A hyperchaotic system from a chaotic system with one saddle and two stable node-foci

This paper presents a 4D new hyperchaotic system which is constructed by a linear controller to a 3D new chaotic system with one saddle and two stable node-foci. Some complex dynamical behaviors such as ultimate boundedness, chaos and hyperchaos of the simple 4D autonomous system are investigated and analyzed. The corresponding bounded hyperchaotic and chaotic attractor is first numerically verified through investigating phase trajectories, Lyapunove exponents, bifurcation path, analysis of power spectrum and Poincaré projections. Finally, two complete mathematical characterizations for 4D Hopf bifurcation are rigorous derived and studied.     

A new hyperchaotic system from the Lü system and its control

This paper presents a 4D new hyperchaotic system which is constructed by a linear controller to a 3D Lü system. Some complex dynamical behaviors such as Hopf bifurcation, chaos and hyperchaos of the simple 4D autonomous system are investigated and analyzed. The corresponding hyperchaotic and chaotic attractor is first numerically verified through investigating phase trajectories, Lyapunove exponents, bifurcation path, analysis of power spectrum and Poincaré projections. Furthermore, the design is illustrated with both simulations and experiments. Finally, the control problem of a new hyperchaotic system is investigated using negative feedback control. Ordinary feedback control, dislocated feedback control and speed feedback control are used to suppress hyperchaos to an unstable equilibrium. Numerical simulations are presented to demonstrate the effectiveness of the proposed controllers.     

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.     

Stability and Hopf Bifurcation of a Type of Protein Synthesis System with Time Delay and Negative Feedback

This paper studied the stability and Hopf bifurcation of a type of protein synthesis system with time delay and negative feedback. Firstly, it is proved theoretically that the time delay, nonlinearity in the protein production and the cooperativity in the negative feedback are key factors to generate circadian oscillation; Taking time delay as a parameter, we obtained the critical value of the time delay that Hopf bifurcation generates. Secondly, based on the center manifold and normal form theorem, we derived the formulas for determining the stability of bifurcating periodic solutions and the supercritical or subcritical Hopf bifurcation. Finally, the matlab program is used to simulate the results.     

A hyperchaotic system from the Rabinovich system

This paper presents a new 4D hyperchaotic system which is constructed by a linear controller to the 3D Rabinovich chaotic system. Some complex dynamical behaviors such as boundedness, chaos and hyperchaos of the 4D autonomous system are investigated and analyzed. A theoretical and numerical study indicates that chaos and hyperchaos are produced with the help of a Liénard-like oscillatory motion around a hypersaddle stationary point at the origin. The corresponding bounded hyperchaotic and chaotic attractors are first numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation path and Poincaré projections. Finally, two complete mathematical characterizations for 4D Hopf bifurcation are rigorously derived and studied.     

Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay

In this paper, we investigated Hopf bifurcation by analyzing the distributed ranges of eigenvalues of characteristic linearized equation. Using communication delay as the bifurcation parameter, linear stability criteria dependent on communication delay have also been derived, and, furthermore, the direction of Hopf bifurcation as well as stability of periodic solution for the exponential RED algorithm with communication delay is studied. We find that the Hopf bifurcation occurs when the communication delay passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, a numerical simulation is presented to verify the theoretical results.     

Hopf bifurcation analysis of a predator-prey system with Holling type IV functional response and time delay

This paper is concerned with a predator-prey system with Holling type IV functional response and time delay. Our aim is to investigate how the time delay affects the dynamics of the predator-prey system. By choosing the delay as a bifurcation parameter, the local asymptotic stability of the positive equilibrium and existence of local Hopf bifurcations are analyzed. Based on the normal form and the center manifold theory, the formulaes for determining the properties of Hopf bifurcation of the predator-prey system are derived. Finally, to support these theoretical results, some numerical simulations are given to illustrate the results.     

Hopf bifurcation analysis in a delayed human respiratory system

This paper considers a delayed human respiratory model. By choosing time delay as a parameter, the stability of the equilibrium of the model is investigated and the conditions which guarantee the existence of local and global Hopf bifurcation are derived. Finally, these results are illustrated by numerical simulations of a specific version of the system.     

具有捕食者相互残杀项时滞系统的Hopf分支

研究了具有捕食者相互残杀项的时滞系统的Hopf分支,通过选择时滞作为一个分支参数,研究了正平衡点的稳定性和正周期解的Hopf分支.而且通过应用规范型和中心流形的理论,得出了确定分支方向的明确的算法.     

