两类偏泛函微分方程的行波解的Hopf分支
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摘要
本文主要借助时滞微分方程的平衡点稳定性的判定方法和Hopf分支理论探讨了时滞量大小对两类偏泛函微分方程的行波解的动力学行为的影响。2001年,Wu Jianghong[1]给出了下面时滞Frisher-Kpp方程行波解的存在性2001年Joseph W.-H. So, Xingfu Zou[2]也给出了下面时滞Nicholson’s Blowfies方程行波解的存在性
当上述两个方程存在行波解u ( x ,t )= ? ( z), z = x + ct和N ( x ,t )= ? ( z), z = x +ct(其中c表示波速)时,我们分别把上述两个转化为时滞微分方程如下:
这时我们借助时滞微分方程中研究平衡点稳定性的判定方法和Hopf分支理论研究了上述两类偏泛函微分方程的行波解在时滞量影响下的定性行为。我们发现时滞量大小对其行波解的定性行为有着很大的影响,当时滞量穿过某个时滞阈值τ0时,其行波解将变为周期行波解。
This paper mainly discussed that delay term had impact on traveling wave solution of two partial functional differential equations employing the method of stability of equilibrium and theory of Hopf bifurcation. In 2001 years, Wu Jianhong[1] acquired the existence about the traveling wave solution of delay Fish-Kpp equation and in 2001years, Joseph W.-H. So, Xingfu Zou[2] also acquired the existence about the traveling wave solution of delay Nicholson’s Blowfies equation
When the above two equations have traveling wave solution u ( x ,t )= ? ( z),z = x + ctand N ( x ,t )= ? ( z), z = x + ct, we can transform the above equations into delay differential equations respectively as follows:
We employed method of equilibrium stability and theory of Hopf bifurcation to research the qualitative behavior of the traveling wave solution of the partial functional differential equations and found that delay term has important impact on the traveling wave solution and when delay term across a certain delay valueτ0it will become periodic traveling solution.
当上述两个方程存在行波解u ( x ,t )= ? ( z), z = x + ct和N ( x ,t )= ? ( z), z = x +ct(其中c表示波速)时,我们分别把上述两个转化为时滞微分方程如下:
这时我们借助时滞微分方程中研究平衡点稳定性的判定方法和Hopf分支理论研究了上述两类偏泛函微分方程的行波解在时滞量影响下的定性行为。我们发现时滞量大小对其行波解的定性行为有着很大的影响,当时滞量穿过某个时滞阈值τ0时,其行波解将变为周期行波解。
This paper mainly discussed that delay term had impact on traveling wave solution of two partial functional differential equations employing the method of stability of equilibrium and theory of Hopf bifurcation. In 2001 years, Wu Jianhong[1] acquired the existence about the traveling wave solution of delay Fish-Kpp equation and in 2001years, Joseph W.-H. So, Xingfu Zou[2] also acquired the existence about the traveling wave solution of delay Nicholson’s Blowfies equation
When the above two equations have traveling wave solution u ( x ,t )= ? ( z),z = x + ctand N ( x ,t )= ? ( z), z = x + ct, we can transform the above equations into delay differential equations respectively as follows:
We employed method of equilibrium stability and theory of Hopf bifurcation to research the qualitative behavior of the traveling wave solution of the partial functional differential equations and found that delay term has important impact on the traveling wave solution and when delay term across a certain delay valueτ0it will become periodic traveling solution.
引文
[1] Jianhong Wu. Traveling wave fronts of reaction-diffusion systems with delay. Journal of Dynamics and Differential Equations.[J]. Vol(13), No.3, 2001.
[2] So. J, W. H, Wu, J. & Zou, X. Traveling waves for the diffusive Nicholson’s blowflies equation, Appl. Math. Comput.[J]. 122(2001): 385-395.
[3] Wei J, Ruan S G. Stability and bifurcation in a neural network model with two delays Physica D. [J]. 1999, 130: 255-2772.
