非线性动力系统规范形理论及应用问题研究
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摘要
规范形理论是研究动力系统、微分方程及非线性振动等领域动力学特征的强有力工具之一。规范形理论又称正规形理论,它的基本思想,是在奇点(或不动点)附近经过光滑变换把向量场(或微分同胚)化成尽可能简单的形式,以便于研究。然而,计算给定系统的最简规范形本身就是一项很复杂的工作,另外,有关Hopf分岔系统、退化Hopf分岔系统及其规范形理论在力学等实际系统的应用研究也越来越受到广大科学工作者的广泛关注。本论文主要研究了规范形理论中最简规范形的计算,规范形理论在力学和生物学系统中的应用,非线性动力系统中的Hopf分岔与混沌等动力学特征。主要创新点有以下几个方面:
(1)在传统规范形的基础上,利用规范形理论和矩阵表示法的思想研究了Hopf分岔系统的最简规范形,给出最简规范形的计算公式,和所选取的非线性变换公式。研究了余维2及高余维退化Hopf分岔系统的最简规范形。提出由于条件的不同,系统具有两种不同的最简规范形形式,并给出计算公式。
(2)利用动力系统中的规范形理论研究Neimark-Sacker系统的最简规范形。指出传统的Neimark-Sacker系统的规范形可以继续化简,计算了余维2及高余维退化Neimark-Sacker系统的最简规范形,得出了五个定理,说明退化Neimark-Sacker系统的最简规范形的振幅方程最多含有两个非线性项,具有两种不同的形式,给出了公式的代数表达。本文提出的方法,为深入研究Hopf分岔系统的稳定性、分岔等复杂动力学行为奠定了基础。
(3)利用Hopf定理和规范形理论,讨论了Furuta旋转倒立摆非线性数学模型的Hopf分岔等动力学特征。给出系统存在Hopf分岔的条件,讨论了周期轨道的稳定性,利用数值模拟,得到系统的相轨迹图。比较严格地证明了系统存在Smale马蹄意义下的混沌现象,并给出发生Silnikov型Smale混沌的条件。为进一步研究旋转倒立摆的复杂动力学行为奠定了基础,同时也为旋转倒立摆的控制和仿真研究,提供了理论依据。
(4)研究了一类新的连续自治三维混沌系统,即Van del Pol Jerk系统。通过理论分析和数值模拟,研究了系统的基本动力学性质。利用Silnikov定理,研究了系统具有混沌现象,通过Cardano公式和微分方程级数解理论,研究了系统的特征值和同宿轨道。比较严格地证明了系统存在Silnikov型Smale马蹄混沌现象。并指出系统存在混沌现象的充分条件。利用数值模拟,验证了本文提出方法的正确性。
(5)讨论了一类具有二重饱和反应速度的生化反应动力系统的动力学特征。利用微分方程定性理论,完整地研究了该系统极限环的不存在性和存在唯一性的充分条件,利用规范形理论,研究了该系统的Hopf分岔,并与具有米氏饱和反应速度的生化模型的定性性质进行了比较。
The normal form theory is one of the useful tools in the fields of dynamical system, ordinary differential equations and nonlinear vibration. Normal form theory plays an important role in the study of dynamical behavior of nonlinear systems near the dynamic equilibrium points because it greatly simplifies the analysis and formulations. This simple form can be used conveniently for analyzing the dynamical behavior of the original system in the vicinity of equilibrium. However, it is not a simple task to calculate the normal form for some given ordinary differential equations. The normal forms of Hopf and generalized Hopf bifurcations, as well as their applications have been extensively studied by many researchers. The research contents and the innovative contributions of this dissertation are as follows:
(1) The Hopf Bifurcations have been studied by normal form theory and the matrix representation method of dynamic system. On the basis of normal form theory, the Hopf bifurcation systems are further simplified to the simplest normal forms. The normal forms of generalized Hopf bifurcations have been extensively studied. Theorems are presented to show that the conventional normal form of generalized Hopf bifurcations is further simplified to the simplest normal form there are at most two terms remaining in the amplitude equation of the simplest normal form up to 2k+1 order. There are two kinds of the simplest normal forms.
(2) The normal forms of generalized Neimark-Sacker bifurcations have been extensively studied using normal form theory of dynamic system. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcations can be further simplified. We calculate the simplest normal forms of generalized Neimark-Sacker bifurcations. There are two kinds of the simplest normal forms. These algebra expression formulas are given.
(3) Study Hopf bifurcations by normal form theory in the Furuta pendulum system. We calculate the normal forms of the Hopf bifurcation systems. The stability of the limit cycle is discussed. The space trajectories are investigated via numerical simulation, which are aslo verified the validity of our analysis. Based on the Silnikov criterion, the chaotic characters of the dynamical systems are discussed. Using Cardano formula and series solution of differential equation, eigenvalue problem and the existence of homoclinic orbit are studied respectively. Furthermore, a rigorous proof for the existence of Silnikov-sense Smale horseshoes chaos is presented and some conditions which lead to the chaos are obtained.
(4) Study a new three-dimensional continuous autonomous chaotic system. The new Van del Pol Jerk system contains a cubic terms and six system parameters. Basic dynamic properties of the new system are studied by means of theoretical analysis and numerical simulation. Based on the Silnikov criterion, the chaotic characters of the dynamical systems are discussed. Using Cardano formula and series solution of differential equation, eigenvalue problem and the existence of homoclinic orbit are studied respectively. Furthermore, a rigorous proof for the existence of Silnikov-sense Smale horseshoes chaos is presented and some conditions which lead to the chaos are obtained. The formation mechanism shows that this chaotic system has impulsive homoclinic chaos, and numerical simulation demonstrates that there is a route to chaos.
