某些延迟微分方程数值方法的分支相容性
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摘要
近几十年来,延迟微分方程已经被广泛地应用到近代物理学、生物学、医学、经济学、人口学、化学反应工程学、自动控制理论等众多科学领域。对这类方程,由于只有少数特殊的方程可以显式求解,因此发展适用的数值方法是必要的。然而,能够正确反映原系统性质的数值方法才具有应用价值,所以研究数值方法能否保持原系统的动力学行为在理论上和应用上都具有十分重要的意义。
本论文分别对几类延迟微分方程,研究了某些数值方法的分支相容性,即方法能否保持原方程分支的性质。
首先,本文应用差分方法求解一类具有负反馈的二阶延迟微分方程,并研究了其数值离散系统的动力学行为。通过分析随着参数的变化,特征根的变化情况,再应用Neimark-Sacker分支定理,本文给出了Neimark-Sacker分支存在的充分条件。利用规范形理论和中心流形定理,我们计算了确定分支方向及闭的不变曲线稳定性的显式公式。通过比较数值离散系统和原系统的分支性质,结果说明了差分方法关于这类二阶延迟微分方程是分支相容的。
其次,针对M.C. Mackey和L. Glass提出的用于描述血循环中粒细胞密度的延迟微分方程,本文考虑了非标准有限差分方法的分支相容性。应用上面类似的方法,我们分析了其数值离散系统正不动点的稳定性,给出了Neimark-Sacker分支存在的充分条件,得到了判断分支方向和闭的不变曲线稳定性的显式表达式。
再次,本文应用中点公式求解一个描述动脉中二氧化碳浓度的延迟微分系统。我们分析了得到的数值离散系统正不动点的稳定性,给出了它经历Neimark-Sacker分支的条件,计算了确定分支方向和闭的不变曲线稳定性的显式表达式。得到的结论与原系统的分支性质比较表明,对于此方程中点公式是分支相容的。
最后,本文研究了一类Runge-Kutta方法对于一类具有一般形式的延迟微分方程的分支相容性。应用隐函数定理,我们证明了如果原方程具有Hopf分支,那么这类Runge-Kutta方法对于该方程是分支相容的,并且如果方法是p阶的,那么Neimark-Sacker分支点收敛于Hopf分支点的收敛阶数也是p。为了验证上述理论结论的正确性,我们应用2级Gauss方法求解延迟Logistic方程,计算得到了Neimark-Sacker分支点收敛于Hopf分支点的阶数是4。
此外,在每章的理论证明之后,我们都进行了相应的数值算例。它们表明了理论结果的正确性。
In the past several decades, delay differential equations have been widely appliedin many fields of science, such as in modern physics, biology, medicine, economics, de-mography, chemical reaction engineering, the theory of automatic control etc. Since theseequations can only be solved explicitly in some special cases, it is necessary to developsome appropriate numerical methods. However, a numerical method is valuable only ifthe method can re?ect exactly the property of the original system. Hence, both in the the-ory and in the applications, it is great significant to study whether the numerical methodscan preserve the dynamical behavior of the original system.
In this dissertation, the bifurcational consistency of some numerical methods forcertain kinds of delay differential equations is studied, that is, whether the numericalmethods could preserve the bifurcation of the original systems is researched.
Firstly, for a class of second order delay differential equations with negative feed-back, the dynamical behavior of the numerical discrete system derived by a differencemethod is investigated. The sufficient conditions under which the Neimark-Sacker bi-furcation exists are derived by analyzing the moving of the characteristic roots withthe changing of the delay parameter and using the Neimark-Sacker bifurcation theorem.Meanwhile, the explicit expressions of determining the direction of the bifurcation and thestability of the closed invariant curve are given by using the normal form theory and thecenter manifold theorem. Through comparing the bifurcation of the origin system withthe numerical discrete system, it is showed that the difference method is bifurcationallyconsistent for the second order delay differential equations.