[4] Liao X F, Cheng G R. Local stability, Hopf and resonant codimension-two bifurcation in a harmonic oscillator with two delays.[J]. International Journal of Bifurcation and Chaos, 2001, 11:2105-2121.
[5] Hong T, Hughes P C. Effect of time delay on the stability of flexible srtucters with rate feedback Control. Journal of Vibration and Control.[J]. 2001, 7:33-49.
[6] Gourley S A, Chaplain M A. Traveling fronts in a food-limited population model with delay. Proceedings of the Royal Society of Edinburgh.[J]. 2002, 132A: 75-89.
[7] Azuma T, Ikeda K, Kondo T, Uchida K. Memory state feedback control synthesis for linear systems with time delay via a finite number of linear matrix inequalities.Computers and Electrical Engineering.[J]. 2002, 28:217-228.
[8]廖晓昕.稳定性的数学理论及应用.[M].武汉:华中师范大学出版社, 2001.
[9] Olgac N, Sipahi R. An exact method for the stability analysis of time-delayed linear time-invariant systems. IEEE Transactions on Automatic Control.[J]. 2002, 47(5):793-797.
[10] Xu J, Chung K W. Effect of time delayed position feedback on a van der pol-duffing oscillator. Physica D.[J]. 2003, 180:17-39.
[11] Buono P L. Belar J. Restrictions and unfolding of double Hopf bifurcation in functional differential equations. J Diff Equa.[J]. 2003, 189(1)234-266.
[12] Chen Y,Wu J. Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neutral network. Physica D.[J]. 1999, 134:185-199.
[13] Raghothama A, Narayanan S. Periodic response and chaos in nonlinear systems with parametric excitation and time delay. Nonlinear Dynamics.[J]. 2002, 27:341-365.
[14] Ucar A. On the chaotic behaviour of a prototype delayed dynamical system. Chaos, Solitons and Fractals.[J]. 2003, 16:187-194.
[15] Das S L, Chatterjee A. Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations. Nonlinear Dynamics.[J]. 2002, 30(4):323-335.
[16] Xu J, Chung K. W. Delay reduced double Hopf bifurcation in a limit cycle oscillator:extension of a perturbation-incremental method. Dynamics of Continuous and Impulsive Systems B.[J]. 2004, 11:136-143.
[17]宋永利,韩茂安.具Allee反应的时滞扩散单种群增长模型的波前解.[J]. 2005, 173-180,数学年刊.
[18] Ma S. Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations.[J].171(2001): 293-314.
[19] Chaaf K. Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Soc.[J]. 302(1987): 587-615.
[20] So J, W. H, Wu, J. & Zou, X.. A reaction-diffusion model for a single species with age structure. Traveling fronts on unbounded domains, Proc. R. Soc. Lond. A.[J]. 457(2001): 1841-1853.
[21] Song,Y. Peng, Y.& Han, M. Traveling wave fronts in the diffusive single species model with Allee effect and distribute delay, Appl. Math. Comput.[J]. 152(2004): 483-497.
[22] Aizik I. Volpert, Vitaly A. Volpert,Vladimir A. Volpert. Traveling Wave Solutions of Parabolic Systems, 1994 by the American Mathematical Society.[J].
[23] Li Zheng-yuan, Liu Ying-dong, Ye Qi-xiao. Strong and Weak Property of Travelling Wave Front Solutions for a Class of Degenerate Parabolic Equations, Northeast. Math.[J]. 17 (2) (2001): 195—204.
[24] Dahe Feng, Junliang Lü, Jibin Li, Tianlan He. Bifurcation studies on traveling wave solutions for nonlinear intensity Klein-Gordon equation, Applied Mathematics and Computation.[J]. 189(2007): 271-284.
[25] J. Shen, W. Xu, Y. Jin. Birfurcation method and traveling wave solution to Whitham-Broer-Kaup equation, Appl. Math. Comput.[J]. 171(2005).
