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HOPF BIFURCATION OF AN OSCILLATOR WITH QUADRATIC AND CUBIC NONLINEARITIES AND WITH DELAYED VELOCITY FEEDBACK 被引量:6
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作者 王怀磊 王在华 胡海岩 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2004年第4期426-434,共9页
This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms,and with linear delayed velocity feedback.The analysis indicates that for a sufficiently large velocity feedback gain,th... This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms,and with linear delayed velocity feedback.The analysis indicates that for a sufficiently large velocity feedback gain,the equilibrium of the system may undergo a number of stability switches with an increase of time delay,and then becomes unstable forever.At each critical value of time delay for which the system changes its stability,a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay.The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability.It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions. 展开更多
关键词 delay differential equation stability switches supercritical hopf bifurcation subcritical hopf bifurcation Fredholm alternative
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Bifurcation Analysis of a Nonlinear Genetic Network Model with Time Delay
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作者 Anael Verdugo 《Journal of Applied Mathematics and Physics》 2023年第8期2252-2266,共15页
This paper presents a bifurcation study of a mRNA-protein network with negative feedback and time delay. The network is modeled as a coupled system of N ordinary differential equations (ODEs) and N delay differential ... This paper presents a bifurcation study of a mRNA-protein network with negative feedback and time delay. The network is modeled as a coupled system of N ordinary differential equations (ODEs) and N delay differential equations (DDEs). Linear analysis of the stable equilibria shows the existence of a critical time delay beyond which limit cycle oscillations are born in a Hopf bifurcation. The Poincaré-Lindstedt perturbation method is applied to the nonlinear system, resulting in closed form approximate expressions for the amplitude and frequency of oscillation. We confirm our perturbation analysis results by numerically simulating the full 2N-dimensional nonlinear system for N = 2, 7, 15, and 30. Our results show that for small perturbations the equilibrium undergoes a supercritical Hopf and the system exhibits stable periodic solutions. Furthermore, our closed form numerical expressions exhibit the importance of the network’s negative feedback and nonlinear effects. 展开更多
关键词 bifurcation Analysis hopf Delay differential Equations
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Hopf Bifurcation of a Positive Feedback Delay Differential Equation
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作者 陈玉明 黄立宏 《Northeastern Mathematical Journal》 CSCD 2003年第3期213-223,共11页
Under some minor technical hypotheses, for each T larger than a certain rS > 0, Krisztin, Walther and Wu showed the existence of a periodic orbit for the positive feedback delay differential equation x(t) = -rμx(t... Under some minor technical hypotheses, for each T larger than a certain rS > 0, Krisztin, Walther and Wu showed the existence of a periodic orbit for the positive feedback delay differential equation x(t) = -rμx(t) + rf(x(t-1)), where r and μ are positive constants and f : R → R satisfies f(0) = 0 and f' > 0. Combining this with a unique result of Krisztin and Walther, we know that this periodic orbit is the one branched out from 0 through Hopf bifurcation. Using the normal form theory for delay differential equations, we show the same result under the condition that f ∈ C3(R,R) is such that f''(0) = 0 and f'''(0) < 0, which is weaker than those of Krisztin and Walther. 展开更多
关键词 delay differential equation positive feedback hopf bifurcation
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退化时滞微分系统的Hopf分支 被引量:1
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作者 杨芳 蒋威 《应用数学》 CSCD 北大核心 2007年第1期53-58,共6页
本文讨论了一般二维退化时滞微分系统当τ≠0时平衡点稳定性的范围,并以滞量τ为分支参数研究系统出现Hopf分支的条件.
关键词 退化时滞微分系统 hopf分支 特征方程
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三维退化时滞微分系统的Hopf分支
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作者 朱玲 蒋威 《大学数学》 北大核心 2008年第2期61-65,共5页
讨论了三维退化时滞微分系统的Hopf分支.通过分析其特征方程,发现当时滞穿越某些值时出现了分支.给出了寻找Hopf分支点的计算方法.
