilibrium is also shown with numerical analysis.     

Bifurcation analysis in a delayed diffusive Nicholson’s blowflies equation

The dynamics of a diffusive Nicholson’s blowflies equation with a finite delay and Dirichlet boundary condition have been investigated in this paper. The occurrence of steady state bifurcation with the changes of parameter is proved by applying phase plane ideas. The existence of Hopf bifurcation at the positive equilibrium with the changes of specify parameters is obtained, and the phenomenon that the unstable positive equilibrium state without dispersion may become stable with dispersion under certain conditions is found by analyzing the distribution of the eigenvalues. By the theory of normal form and center manifold, an explicit algorithm for determining the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived.