The double Hopf bifurcation of a composite laminated piezoelectric plate with combined external and internal excitations is studied. Using a multiple scale method, the average equations are obtained in two coordinates...The double Hopf bifurcation of a composite laminated piezoelectric plate with combined external and internal excitations is studied. Using a multiple scale method, the average equations are obtained in two coordinates. The bifurcation response equations of the composite laminated piezoelectric plate with the primary parameter resonance, i.e., 1：3 internal resonance, are achieved. Then, the bifurcation feature of bifurcation equations is considered using the singularity theory. A bifurcation diagram is obtained on the parameter plane. Different steady state solutions of the average equations are analyzed. By numerical simulation, periodic vibration and quasi-periodic vibration responses of the Composite laminated piezoelectric plate are obtained.展开更多
The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf...The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf bifurcation is given. Both the period-doubling bifurcation and saddle-node bifurcation of periodical solutions are computed since the observed floquet multiplier overpass the unit circle by DDE-Biftool software in Matlab. The continuation of saddle-node bifurcation line or period-doubling curve is carried out as varying free parameters and time delays. Two different transition modes of saddle-node bifurcation are discovered which is verified by numerical simulation work with aids of DDE-Biftool.展开更多
The mathematical model of stem cells is discussed with its motivation to describe the tissue relationship by technically introducing a two compartments model. The clear link between the proliferation phase of stem cel...The mathematical model of stem cells is discussed with its motivation to describe the tissue relationship by technically introducing a two compartments model. The clear link between the proliferation phase of stem cells and the circulating neutrophil phase is set forth after delay feedback control of the state variable of stem cells. Hopf bifurcation is discussed with varying free parameters and time delays. Based on the center manifold theory, the normal form near the critical point is computed and the stability of bifurcating periodical solution is rigorously discussed. With the aids of the artificial tool on-hand which implies how much tedious work doing by DDE-Biftool software, the bifurcating periodic solution after Hopf point is continued by varying time delay.展开更多
An analytical method is introduced to investigate double Hopf bifurcations induced by two delays qualitatively and quantitatively.As an illustrative example,the clear procedure is demonstrated to study delay-induced w...An analytical method is introduced to investigate double Hopf bifurcations induced by two delays qualitatively and quantitatively.As an illustrative example,the clear procedure is demonstrated to study delay-induced weak resonant double Hopf bifurcation in a nonlinear system with multiple delays.When two delays are close to double Hopf bifurcation point,all solutions derived from the bifurcation are classified qualitatively and expressed explicitly.Numerical simulations are a good agreement with our theoretical analysis,and also already work in references.The results show that our work in this paper proposes a simple and valid method for investigating delay-induced double Hopf bifurcations.The important feature of our work is that the explicit expression of periodic solutions is easy to be obtained by solving algebraic equations.展开更多
An HIV infection model with saturated infection rate and double delays is investigated. First, the existence of the infection-free equilibrium E0, the immune-exhausted equilibrium E1 and the infected equilibrium E2 wi...An HIV infection model with saturated infection rate and double delays is investigated. First, the existence of the infection-free equilibrium E0, the immune-exhausted equilibrium E1 and the infected equilibrium E2 with immunity in different conditions is shown. By analyzing the characteristic equation, we study the locally asymptotical stability of the trivial equilibrium, and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold. Then with suitable Lyapunov function and LaSalle＇s invariance principle, the global stability of the three equilibriums is obtained. Finally, numerical simulations are presented to illustrate the main mathematical results.展开更多
研究了具有时滞耦合的n个van der Pol振子系统中发生的弱共振双Hopf分岔.应用改进的多尺度方法,得到了2∶5共振的复振幅方程.通过将复振幅设为极坐标形式,将复振幅方程转化为一个二维的实振幅系统.通过研究实振幅方程的平衡点及其稳定性...研究了具有时滞耦合的n个van der Pol振子系统中发生的弱共振双Hopf分岔.应用改进的多尺度方法,得到了2∶5共振的复振幅方程.通过将复振幅设为极坐标形式,将复振幅方程转化为一个二维的实振幅系统.通过研究实振幅方程的平衡点及其稳定性,对系统在2∶5共振点附近的动力学行为进行了开折和分类.得到了一些有趣的动力学现象,如振幅死区、周期解和双稳态解等,相应的数值模拟验证了理论结果的正确性.展开更多
研究时滞反馈van der Pol-Duffing系统的共振双Hopf分岔,讨论时滞量和位移反馈增益变化对双Hopf分岔的影响。利用Hopf分岔定理得到系统出现1∶2共振双Hopf分岔的充要条件;借助中心流形定理和平均化方法约化了系统,从理论上分析共振双Hop...研究时滞反馈van der Pol-Duffing系统的共振双Hopf分岔,讨论时滞量和位移反馈增益变化对双Hopf分岔的影响。利用Hopf分岔定理得到系统出现1∶2共振双Hopf分岔的充要条件;借助中心流形定理和平均化方法约化了系统,从理论上分析共振双Hopf分岔点附近的动力学行为,得到共振双Hopf分岔引起的各种周期解的近似解析解和稳定性条件;通过数值实验,验证了理论分析的正确性。结果表明,时滞和位移反馈增益不仅导致共振双Hopf分岔,而且会使系统出现多稳态周期运动。展开更多
研究Runge-Kutta方法对以时滞为参数的双时滞van der Pol方程的数值Hopf分支问题。证明当该方程分支参数值在τ1=τ01处产生Hopf分支时,其数值解相应地在分支参数值τ1*=τ01+O(hp)处产生Hopf分支(p为Runge-Kutta方法的方法阶),且以解...研究Runge-Kutta方法对以时滞为参数的双时滞van der Pol方程的数值Hopf分支问题。证明当该方程分支参数值在τ1=τ01处产生Hopf分支时,其数值解相应地在分支参数值τ1*=τ01+O(hp)处产生Hopf分支(p为Runge-Kutta方法的方法阶),且以解析解的分支参数值为极限,从而论证了双时滞van der Pol方程数值解保持其原解析解的动力学特性。展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.11402127,11290152 and 11072008)
文摘The double Hopf bifurcation of a composite laminated piezoelectric plate with combined external and internal excitations is studied. Using a multiple scale method, the average equations are obtained in two coordinates. The bifurcation response equations of the composite laminated piezoelectric plate with the primary parameter resonance, i.e., 1：3 internal resonance, are achieved. Then, the bifurcation feature of bifurcation equations is considered using the singularity theory. A bifurcation diagram is obtained on the parameter plane. Different steady state solutions of the average equations are analyzed. By numerical simulation, periodic vibration and quasi-periodic vibration responses of the Composite laminated piezoelectric plate are obtained.
文摘The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf bifurcation is given. Both the period-doubling bifurcation and saddle-node bifurcation of periodical solutions are computed since the observed floquet multiplier overpass the unit circle by DDE-Biftool software in Matlab. The continuation of saddle-node bifurcation line or period-doubling curve is carried out as varying free parameters and time delays. Two different transition modes of saddle-node bifurcation are discovered which is verified by numerical simulation work with aids of DDE-Biftool.
文摘The mathematical model of stem cells is discussed with its motivation to describe the tissue relationship by technically introducing a two compartments model. The clear link between the proliferation phase of stem cells and the circulating neutrophil phase is set forth after delay feedback control of the state variable of stem cells. Hopf bifurcation is discussed with varying free parameters and time delays. Based on the center manifold theory, the normal form near the critical point is computed and the stability of bifurcating periodical solution is rigorously discussed. With the aids of the artificial tool on-hand which implies how much tedious work doing by DDE-Biftool software, the bifurcating periodic solution after Hopf point is continued by varying time delay.
基金supported by the National Natural Science Foundation of China(Grant Nos.11872175,11572224,61603125 and 21130010)Young Talents Fund of Henan University of Economics and Law+1 种基金National Key Project Cultivation Project of Henan University of Economics and LawKey Research Project of Higher Education Institutions of Henan Province(Grant No.18A110003)。
文摘An analytical method is introduced to investigate double Hopf bifurcations induced by two delays qualitatively and quantitatively.As an illustrative example,the clear procedure is demonstrated to study delay-induced weak resonant double Hopf bifurcation in a nonlinear system with multiple delays.When two delays are close to double Hopf bifurcation point,all solutions derived from the bifurcation are classified qualitatively and expressed explicitly.Numerical simulations are a good agreement with our theoretical analysis,and also already work in references.The results show that our work in this paper proposes a simple and valid method for investigating delay-induced double Hopf bifurcations.The important feature of our work is that the explicit expression of periodic solutions is easy to be obtained by solving algebraic equations.
基金This work was supported by National Natural Science Foundation of China （61174209, 11471034）.
文摘An HIV infection model with saturated infection rate and double delays is investigated. First, the existence of the infection-free equilibrium E0, the immune-exhausted equilibrium E1 and the infected equilibrium E2 with immunity in different conditions is shown. By analyzing the characteristic equation, we study the locally asymptotical stability of the trivial equilibrium, and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold. Then with suitable Lyapunov function and LaSalle＇s invariance principle, the global stability of the three equilibriums is obtained. Finally, numerical simulations are presented to illustrate the main mathematical results.
文摘研究了具有时滞耦合的n个van der Pol振子系统中发生的弱共振双Hopf分岔.应用改进的多尺度方法,得到了2∶5共振的复振幅方程.通过将复振幅设为极坐标形式,将复振幅方程转化为一个二维的实振幅系统.通过研究实振幅方程的平衡点及其稳定性,对系统在2∶5共振点附近的动力学行为进行了开折和分类.得到了一些有趣的动力学现象,如振幅死区、周期解和双稳态解等,相应的数值模拟验证了理论结果的正确性.
文摘研究时滞反馈van der Pol-Duffing系统的共振双Hopf分岔,讨论时滞量和位移反馈增益变化对双Hopf分岔的影响。利用Hopf分岔定理得到系统出现1∶2共振双Hopf分岔的充要条件;借助中心流形定理和平均化方法约化了系统,从理论上分析共振双Hopf分岔点附近的动力学行为,得到共振双Hopf分岔引起的各种周期解的近似解析解和稳定性条件;通过数值实验,验证了理论分析的正确性。结果表明,时滞和位移反馈增益不仅导致共振双Hopf分岔,而且会使系统出现多稳态周期运动。
文摘研究Runge-Kutta方法对以时滞为参数的双时滞van der Pol方程的数值Hopf分支问题。证明当该方程分支参数值在τ1=τ01处产生Hopf分支时,其数值解相应地在分支参数值τ1*=τ01+O(hp)处产生Hopf分支(p为Runge-Kutta方法的方法阶),且以解析解的分支参数值为极限,从而论证了双时滞van der Pol方程数值解保持其原解析解的动力学特性。
