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摘要：为了研究物种的稳定性问题，要求缩小或者扩大生物系统的稳定区域，通过混合控制欧拉法研究了一个时滞Gompertz模型，运用状态反馈和参数扰动控制得到了Neimark-Sacker分支的理想结果。根据Hopf分支理论得到了连续系统平衡点的稳定性，通过混合控制欧拉算法得到了离散系统在要求的分支点所产生的Neimark-Sacker分支，利用中心流形定理和正规形方法，给出了确定分支周期解的分支方向与稳定性的计算公式。采用数值模拟验证了所得结果的正确性。研究结果表明，对于延迟Gompertz模型系统，如果选择合适的控制参数，就能够使分支点提前或者延迟。研究方法在理论和数值模拟方面都得到了良好的预期结果，为解决相关的控制问题提供了新的方法，对其他领域的控制问题研究具有一定的借鉴意义。
关键词：常微分方程数值解; Gompertz模型; 混合控制; 歐拉法; 延迟; Neimark-Sacker分支
中图分类号：O1891文献标志码：A
Abstract： In order to study the stability of species， the biological systems are required to reduce or expand the stable region. For a Gompertz model with time delay， a hybrid control Euler method is proposed in which state feedback and parameter perturbation are used to control the Neimark-Sacker bifurcation. The local stability of the equilibria is discussed according to Hopf bifurcation theory. For controlling Neimark-Sacker bifurcation， the hybrid control numerical algorithm is introduced to generate the Neimark-Sacker bifurcation at a desired bifurcation point. The explicit algorithms for determining the direction of the bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form method and center manifold theorem. Numerical examples are provided to illustrate the theoretical results. The research results show that the branch point can be in advance or delay for the delay Gompertz model system through choosing appropriate control parameters. The algorithm has obtained good results both in theory and numerical performance， which provides a new method and has certain theoretical significance for its application in many control problems.
Keywords：numerical solution of ordinary differential equation; Gompertz model; hybrid control; Euler method; delay; Neimark-Sacker bifurcation
5结论
为了扩大或者缩小控制区域，给出了应用状态反馈和参数扰动的混合控制数值欧拉法得到了Neimark-Sacker分支。对Gompertz连续系统实施混合控制得到了Hopf分支;通过选择合适的控制参数，实施混合控制数值算法延迟了原来分支点的出现，应用混合控制欧拉法，对充分小的步长给出了保持分支的结果。通过理论和数值模拟验证了所得结果，得到了延迟Gompertz模型系统通过选择合适的控制参数，分支点可能提前或者延迟。在将来的研究计划中，笔者将设计更好的数值控制方法，达到更好的控制效果。
参考文献/References：
[1]LOPEZ-GOMEZ J， ORTEGA R， TINEO A. The periodic predator-prey Lotka-Volterra model[J]. Advances in Differential Equations， 1996，1（3）： 403-423.
[2]PIOTROWSKA M J， FORYS U. Analysis of the Hopf bifurcation for the family of angiogenesis models[J].Journal of Mathematical Analysis Applications，2011，382（1）：180-203.
[3]JIA Jianwen， LI Chunhua. A Predator-Prey Gompertz model with time delay and impulsive perturbations on the prey[J]. Discrete Dynamics in Nature Society， 2009（1026）：332-337. [4]DONG Lingzhen， CHEN Lansun， SUN Lihua. Optimal harvesting policies for periodic Gompertz systems[J]. Nonlinear Analysis Real World Applications，2007，8（2）：572-578.
[5]沈啟宏，魏俊杰. 具时滞的人类呼吸系统模型的稳定性与分支[J].应用数学和力学， 2004，25（11）：1169-1181.
SHENG Qihong， WEI Junjie. Stability and bifurcation of a human respiratory system model with time delay[J]. Applied Mathematics and Mechanics， 2004，25（11）：1169-1181.
[6]魏俊杰，张春蕊，李秀玲.具时滞的二维神经网络模型的分支[J].应用数学和力学，2005，26（2）：193-200.
WEI Junjie， ZHENG Chunrui， LI Xiuling. Bifurcation in a two-dimensional neural network model with delay[J]. Applied Mathematics and Mechanics， 2005，26（2）：193-200.
[7]YU Pei， CHEN Guanrong. Hopf bifurcation control using nonlinear feedback with polynomial functions[J]. International Journal of Bifurcation Chaos， 2004，14（5）： 1683-1704.
[8]YU Pei. Bifurcation control for a class of Lorenze-like systems[J].International Journal of Bifurcation Chaos，2011，21（9）： 2647-2664.
[9]CHEN G， MOIOLA J L， WANG H O. Bifurcation control： theories， methods， and applications[J]. International Journal of Bifurcation Chaos，2000，10（3）： 511-548.
[10]HILL D J， HISKENS I A， WANG Y. Robust， adaptive or nonlinear control for modern power systems[C]// Proceedings of the 32nd IEEE Conference on Decision and Control. San Antonio：IEEE Xplore，1993：2335-2340.