Bifurcation analysis of a non-standard finite difference scheme for a time-delayed model of asset prices

In this paper, we apply a non-standard finite difference scheme to a time-delayed model of speculative asset markets and discuss the effect of time delay on the dynamics of asset prices. Firstly, the stability of the positive equilibrium of the system is investigated by analysing the characteristic equation. By choosing the time delay as a bifurcation parameter, we prove that Hopf bifurcations occur when the delay passes a sequence of critical values. Then, the explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived. Finally, some numerical simulations are given to verify the theoretical analysis.     

带有非线性收获和妊娠时滞的微分代数捕食者-食饵系统的稳定性及Hopf分支

本文研究了一类同时带有非线性食饵收获和捕食者妊娠时滞的微分代数捕食者-食饵系统的稳定性及Hopf分支问题.利用了分支理论和稳定性理论,以捕食者妊娠时滞作为系统的分支参数,获得了所提出的新系统在正平衡点处系统稳定性的相关判据条件和Hopf分支的产生条件.推广了一般带有线性收获和时滞的微分代数捕食者-食饵系统的结论.     

Dynamics of a congestion control model in a wireless access network

In this paper, a congestion control algorithm with heterogeneous delays in a wireless access network is considered. We regard the communication time delay as a bifurcating parameter to study the dynamical behaviors, i.e., local asymptotical stability, Hopf bifurcation and resonant codimension-two bifurcation. By analyzing the associated characteristic equation, the Hopf bifurcation occurs when the delay passes through a sequence of critical value. Furthermore, the direction and stability of the bifurcating periodic solutions are derived by applying the normal form theory and the center manifold theorem. In the meantime, the resonant codimension-two bifurcation is also found in this model. Some numerical examples are finally performed to verify the theoretical results.     

Stability and bifurcation of a Cohen-Grossberg neural network with discrete delays

A Cohen-Grossberg neural network with discrete delays is investigated in this paper. The qualitative analysis is given for the system and it is found that the system undergoes a sequence of Hopf bifurcations by choosing the discrete time delay as a bifurcation parameter. Moreover, by applying the normal form theory and the center manifold theorem, the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained. Numerical simulations are given to illustrate the obtained results.     

Influence of discrete delay on pattern formation in a ratio-dependent prey–predator model

In this paper we explore how the two mechanisms, Turing instability and Hopf bifurcation, interact to determine the formation of spatial patterns in a ratio-dependent prey–predator model with discrete time delay. We conduct both rigorous analysis and extensive numerical simulations. Results show that four types of patterns, cold spot, labyrinthine, chaotic as well as mixture of spots and labyrinthine can be observed with and without time delay. However, in the absence of time delay, the two aforementioned mechanisms have a significant impact on the emergence of spatial patterns, whereas only Hopf bifurcation threshold is derived by considering the discrete time delay as the bifurcation parameter. Moreover, time delay promotes the emergence of spatial patterns via spatio-temporal Hopf bifurcation compared to the non-delayed counterpart, implying the destabilizing role of time delay. In addition, the destabilizing role is prominent when the magnitude of time delay and the ratio of diffusivity are comparatively large.     

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.     

Local and global Hopf bifurcation in a neutral population model with age structure

In this paper, an age‐structured population model with the form of neutral functional differential equation is studied. We discuss the stability of the positive equilibrium by analyzing the characteristic equation. Local Hopf bifurcation results are also obtained by choosing the mature delay as bifurcation parameter. On the center manifold, the normal form of the Hopf bifurcation is derived, and explicit formulae for determining the criticality of bifurcation are theoretically given. Moreover, the global continuation of Hopf bifurcating periodic solutions is investigated by using the global Hopf bifurcation theory of neutral equations. Finally, some numerical examples are carried out to support the main results.     

一类带无选择性捕获和时滞的捕食食饵系统的Hopf分支分析

本文主要研究了一个改进的带时滞和无选择捕获函数的捕食-食饵生态经济系统的稳定性和Hopf分支.利用微分代数系统的稳定性理论和分支理论,得到了系统正平衡点稳定性的条件,以及当时滞τ作为分支参数时系统产生Hopf分支的条件.对Leslie-Gower捕食-食饵模型进行了一定程度的完善,使得建立的模型更符合实际情况,因此得到的结论也更加科学.     

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 bifurcation in a harvested one-predator–two-prey model with delays

It is known that one-predator–two-prey system with constant rate harvesting can exhibit very rich dynamics. If such a system contains time delayed component, it can have more interesting behavior. In this paper we study the effects of the time delay on the dynamics of the harvested one-predator–two-prey model. It is shown that time delay can cause a stable equilibrium to become unstable. By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay τ crosses some critical values. The direction and stability of the Hopf bifurcation are investigated by following the procedure of deriving normal form given by Faria and Magalhães. An example is given and numerical simulations are finally performed for justifying the theoretical results.     

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.