关键词 时滞 退化微分系统 hopf分支 稳定性
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A perturbation-incremental scheme for studying Hopf bifurcation in delayed differential systems 被引量:15
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作者 CHUNG Kwok Wai 《Science China(Technological Sciences)》 SCIE EI CAS 2009年第3期698-708,共11页
A new method, called perturbation-incremental scheme (PIS), is presented to investigate the periodic solution derived from Hopf bifurcation due to time delay in a system of first-order delayed differential equations. ... A new method, called perturbation-incremental scheme (PIS), is presented to investigate the periodic solution derived from Hopf bifurcation due to time delay in a system of first-order delayed differential equations. The method is summarized as three steps, namely linear analysis at critical value, perturba- tion and increment for continuation. The PIS can bypass and avoid the tedious calculation of the center manifold reduction (CMR) and normal form. Meanwhile, the PIS not only inherits the advantages of the method of multiple scales (MMS) but also overcomes the disadvantages of the incremental harmonic balance (IHB) method. Three delayed systems are used as illustrative examples to demonstrate the validity of the present method. The periodic solution derived from the delay-induced Hopf bifurcation is obtained in a closed form by the PIS procedure. The validity of the results is shown by their consis- tency with the numerical simulation. Furthermore, an approximate solution can be calculated in any required accuracy. 展开更多
关键词 DELAYED differential equation perturbation-incremental scheme hopf bifurcation synchronization center MANIFOLD
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Hopf Bifurcations of Second-Order Differential Equations With Finite Delay and Their Applications
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作者 魏俊杰 黄启昌 《Chinese Science Bulletin》 SCIE EI CAS 1994年第5期358-362,共5页
In recent years, many papers have dealt with the Hopf bifurcations of somespecial delay differential equations. However, only a few scholars have paidattention to the Hopf bifurcations of general delay equations. The ... In recent years, many papers have dealt with the Hopf bifurcations of somespecial delay differential equations. However, only a few scholars have paidattention to the Hopf bifurcations of general delay equations. The purpose of thisnote is to study the Hopf bifurcations of a kind of general second-order differentialequations with finite delay. Our results can also be used to discuss theHopf bifurcations of some third-order delay equations. 展开更多
关键词 FINITE DELAY differential EQUATIONS hopf bifurcations.
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A multi-delay model for pest control with awareness induced interventions — Hopf bifurcation and optimal control analysis
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作者 Fahad Al Basir 《International Journal of Biomathematics》 SCIE 2020年第6期113-134,共22页
Farming awareness is an important measure for pest controlling in agricultural practice.Time delay in controlling pest may affect the system.Time delay occurs in organizing awareness campaigns,also time delay may take... Farming awareness is an important measure for pest controlling in agricultural practice.Time delay in controlling pest may affect the system.Time delay occurs in organizing awareness campaigns,also time delay may takes place in becoming aware of the control strategies or implementing suitable controlling methods informed through social media.Thus we have derived a mathematical model incorporating two time delays into the system and Holling type-II functional response.The existence and the stability criteria of the equilibria are obtained in terms of the basic reproduction number and time delays.Stability changes occur through Hopf-bifurcation when time delays cross the critical values.Optimal control theory has been applied for cost-effectiveness of the delayed system.Numerical simulations are carried out to justify the analytical results.This study shows that optimal farming awareness through radio,TV etc.can control the delay induced bifurcation in a cost-effective way. 展开更多
关键词 Mathematical model Holling type-II functional response delay differential equation(DDE) basic reproduction number hopf bifurcation optimal control
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NUMERICAL HOPF BIFURCATION OF DELAY-DIFFERENTIAL EQUATIONS
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作者 Zhang Chunrui (Dept. of Math., Northeast Forestry University, Harbin 150040) Zheng Baodong (Dept. of Math., Harbin Institute of Technology, Harbin 150001) 《Annals of Differential Equations》 2006年第3期436-441,共6页
In this paper we consider the numerical solution of some delay differential equations undergoing a Hopf bifurcation. We prove that if the delay differential equations have a Hopf bifurcation point atλ=λ*, then the n... In this paper we consider the numerical solution of some delay differential equations undergoing a Hopf bifurcation. We prove that if the delay differential equations have a Hopf bifurcation point atλ=λ*, then the numerical solution of the equation also has a Hopf bifurcation point atλh =λ* + O(h). 展开更多
关键词 delay differential equations Euler-method numerical approximation hopf bifurcation
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Suppression of oscillatory congestion via trunk link bandwidth and control gain in star network 被引量:1
