Bifurcations and chaos in a nonlinear discrete time Cournot duopoly game

A nonlinear discrete time Cournot duopoly game is investigated in this paper. The conditions of existence for saddle-node bifurcation, transcritical bifurcation and flip bifurcation are derived using the center manifold theorem and the bifurcation theory. We prove that there exists chaotic behavior in the sense of Marotto＇s definition of chaos. The numerical simulations not only show the consistence with our theoretical analysis, but also exhibit the complex but interesting dynamical behaviors of the model. The computation of maximum Lyapunov exponents confirms the theoretical analysis of the dynamical behaviors of the system.     

Delay driven Hopf bifurcation and chaos in a diffusive toxin producing phytoplankton‐zooplankton model

This paper deals with a diffusive toxin producing phytoplankton‐zooplankton model with maturation delay. By analyzing eigenvalues of the characteristic equation associated with delay parameter, the stability of the positive equilibrium and the existence of Hopf bifurcation are studied. Explicit results are derived for the properties of bifurcating periodic solutions by means of the normal form theory and the center manifold reduction for partial functional differential equations. Numerical simulations not only agree with the theoretical analysis but also exhibit the complex behaviors such as the period‐3, 5, 6, 7, 8, 11, and 12 solutions, cascade of period‐doubling bifurcation in period‐2, 4, quasi‐periodic solutions, and chaos. The key observation is that time delay may control harmful algae blooms (HABs). Moreover, numerical simulations show that the chaotic states induced by the period‐doubling bifurcation are purely temporal, which is stationary in space and oscillatory in time. The investigations may provide some new insights on harmful phytoplankton blooms.     

Complex dynamics in a discrete-time predator-prey system without Allee effect

In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos, in the sense of Marotto, is also proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and richer dynamics behaviors. More specifically, this paper presents the finding of period-one orbit, period-three orbits, and chaos in the sense of Marotto, complete period-doubling bifurcation and invariant circle leading to chaos with a great abundance period-windows, simultaneous occurrance of two different routes (invariant circle and inverse period- doubling bifurcation, and period-doubling bifurcation and inverse period-doubling bifurcation) to chaos for a given bifurcation parameter, period doubling bifurcation with period-three orbits to chaos, suddenly appearing or disappearing chaos, different kind of interior crisis, nice chaotic attractors, coexisting (2,3,4) chaotic sets, non-attracting chaotic set, and so on, in the discrete-time predator-prey system. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding is given of the discrete-time predator-prey systems with Allee effect and without Allee effect.     

Bifurcation,chaos, and their control in a wheelset model

In this paper, we present an improved wheelset motion model with two degrees of freedom and study the dynamic behaviors of the system including the symmetry, the existence and uniqueness of the solution, continuous dependence on initial conditions, and Hopf bifurcation. The dynamic characteristics of the wheelset motion system under a nonholonomic constraint are investigated. These results generalize and improve some known results about the wheelset motion system. Meanwhile, based on multiple equilibrium analysis, calculation of Lyapunov exponents and Poincaré section, the chaotic behaviors of the wheelset system are discussed, which indicates that there are more complex dynamic behaviors in the railway wheelset system with higher order terms of Taylor series of trigonometric functions. This paper has also realized the chaos control and bifurcation control for the wheelset motion system by adaptive feedback control method and linear feedback control. The results show that the chaotic wheelset system and bifurcation wheelset system are all well controlled, whether by controlling the yaw angle and the lateral displacement or only by controlling the yaw angle. Numerical simulations are carried out to further verify theoretical analyses.     

近哈密顿系统极限环的Hopf分支与同宿分支

利用Hopf与同宿两种分支中出现的系数研究了近哈密顿系统Hopf和同宿分支产生的极限环的个数与分布,得到了全局分支产生极限环的一个新的充分条件.     

具有双参数的培养器系统的Hopf分支

We study a chemostat system with two parameters, So-initial density and D-flow-speed of the solution. At first, a generalization of the traditional Hopf bifurcation theorem is given. Then, an existence theorem for the Hopf bifurcation of the chemostat system is presented.     

Bifurcations and distribution of limit cycles for near-Hamiltonian polynomial systems

This paper concerns with limit cycles through Hopf and homoclinic bifurcations for near-Hamiltonian systems. By using the coefficients appeared in Melnikov functions at the centers and homoclinic loops, some sufficient conditions are obtained to find limit cycles.     