Secondly, for a delay differential equation constructed by M.C. Mackey and L.Glass, which characterizes the regulation of the density of mature cells in blood circu-lation, the bifurcational consistency of a non-standard finite difference method is consid-ered. It follows the similar way as in the above problem. The stability of the positivefixed point of the numerical discrete system is analyzed. The conditions which guaranteethe existence of the Neimark-Sacker bifurcation are given. The explicit expressions fordetermining the direction of the bifurcation and the stability of the closed invariant curveare obtained.
Thirdly, for an arterial carbon dioxide control system, a delay differential equation,the Midpoint rule is applied to solve the numerical solutions. It is analyzed the stabilityof the positive fixed point of the numerical discrete system. The conditions under whichthe discrete system undergoes a Neimark-Sacker bifurcation are given. The explicit ex-pressions for determining the direction of the bifurcation and the stability of the closedinvariant curve are obtained. It is illustrated that the Midpoint rule for the system is bi-furcationally consistent, by comparing the results with the dynamical behaviors of thecontrol system.
At last, a class of Runge-Kutta methods for some general delay differential equationsare studied. It is proved by employing the implicit function theorem that the Runge-Kuttamethods are bifurcationally consistent for the equations which undergo Hopf bifurcation,and that the Neimark-Sacker bifurcation point converges to the Hopf bifurcation pointwith order p if the Runge-Kutta method is of order p. To illustrate the correctness of theresults, the 2-stage Gauss method is used to solve the delay Logistic equation and it isseen that the Neimark-Sacker bifurcation point converges to the Hopf bifurcation pointwith order 4.
Moreover, after the theoretical proof in every chapter, some numerical examples aregiven which illustrate the correctness of the theoretical results.
本论文分别对几类延迟微分方程,研究了某些数值方法的分支相容性,即方法能否保持原方程分支的性质。
首先,本文应用差分方法求解一类具有负反馈的二阶延迟微分方程,并研究了其数值离散系统的动力学行为。通过分析随着参数的变化,特征根的变化情况,再应用Neimark-Sacker分支定理,本文给出了Neimark-Sacker分支存在的充分条件。利用规范形理论和中心流形定理,我们计算了确定分支方向及闭的不变曲线稳定性的显式公式。通过比较数值离散系统和原系统的分支性质,结果说明了差分方法关于这类二阶延迟微分方程是分支相容的。
其次,针对M.C. Mackey和L. Glass提出的用于描述血循环中粒细胞密度的延迟微分方程,本文考虑了非标准有限差分方法的分支相容性。应用上面类似的方法,我们分析了其数值离散系统正不动点的稳定性,给出了Neimark-Sacker分支存在的充分条件,得到了判断分支方向和闭的不变曲线稳定性的显式表达式。
再次,本文应用中点公式求解一个描述动脉中二氧化碳浓度的延迟微分系统。我们分析了得到的数值离散系统正不动点的稳定性,给出了它经历Neimark-Sacker分支的条件,计算了确定分支方向和闭的不变曲线稳定性的显式表达式。得到的结论与原系统的分支性质比较表明,对于此方程中点公式是分支相容的。
最后,本文研究了一类Runge-Kutta方法对于一类具有一般形式的延迟微分方程的分支相容性。应用隐函数定理,我们证明了如果原方程具有Hopf分支,那么这类Runge-Kutta方法对于该方程是分支相容的,并且如果方法是p阶的,那么Neimark-Sacker分支点收敛于Hopf分支点的收敛阶数也是p。为了验证上述理论结论的正确性,我们应用2级Gauss方法求解延迟Logistic方程,计算得到了Neimark-Sacker分支点收敛于Hopf分支点的阶数是4。
此外,在每章的理论证明之后,我们都进行了相应的数值算例。它们表明了理论结果的正确性。
In the past several decades, delay differential equations have been widely appliedin many fields of science, such as in modern physics, biology, medicine, economics, de-mography, chemical reaction engineering, the theory of automatic control etc. Since theseequations can only be solved explicitly in some special cases, it is necessary to developsome appropriate numerical methods. However, a numerical method is valuable only ifthe method can re?ect exactly the property of the original system. Hence, both in the the-ory and in the applications, it is great significant to study whether the numerical methodscan preserve the dynamical behavior of the original system.