[26] J.Shen, W. Xu. Birfurcation of traveling wave solutions in a model of the hydrogen-bondedsystems, Appl. Math. Comput.[J]. 171(2005): 677-702.
[27] Jack Hale. Theory of functional differential equations.[M]. 1977 by Springer-Verlag New York Inc.
[28] John Guckenheimer Philip Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields.[M]. 1983 by Springer-Verlag New York Inc.
[29] Hassard B, Kazarinoff N, Wan Y. Theory and Application of Hopf bifurcation.[M]. Cambridge: Cambridge University Press, 1981.
[30]廖晓峰,吴中福,王康.带分布时延神经网络:从稳定到振荡再到稳定的动力学现象.电子与信息学报.[J].687-692, (2001).
[31]徐鉴.非线行时滞、时变系统分岔和非线性模态研究(博士后工作报告).[D].北京,北京航空航天大学, 1996.
[32]徐鉴,陆启韶,黄克累.一类Van der Pol型时滞系统的稳定性和Hopf分岔见:庄逢甘编现代力学与科技进步——庆祝力学学会成立40周年,现代力学与科技进步学术大会.[C].北京1997-08-26-28北京:清华大学出版社, 1997, 1551-1555.
[33] Xinye Li, J.C. Ji , Colin H. Hansen. Dynamics of two delay coupled van derPoloscillator Mechanics Research Communications.[J]. 33 (2006): 614–627.
[34] Yang Gaoxiang, Shu Yonglu. Hopf bifurcation of the traveling wave solution of delayed Fisher-KPP equation, Applied Mathematics and Computation.[J]. (to submit).
[35] Yang Gaoxiang, Shu Yonglu. Hopf bifurcation of the traveling wave solution of Nicholson blowflies equation. (to submit).
[2] So. J, W. H, Wu, J. & Zou, X. Traveling waves for the diffusive Nicholson’s blowflies equation, Appl. Math. Comput.[J]. 122(2001): 385-395.
[3] Wei J, Ruan S G. Stability and bifurcation in a neural network model with two delays Physica D. [J]. 1999, 130: 255-2772.
[4] Liao X F, Cheng G R. Local stability, Hopf and resonant codimension-two bifurcation in a harmonic oscillator with two delays.[J]. International Journal of Bifurcation and Chaos, 2001, 11:2105-2121.
[5] Hong T, Hughes P C. Effect of time delay on the stability of flexible srtucters with rate feedback Control. Journal of Vibration and Control.[J]. 2001, 7:33-49.
[6] Gourley S A, Chaplain M A. Traveling fronts in a food-limited population model with delay. Proceedings of the Royal Society of Edinburgh.[J]. 2002, 132A: 75-89.
[7] Azuma T, Ikeda K, Kondo T, Uchida K. Memory state feedback control synthesis for linear systems with time delay via a finite number of linear matrix inequalities.Computers and Electrical Engineering.[J]. 2002, 28:217-228.
[8]廖晓昕.稳定性的数学理论及应用.[M].武汉:华中师范大学出版社, 2001.
[9] Olgac N, Sipahi R. An exact method for the stability analysis of time-delayed linear time-invariant systems. IEEE Transactions on Automatic Control.[J]. 2002, 47(5):793-797.
[10] Xu J, Chung K W. Effect of time delayed position feedback on a van der pol-duffing oscillator. Physica D.[J]. 2003, 180:17-39.
[11] Buono P L. Belar J. Restrictions and unfolding of double Hopf bifurcation in functional differential equations. J Diff Equa.[J]. 2003, 189(1)234-266.
[12] Chen Y,Wu J. Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neutral network. Physica D.[J]. 1999, 134:185-199.
[13] Raghothama A, Narayanan S. Periodic response and chaos in nonlinear systems with parametric excitation and time delay. Nonlinear Dynamics.[J]. 2002, 27:341-365.