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作者 Sainan WANG Shu ZHANG Jian XU 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2019年第1期25-48,共24页
The time delay-induced instability in an Internet congestion control model is investigated. The star topology is considered, and the link bandwidth ratio and the control gain are selected as the tunable parameters for... The time delay-induced instability in an Internet congestion control model is investigated. The star topology is considered, and the link bandwidth ratio and the control gain are selected as the tunable parameters for congestion suppression. The stability switch boundary is obtained by the eigenvalue analysis for the linearized system around the equilibrium. To investigate the oscillatory congestion when the equilibrium becomes unstable, the center manifold reduction and the normal form theory are used to study the periodic oscillation induced by the delay. The theoretical analysis and numerical simulation show that the ratio between bandwidths of the trunk link and the regular link,rather than these bandwidths themselves, is crucial for the stability of the congestion control system. The present results demonstrate that it is not always effective to increase the link bandwidth ratio for stabilizing the system, and for some certain delays, adjusting the control gain is more efficient. 展开更多
关键词 Internet CONGESTION control hopf bifurcation DELAYED differential equation stability normal form
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THE GLOBAL STABILITY OF A DIFFERENTIAL DELAY MODEL
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作者 李林 《Annals of Differential Equations》 2002年第3期226-234,共9页
The Gierer-Meinhardt's Model with a time delaydx(t)/dt=Co-bx(t)+cx2(t-τ)/y(t)(1+kx2(t-τ)),dy(t)/dt=x2(t)-ay(t).is studied. It is proved that there exists a Hopf bifurcation. Some conditions are established und... The Gierer-Meinhardt's Model with a time delaydx(t)/dt=Co-bx(t)+cx2(t-τ)/y(t)(1+kx2(t-τ)),dy(t)/dt=x2(t)-ay(t).is studied. It is proved that there exists a Hopf bifurcation. Some conditions are established under which the equilibrium is globally stable. 展开更多
关键词 differential delay equations EQUILIBRIUM STABILITY hopf bifurcation
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Chaotic dynamics of a delayed tumor immune interaction model 被引量:2
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作者 Subhas Khajanchi 《International Journal of Biomathematics》 SCIE 2020年第2期1-33,共33页
Due to the unpredictable growth of tumor cells,the tumor-immune interactive dynamics continues to draw attention from both applied mathematicians and oncologists.Math-ematical modeling is a powerful tool to improve ou... Due to the unpredictable growth of tumor cells,the tumor-immune interactive dynamics continues to draw attention from both applied mathematicians and oncologists.Math-ematical modeling is a powerful tool to improve our understanding of the complicated biological system for tumor growth.With this goal,we report a mathematical model which describes how turmor cells evolve and survive the brief encounter with the immune system mediated by immune effector cells and host cells which includes discrete time delay.We analyze the basic mathematical properties of the considered model such as positivity of the system and the boundedness of the solutions.By analyzing the distri-bution of eigenvalucs,local stability analysis of the biologically feasible equilibria and the existence of Hopf bifurcation are obtained in which discrete time delay is used as a bifurcation parameter.Based on the normal form theory and center manifold theorem,we obtain explicit expressions to determine the direction of Hopf bifurcation and the stability of Hopf bifurcating periodic solutions.Numerical simulations are carried out to illustrate the rich dynamical behavior of the delayed tumor model.Our model simula-tions demonstrate that the delayed tumor model exhibits regular and irregular periodic oscillations or chaotic behaviors,which indicate the scenario of long-term tumor relapse. 展开更多
关键词 Tumor-immune interactions delay differential equations hopf bifurcation chaos limit cycle
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A NONLINEAR. DELAY MODEL DESCRIBING THE GROWTH OF TUMOR CELLS UNDER IMMUNE SURVEILLANCE AGAINST CANCER AND ITS STABILITY ANALYSIS 被引量:3
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作者 LING CHEN WANBIAO MA 《International Journal of Biomathematics》 2012年第3期283-295,共13页
In this paper, based on some biological meanings and a model which was proposed by Lefever and Garay (1978), a nonlinear delay model describing the growth of tumor cells under immune surveillance against cancer is g... In this paper, based on some biological meanings and a model which was proposed by Lefever and Garay (1978), a nonlinear delay model describing the growth of tumor cells under immune surveillance against cancer is given. Then, boundedness of the solutions, local stability of the equilibria and Hopf bifurcation of the model are discussed in details. The existence of periodic solutions explains the restrictive interactions between immune surveillance and the growth of the tumor cells. 展开更多
关键词 Immune surveillance differential equation time delay local asymptotic stability hopf bifurcation.
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