[11]CHEN Z， YU P. Hopf bifurcation control for an internet congestion model[J]. International Journal of Bifurcation Chaos， 2005，15（8）：2643-2651.
[12]LIU Zengrong， CHUNG K W. Hybrid control of bifurcation in continuous nonlinear dynamical systems[J]. International Journal of Bifurcation Chaos， 2005，15（12）： 3895-3903.
[13]CHENG Zunshui， CAO Jinde. Hybrid control of Hopf bifurcation in complex networks with delays[J]. Neuro Computing，2014，131：164-170.
[14]SU Huan， DING Xiaohua. Dynamics of a nonstandard finite-difference scheme for Mackey-Glass system[J]. Journal of Mathematical Analysis and Applications， 2008，344（2）： 932-941.
[15]DING Xiaohua， FAN Dejun， LIU Mingzhu. Stability and bifurcation of a numerical discretization Mackey-Glass system[J]. Chaos， Solitons， Fractals， 2007，34（2）： 383-393.
[16]张春蕊，刘明珠.双时滞神经网络模型分支性的数值逼近[J]. 系统仿真学报，2004，16，（4）：797-799.
ZHANG Chunrui， LIU Mingzhu. Hopf bifurcations in numerical approximation for neural network model with two delays[J]. Journal of System Simulation， 2004，16（4）：797-799.
[17]WANG Yuanyuan. Dynamics of a nonstandard finite-difference scheme for delay differential equations with unimodal feedback[J]. Communications in Nonlinear Science Numerical Simulation， 2012，17（10）： 3967-3978. [18]SU Huan， MAO Xuerong， LI Wenxue. Hopf bifurcation control for a class of delay differential systems with discrete-time delayed feedback controller[J]. Chaos， 2016， 26（11）： 113120.
[19]WULF V， FORD N. J. Numerical Hopf bifurcation for a class of delay differential equation[J]. Journal of Computational and Applied Mathematics. 2000，115（1）： 601-616.
[20]RUAN Shigui， WEI Junjie. On the zeros of transcendental functions with applications to stability of delay differential equations with two delays[J]. Dynamics of Continuous Discrete Impulsive Systems， 2003，10（6）： 863-874.
[21]YURI A K. Elements of Applied Bifurcation Theory[M]. New York：Springer-Verlag， 1995.
[22]KUZNETSOV Y. Elements of Applied Bifurcation Theory[M]. New York：Springer-Verlag， 1995.
[23]HALE J. Theory of Functional Differential Equations[M]. New York：Springer-Verlag， 1977.
[24]WULF V. Numerical Analysis of Delay Differential Equations Undergoing a Hopf Bifurcation[D].Liverpool： University of Liverpool， 1999.
[25]WIGGINS S. Introduction to Applied Nonlinear Dynamical System and Chaos[M]. New York：Springer-Verlag， 1990.第40卷第2期河北科技大學学报Vol.40，No.2
2019年4月Journal of Hebei University of Science and TechnologyApr. 2019
关键词：常微分方程数值解; Gompertz模型; 混合控制; 歐拉法; 延迟; Neimark-Sacker分支
中图分类号：O1891文献标志码：A
Abstract： In order to study the stability of species， the biological systems are required to reduce or expand the stable region. For a Gompertz model with time delay， a hybrid control Euler method is proposed in which state feedback and parameter perturbation are used to control the Neimark-Sacker bifurcation. The local stability of the equilibria is discussed according to Hopf bifurcation theory. For controlling Neimark-Sacker bifurcation， the hybrid control numerical algorithm is introduced to generate the Neimark-Sacker bifurcation at a desired bifurcation point. The explicit algorithms for determining the direction of the bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form method and center manifold theorem. Numerical examples are provided to illustrate the theoretical results. The research results show that the branch point can be in advance or delay for the delay Gompertz model system through choosing appropriate control parameters. The algorithm has obtained good results both in theory and numerical performance， which provides a new method and has certain theoretical significance for its application in many control problems.
Keywords：numerical solution of ordinary differential equation; Gompertz model; hybrid control; Euler method; delay; Neimark-Sacker bifurcation
5结论
为了扩大或者缩小控制区域，给出了应用状态反馈和参数扰动的混合控制数值欧拉法得到了Neimark-Sacker分支。对Gompertz连续系统实施混合控制得到了Hopf分支;通过选择合适的控制参数，实施混合控制数值算法延迟了原来分支点的出现，应用混合控制欧拉法，对充分小的步长给出了保持分支的结果。通过理论和数值模拟验证了所得结果，得到了延迟Gompertz模型系统通过选择合适的控制参数，分支点可能提前或者延迟。在将来的研究计划中，笔者将设计更好的数值控制方法，达到更好的控制效果。
参考文献/References：
[1]LOPEZ-GOMEZ J， ORTEGA R， TINEO A. The periodic predator-prey Lotka-Volterra model[J]. Advances in Differential Equations， 1996，1（3）： 403-423.