[14] Ucar A. On the chaotic behaviour of a prototype delayed dynamical system. Chaos, Solitons and Fractals.[J]. 2003, 16:187-194.
[15] Das S L, Chatterjee A. Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations. Nonlinear Dynamics.[J]. 2002, 30(4):323-335.
[16] Xu J, Chung K. W. Delay reduced double Hopf bifurcation in a limit cycle oscillator:extension of a perturbation-incremental method. Dynamics of Continuous and Impulsive Systems B.[J]. 2004, 11:136-143.
[17]宋永利,韩茂安.具Allee反应的时滞扩散单种群增长模型的波前解.[J]. 2005, 173-180,数学年刊.
[18] Ma S. Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations.[J].171(2001): 293-314.
[19] Chaaf K. Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Soc.[J]. 302(1987): 587-615.
[20] So J, W. H, Wu, J. & Zou, X.. A reaction-diffusion model for a single species with age structure. Traveling fronts on unbounded domains, Proc. R. Soc. Lond. A.[J]. 457(2001): 1841-1853.
[21] Song,Y. Peng, Y.& Han, M. Traveling wave fronts in the diffusive single species model with Allee effect and distribute delay, Appl. Math. Comput.[J]. 152(2004): 483-497.
[22] Aizik I. Volpert, Vitaly A. Volpert,Vladimir A. Volpert. Traveling Wave Solutions of Parabolic Systems, 1994 by the American Mathematical Society.[J].
[23] Li Zheng-yuan, Liu Ying-dong, Ye Qi-xiao. Strong and Weak Property of Travelling Wave Front Solutions for a Class of Degenerate Parabolic Equations, Northeast. Math.[J]. 17 (2) (2001): 195—204.
[24] Dahe Feng, Junliang Lü, Jibin Li, Tianlan He. Bifurcation studies on traveling wave solutions for nonlinear intensity Klein-Gordon equation, Applied Mathematics and Computation.[J]. 189(2007): 271-284.
[25] J. Shen, W. Xu, Y. Jin. Birfurcation method and traveling wave solution to Whitham-Broer-Kaup equation, Appl. Math. Comput.[J]. 171(2005).
[26] J.Shen, W. Xu. Birfurcation of traveling wave solutions in a model of the hydrogen-bondedsystems, Appl. Math. Comput.[J]. 171(2005): 677-702.
[27] Jack Hale. Theory of functional differential equations.[M]. 1977 by Springer-Verlag New York Inc.
[28] John Guckenheimer Philip Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields.[M]. 1983 by Springer-Verlag New York Inc.
[29] Hassard B, Kazarinoff N, Wan Y. Theory and Application of Hopf bifurcation.[M]. Cambridge: Cambridge University Press, 1981.
[30]廖晓峰,吴中福,王康.带分布时延神经网络:从稳定到振荡再到稳定的动力学现象.电子与信息学报.[J].687-692, (2001).
[31]徐鉴.非线行时滞、时变系统分岔和非线性模态研究(博士后工作报告).[D].北京,北京航空航天大学, 1996.
[32]徐鉴,陆启韶,黄克累.一类Van der Pol型时滞系统的稳定性和Hopf分岔见:庄逢甘编现代力学与科技进步——庆祝力学学会成立40周年,现代力学与科技进步学术大会.[C].北京1997-08-26-28北京:清华大学出版社, 1997, 1551-1555.
[33] Xinye Li, J.C. Ji , Colin H. Hansen. Dynamics of two delay coupled van derPoloscillator Mechanics Research Communications.[J]. 33 (2006): 614–627.
[34] Yang Gaoxiang, Shu Yonglu. Hopf bifurcation of the traveling wave solution of delayed Fisher-KPP equation, Applied Mathematics and Computation.[J]. (to submit).
[35] Yang Gaoxiang, Shu Yonglu. Hopf bifurcation of the traveling wave solution of Nicholson blowflies equation. (to submit).