In this dissertation, the bifurcational consistency of some numerical methods forcertain kinds of delay differential equations is studied, that is, whether the numericalmethods could preserve the bifurcation of the original systems is researched.
Firstly, for a class of second order delay differential equations with negative feed-back, the dynamical behavior of the numerical discrete system derived by a differencemethod is investigated. The sufficient conditions under which the Neimark-Sacker bi-furcation exists are derived by analyzing the moving of the characteristic roots withthe changing of the delay parameter and using the Neimark-Sacker bifurcation theorem.Meanwhile, the explicit expressions of determining the direction of the bifurcation and thestability of the closed invariant curve are given by using the normal form theory and thecenter manifold theorem. Through comparing the bifurcation of the origin system withthe numerical discrete system, it is showed that the difference method is bifurcationallyconsistent for the second order delay differential equations.
Secondly, for a delay differential equation constructed by M.C. Mackey and L.Glass, which characterizes the regulation of the density of mature cells in blood circu-lation, the bifurcational consistency of a non-standard finite difference method is consid-ered. It follows the similar way as in the above problem. The stability of the positivefixed point of the numerical discrete system is analyzed. The conditions which guaranteethe existence of the Neimark-Sacker bifurcation are given. The explicit expressions fordetermining the direction of the bifurcation and the stability of the closed invariant curveare obtained.
Thirdly, for an arterial carbon dioxide control system, a delay differential equation,the Midpoint rule is applied to solve the numerical solutions. It is analyzed the stabilityof the positive fixed point of the numerical discrete system. The conditions under whichthe discrete system undergoes a Neimark-Sacker bifurcation are given. The explicit ex-pressions for determining the direction of the bifurcation and the stability of the closedinvariant curve are obtained. It is illustrated that the Midpoint rule for the system is bi-furcationally consistent, by comparing the results with the dynamical behaviors of thecontrol system.
At last, a class of Runge-Kutta methods for some general delay differential equationsare studied. It is proved by employing the implicit function theorem that the Runge-Kuttamethods are bifurcationally consistent for the equations which undergo Hopf bifurcation,and that the Neimark-Sacker bifurcation point converges to the Hopf bifurcation pointwith order p if the Runge-Kutta method is of order p. To illustrate the correctness of theresults, the 2-stage Gauss method is used to solve the delay Logistic equation and it isseen that the Neimark-Sacker bifurcation point converges to the Hopf bifurcation pointwith order 4.
Moreover, after the theoretical proof in every chapter, some numerical examples aregiven which illustrate the correctness of the theoretical results.