(5) Study a class of multimolecules saturated reaction model. By using the qualitative theory of ordinary differential equations, completely discuss the existence, nonexistence and uniqueness of limit cycles of the system. By using the normal form theory, the Hopf bifurcation behavior of the dynamic system is exploited. We compared qualitative property with the system with saturated reaction speed.
(1)在传统规范形的基础上,利用规范形理论和矩阵表示法的思想研究了Hopf分岔系统的最简规范形,给出最简规范形的计算公式,和所选取的非线性变换公式。研究了余维2及高余维退化Hopf分岔系统的最简规范形。提出由于条件的不同,系统具有两种不同的最简规范形形式,并给出计算公式。
(2)利用动力系统中的规范形理论研究Neimark-Sacker系统的最简规范形。指出传统的Neimark-Sacker系统的规范形可以继续化简,计算了余维2及高余维退化Neimark-Sacker系统的最简规范形,得出了五个定理,说明退化Neimark-Sacker系统的最简规范形的振幅方程最多含有两个非线性项,具有两种不同的形式,给出了公式的代数表达。本文提出的方法,为深入研究Hopf分岔系统的稳定性、分岔等复杂动力学行为奠定了基础。
(3)利用Hopf定理和规范形理论,讨论了Furuta旋转倒立摆非线性数学模型的Hopf分岔等动力学特征。给出系统存在Hopf分岔的条件,讨论了周期轨道的稳定性,利用数值模拟,得到系统的相轨迹图。比较严格地证明了系统存在Smale马蹄意义下的混沌现象,并给出发生Silnikov型Smale混沌的条件。为进一步研究旋转倒立摆的复杂动力学行为奠定了基础,同时也为旋转倒立摆的控制和仿真研究,提供了理论依据。
(4)研究了一类新的连续自治三维混沌系统,即Van del Pol Jerk系统。通过理论分析和数值模拟,研究了系统的基本动力学性质。利用Silnikov定理,研究了系统具有混沌现象,通过Cardano公式和微分方程级数解理论,研究了系统的特征值和同宿轨道。比较严格地证明了系统存在Silnikov型Smale马蹄混沌现象。并指出系统存在混沌现象的充分条件。利用数值模拟,验证了本文提出方法的正确性。
(5)讨论了一类具有二重饱和反应速度的生化反应动力系统的动力学特征。利用微分方程定性理论,完整地研究了该系统极限环的不存在性和存在唯一性的充分条件,利用规范形理论,研究了该系统的Hopf分岔,并与具有米氏饱和反应速度的生化模型的定性性质进行了比较。
The normal form theory is one of the useful tools in the fields of dynamical system, ordinary differential equations and nonlinear vibration. Normal form theory plays an important role in the study of dynamical behavior of nonlinear systems near the dynamic equilibrium points because it greatly simplifies the analysis and formulations. This simple form can be used conveniently for analyzing the dynamical behavior of the original system in the vicinity of equilibrium. However, it is not a simple task to calculate the normal form for some given ordinary differential equations. The normal forms of Hopf and generalized Hopf bifurcations, as well as their applications have been extensively studied by many researchers. The research contents and the innovative contributions of this dissertation are as follows:
(1) The Hopf Bifurcations have been studied by normal form theory and the matrix representation method of dynamic system. On the basis of normal form theory, the Hopf bifurcation systems are further simplified to the simplest normal forms. The normal forms of generalized Hopf bifurcations have been extensively studied. Theorems are presented to show that the conventional normal form of generalized Hopf bifurcations is further simplified to the simplest normal form there are at most two terms remaining in the amplitude equation of the simplest normal form up to 2k+1 order. There are two kinds of the simplest normal forms.
(2) The normal forms of generalized Neimark-Sacker bifurcations have been extensively studied using normal form theory of dynamic system. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcations can be further simplified. We calculate the simplest normal forms of generalized Neimark-Sacker bifurcations. There are two kinds of the simplest normal forms. These algebra expression formulas are given.
(3) Study Hopf bifurcations by normal form theory in the Furuta pendulum system. We calculate the normal forms of the Hopf bifurcation systems. The stability of the limit cycle is discussed. The space trajectories are investigated via numerical simulation, which are aslo verified the validity of our analysis. Based on the Silnikov criterion, the chaotic characters of the dynamical systems are discussed. Using Cardano formula and series solution of differential equation, eigenvalue problem and the existence of homoclinic orbit are studied respectively. Furthermore, a rigorous proof for the existence of Silnikov-sense Smale horseshoes chaos is presented and some conditions which lead to the chaos are obtained.
(4) Study a new three-dimensional continuous autonomous chaotic system. The new Van del Pol Jerk system contains a cubic terms and six system parameters. Basic dynamic properties of the new system are studied by means of theoretical analysis and numerical simulation. Based on the Silnikov criterion, the chaotic characters of the dynamical systems are discussed. Using Cardano formula and series solution of differential equation, eigenvalue problem and the existence of homoclinic orbit are studied respectively. Furthermore, a rigorous proof for the existence of Silnikov-sense Smale horseshoes chaos is presented and some conditions which lead to the chaos are obtained. The formation mechanism shows that this chaotic system has impulsive homoclinic chaos, and numerical simulation demonstrates that there is a route to chaos.
(5) Study a class of multimolecules saturated reaction model. By using the qualitative theory of ordinary differential equations, completely discuss the existence, nonexistence and uniqueness of limit cycles of the system. By using the normal form theory, the Hopf bifurcation behavior of the dynamic system is exploited. We compared qualitative property with the system with saturated reaction speed.
引文
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