[2]PIOTROWSKA M J， FORYS U. Analysis of the Hopf bifurcation for the family of angiogenesis models[J].Journal of Mathematical Analysis Applications，2011，382（1）：180-203.
[3]JIA Jianwen， LI Chunhua. A Predator-Prey Gompertz model with time delay and impulsive perturbations on the prey[J]. Discrete Dynamics in Nature Society， 2009（1026）：332-337. [4]DONG Lingzhen， CHEN Lansun， SUN Lihua. Optimal harvesting policies for periodic Gompertz systems[J]. Nonlinear Analysis Real World Applications，2007，8（2）：572-578.
[5]沈啟宏，魏俊杰. 具时滞的人类呼吸系统模型的稳定性与分支[J].应用数学和力学， 2004，25（11）：1169-1181.
SHENG Qihong， WEI Junjie. Stability and bifurcation of a human respiratory system model with time delay[J]. Applied Mathematics and Mechanics， 2004，25（11）：1169-1181.
[6]魏俊杰，张春蕊，李秀玲.具时滞的二维神经网络模型的分支[J].应用数学和力学，2005，26（2）：193-200.
WEI Junjie， ZHENG Chunrui， LI Xiuling. Bifurcation in a two-dimensional neural network model with delay[J]. Applied Mathematics and Mechanics， 2005，26（2）：193-200.
[7]YU Pei， CHEN Guanrong. Hopf bifurcation control using nonlinear feedback with polynomial functions[J]. International Journal of Bifurcation Chaos， 2004，14（5）： 1683-1704.
[8]YU Pei. Bifurcation control for a class of Lorenze-like systems[J].International Journal of Bifurcation Chaos，2011，21（9）： 2647-2664.
[9]CHEN G， MOIOLA J L， WANG H O. Bifurcation control： theories， methods， and applications[J]. International Journal of Bifurcation Chaos，2000，10（3）： 511-548.
[10]HILL D J， HISKENS I A， WANG Y. Robust， adaptive or nonlinear control for modern power systems[C]// Proceedings of the 32nd IEEE Conference on Decision and Control. San Antonio：IEEE Xplore，1993：2335-2340.
[11]CHEN Z， YU P. Hopf bifurcation control for an internet congestion model[J]. International Journal of Bifurcation Chaos， 2005，15（8）：2643-2651.
[12]LIU Zengrong， CHUNG K W. Hybrid control of bifurcation in continuous nonlinear dynamical systems[J]. International Journal of Bifurcation Chaos， 2005，15（12）： 3895-3903.
[13]CHENG Zunshui， CAO Jinde. Hybrid control of Hopf bifurcation in complex networks with delays[J]. Neuro Computing，2014，131：164-170.
[14]SU Huan， DING Xiaohua. Dynamics of a nonstandard finite-difference scheme for Mackey-Glass system[J]. Journal of Mathematical Analysis and Applications， 2008，344（2）： 932-941.
[15]DING Xiaohua， FAN Dejun， LIU Mingzhu. Stability and bifurcation of a numerical discretization Mackey-Glass system[J]. Chaos， Solitons， Fractals， 2007，34（2）： 383-393.
[16]张春蕊，刘明珠.双时滞神经网络模型分支性的数值逼近[J]. 系统仿真学报，2004，16，（4）：797-799.
ZHANG Chunrui， LIU Mingzhu. Hopf bifurcations in numerical approximation for neural network model with two delays[J]. Journal of System Simulation， 2004，16（4）：797-799.
[17]WANG Yuanyuan. Dynamics of a nonstandard finite-difference scheme for delay differential equations with unimodal feedback[J]. Communications in Nonlinear Science Numerical Simulation， 2012，17（10）： 3967-3978. [18]SU Huan， MAO Xuerong， LI Wenxue. Hopf bifurcation control for a class of delay differential systems with discrete-time delayed feedback controller[J]. Chaos， 2016， 26（11）： 113120.
[19]WULF V， FORD N. J. Numerical Hopf bifurcation for a class of delay differential equation[J]. Journal of Computational and Applied Mathematics. 2000，115（1）： 601-616.
[20]RUAN Shigui， WEI Junjie. On the zeros of transcendental functions with applications to stability of delay differential equations with two delays[J]. Dynamics of Continuous Discrete Impulsive Systems， 2003，10（6）： 863-874.
[21]YURI A K. Elements of Applied Bifurcation Theory[M]. New York：Springer-Verlag， 1995.
[22]KUZNETSOV Y. Elements of Applied Bifurcation Theory[M]. New York：Springer-Verlag， 1995.
[23]HALE J. Theory of Functional Differential Equations[M]. New York：Springer-Verlag， 1977.
[24]WULF V. Numerical Analysis of Delay Differential Equations Undergoing a Hopf Bifurcation[D].Liverpool： University of Liverpool， 1999.
[25]WIGGINS S. Introduction to Applied Nonlinear Dynamical System and Chaos[M]. New York：Springer-Verlag， 1990.第40卷第2期河北科技大學学报Vol.40，No.2
2019年4月Journal of Hebei University of Science and TechnologyApr. 2019