引文
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100 C.R. Zhang, B.D. Zheng. Hopf Bifurcation in Numerical Approximation of a n-Dimension Neural Network Model with Multi-Delays. Chaos Solitons Fractals.2005, 25: 129～146
101 C.R. Zhang, M.Z. Liu, B.D. Zheng. Hopf Bifurcation in Numerical Approximationfor Delay Differential Equations. J. Appl. Math. Computing. 2004, 14: 319～328
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119 J.J. Wei. Bifurcation Analysis in A Scalar Delay Differential Equation. Nonlinear-ity. 2007, 20(11): 2483～2498
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100 C.R. Zhang, B.D. Zheng. Hopf Bifurcation in Numerical Approximation of a n-Dimension Neural Network Model with Multi-Delays. Chaos Solitons Fractals.2005, 25: 129～146
101 C.R. Zhang, M.Z. Liu, B.D. Zheng. Hopf Bifurcation in Numerical Approximationfor Delay Differential Equations. J. Appl. Math. Computing. 2004, 14: 319～328
102 X.H. Ding, D. Fan, M.Z. Liu. Stability and Bifurcation of a Numerical Discretiza-tion Mackey-Glass System. Chaos Solitons Fractals. 2007, 34: 383～393
103 X.H. Ding, W.X. Li. Stability and Bifurcation of Numerical Discretization Nichol-son Blow?ies Equation with Delay. Discrete Dyn. Nat. Soc. 2006, Article Number:19413
104 Y.A. Kuznetsov. Elements of Applied Bifurcation Theory. Second Edition.Springer-Verlag. 1998, 57～63 79～104
105 S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos.Spring-Verlag. 1990, 253～381 374～381
106 http://www.scholarpedia.org/article/Neimark-Sacker bifurcation
107 X. Wang, E.K. Blum, Q.N. Li. Consistency of Local Dynamics and Bifurcationof Continuous-Time Dynamical Systems and Their Numerical Discretizations. J.Differ. Equ. Appl. 1998, 4(1): 29～57
108 C.R. Zhang, Y.G. Zu, B.D. Zheng. Stability and Bifurcation of a Discrete Red BloodCell Survival Model. Chaos Solitons Fractals. 2006, 28: 386～394
109 A. Beuter. J. Belair, C. Labrie. Feedback and Delays in Neurological Diseases: aModeling Study Using Dynamical Systems. Bull. Math. Biol. 1993, 55: 525～541
110 R. Valle′e, M. Dubois, M. Cote′, C. Delisle. Second-Order Differential-Delay Equa-tion to Describe a Hybrid Bistable Device. Phys. Rev. A. 1987, 36: 1327～1332
111 U. an der Heiden, A. Longtin, M. C. Mackey, J. G. Milton, R. Scholl. OscillatoryModes in a Nonlinear Second-Order Differential Equation With Delay. J. Dyn. Diff.Eq. 1990, 2: 423～449
112 S.A. Campbell, J. Belair, T. Ohira, J. Milton. Limit Cycles, Tori and Complex Dy-namics in a Second-Order Differential Equation with Delayed Negative Feedback.J. Dyn. Diff. Eq. 1995, 7(1): 213～236
113 E. Boe, H.C. Chang. Dynamics of Delayed Systems under Feedback Control.Chem. Eng. Sci. 1989, 44: 1281～1294
114 K.L. Cooke, Z. Grossman. Discrete Delay, Distributed Delay and StabilitySwitches. J. Math. AnaL. Appl. 1982, 86: 592～627
115 F.G. Boese. The Stability Chart for the Linearized Cushing Equation with a Dis-crete Delay and with Gamma-Distributed Delays. J. Math. Anal. Appl. 1989, 140:510～536
116 W. Rudin. Real and Complex Analysis. Third Edition. Amer Math Soc. Providence,RI. 2004: 225～229
117 M.C. Mackey, L. Glass. Oscillation and Chaos in Physiological Control Systems.Science. 1977, 197: 287～289
118 M.A. Kramer, B.A. Lopour, H.E. Kirsch, A.J. Szeri. Bifurcation Control of a Seiz-ing Human Cortex. Phys. Rev. E. 2006, 73: 041928
119 J.J. Wei. Bifurcation Analysis in A Scalar Delay Differential Equation. Nonlinear-ity. 2007, 20(11): 2483～2498
120 K.C. Patidar. On the Use of Nonstandard Finite Difference Methods. J. Differ. Equ.Appl. 2005, 11(8): 735～758
121 D.T. Dimitrov, H.V. Kojouharov. Nonstandard Finite-Difference Methods forPredator-Prey Models with General Functional Response. Math. Comput. Simul.2008, 78: 1～11
122 S.M. Moghadas, M.E. Alexander, B.D. Corbett. A Non-Standard NumericalScheme for a Generalized Gause-Type Predator-Prey Model. Physica D. 2004, 188:134～151
123 E. Hairer, C. Lubich, G. Wanner. Geometric Numerical Integration. Springer-Verlag. 2002, 120～121