Stability and bifurcation analysis for a neural network model with discrete and distributed delays

In this paper, a two‐neuron network with both discrete and distributed delays is considered. With the corresponding characteristic equation analyzed, the local stability of the trivial equilibrium is investigated. With the discrete time delay taken as a bifurcation parameter, the existence of Hopf bifurcation is established. Moreover, formulae for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are derived. Finally, numerical simulations are carried out to illustrate the main results and further to exhibit that there is a characteristic sequence of bifurcations leading to a chaotic dynamics, which implies that the system admits rich and complex dynamics. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Dynamical behaviors of a discrete two-dimensional competitive system exactly driven by the large centre

In this paper, a new discrete large-sub-center system is obtained by using the Euler and nonstandard discretization methods for the corresponding continuous system. It is surprised that all dynamic behaviors of the discrete system are exactly driven by the large-center equation, for example, the stabilities, the bifurcations, the period-doubling orbits, and the chaotic dynamics, etc. Additionally, the global asymptotical stability, the existence of exact 2-periodic solutions, the flip bifurcation theorem, and the invariant set of the sub-center equation is also given. These results reveal far richer dynamics of the discrete model compared with the continuous model. Through numerical simulation, we can observe some complex dynamic behaviors, such as period-doubling cascade, periodic windows, chaotic dynamics, etc. Especially, our theoretical results are also showed by those numerical simulations.     

一类耦合非线性波方程的行波解分支

利用动力系统的Hopf分支理论来研究耦合非线性波方程周期行波解的存在性和稳定性．应用行波法把一类耦合非线性波方程转换为三维动力系统来研究,从而给在不同的参数条件下给出了周期解存在和稳定性的充分条件．     

Bifurcations for a predator-prey system with two delays

In this paper, a predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we show that Hopf bifurcations can occur as τ crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation result of [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799-4838], we may show the global existence of periodic solutions.     

Neimark-Sacker bifurcation and chaos control in Hassell-Varley model

We investigate the complex behaviour of a modified Nicholson–Bailey model. The modification is proposed by Hassel and Varley taking into account that interaction between parasitoids is taken in such a way that the searching area per parasitoid is inversely proportional to the m-th power of the population density of parasitoids. Under certain parametric conditions the unique positive equilibrium point of system is locally asymptotically stable. Moreover, it is proved that system undergoes Neimark-Sacker bifurcation for small range of parameters by using standard mathematical techniques of bifurcation theory. In order to control Neimark-Sacker bifurcation, we apply simple feedback control strategy and pole-placement technique which is a modification of OGY method. Moreover, the hybrid control methodology is also implemented for chaos controlling. Numerical simulations are provided to illustrate theoretical discussion.     

Dynamical behavior analysis of a two-dimensional discrete predator-prey model with prey refuge and fear factor

This paper investigates the dynamics of an improved discrete Leslie-Gower predator-prey model with prey refuge and fear factor. First, a discrete Leslie-Gower predator-prey model with prey refuge and fear factor has been introduced. Then, the existence and stability of fixed points of the model are analyzed. Next, the bifurcation behaviors are discussed, both flip bifurcation and Neimark-Sacker bifurcation have been studied. Finally, some simulations are given to show the effectiveness of the theoretical results.     

Bifurcation and chaos in an epidemic model with nonlinear incidence rates

This paper investigates a discrete-time epidemic model by qualitative analysis and numerical simulation. It is verified that there are phenomena of the transcritical bifurcation, flip bifurcation, Hopf bifurcation types and chaos. Also the largest Lyapunov exponents are numerically computed to confirm further the complexity of these dynamic behaviors. The obtained results show that discrete epidemic model can have rich dynamical behavior.     

Complex dynamic behaviors of a discrete-time predator-prey system

The dynamics of a discrete-time predator-prey system is investigated in detail in this paper. It is shown that the system undergoes flip bifurcation and Hopf bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto''s chaos is proved when some certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-6, 7, 8, 10, 14, 18, 24, 36, 50 orbits, attracting invariant cycles, quasi-periodic orbits, nice chaotic behaviors which appear and disappear suddenly, coexisting chaotic attractors, etc. These results reveal far richer dynamics of the discrete-time predator-prey system. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

Bifurcations in a diffusive predator–prey model with predator saturation and competition response

A diffusive predator–prey model with predator saturation and competition response subject to homogeneous Neumann boundary conditions is considered in this paper. We find that the spatially homogeneous and non‐homogeneous periodic solutions through all parameters of the system are spatially homogeneous. To verify our theoretical results, some numerical simulations are also carried out. Copyright © 2014 John Wiley & Sons, Ltd.     

一个具有唯一鞍焦点的三维混沌系统分析

讨论了一个具有唯一鞍焦点的多参数三维混沌系统,该系统包含了Sprott提出的一个最简混沌模型.在特定的条件下得到了Hopf分岔的存在性条件;进一步利用规范型理论获得了决定Hopf分岔方向和分支周期解稳定性的公式,同时利用计算机模拟证实本文的理论分析结果.