一类非线性结构混沌运动的研究
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摘要
梁、拱、板、壳等非线性结构的受迫振动问题，经过Galerkin原理的转换，均可归结为如下形式的非线性动力方程老+d窝+2pJ’+gr’’占g(X，i，／) (1)
对方程(1)的动力学性质的研究，是当前固体力学的研究领域中前沿的研究内容。对方程(1)的研究的解析结果，能有效的分析、计算和掌握梁、板、壳、拱等非线性弹性动力系统的发展演化规律，更好的认识、理解和实现对这类非线性系统的预测和控制。
方程(1)是含有二次非线性项和三次非线性项的动力学方程。当
9：0时，方程 (1) 成为
~+ax+Tx’’占g(x，i,t) (2)
方程(2)是Duffing方程，其特点是方程等式左边中的非线性项为三次幂。前人对Duffing方程(2)描述的系统进行了许多研究，但很少见到用解析方法研究方程(1)。二次非线性项和三次非线性项共同存在于方程(1)中，使得用解析方法研究这类系统的难度增大，对应Hamilton系统中的同宿轨道或异宿轨道的解析表达式的求解相当困难。本文用Melinkov方法对方程(1)的混沌运动进行了全面和详细地研究。主要工作和结果如下：
1．分析了方程(1)建立的平面Poincare映射的奇点性质，讨论了此类方程对应的Hamilton系统的同宿轨道和异宿轨道与三个参数O、 夕、尸的关系，给出了Hamilton系统存在同宿轨道或异宿轨道的充分必要条件。
2．得出了同宿轨道或异宿轨道的解析表达式。应用Melnikov方法，计算并建立了同宿轨道或异宿轨道的Melnikov函数。给出了Poincare映射出现Smale马蹄混沌的临界值。
3．得到了同宿轨道或异宿轨道内的，围绕中心型奇点的一族周期轨道的解析表达式。计算并建立了次谐周期轨道的Melnikov函数，给出了Poincare映射出现周期m点的判据。
4．讨论了系统经过次谐分叉进入Smale马蹄混沌的具体途径。
文中的各个结果均以具体的解析形式给出，其中包括同宿轨道或异宿轨道的解析表达式及其Melnikov函数；同(异)宿轨道内围绕中心型奇点的周期轨道的解析表达式及其Melnikov函数；出现周期m点的临界值；出现Smale马蹄混沌的临界值等。这些结果对于分析和研究方程(1)的Smale马蹄
西南交通大学博士研究生学位论文 第11页
混浊运动具有重要意义。
本文进行的研究讨论工作始终考虑方程中三个参数a、p、厂对系统的
影响，出于参数。、g、厂决定着确定系统的动力学行为，因而文中所得’
到的各个结果具有一般性和普适性。至此，本文基本解诀了方程 门）的关
于Smale马蹄混炖的判别及相关问题。
Converted by Galerkin principle, problems of forced vibration of nonlinear structure such as beam, arch, slab and shell can be converted into the following nonlinear dynamic equation
Researches on dynamic properties of equation (1) are the leading work of present research activities of Solid Mechanics. Through the analytical research result of equation (1), evolution rules of nonlinear elastic dynamic system such as beam, slab, shell and arch can be effectively analyzed, calculated and mastered. Therefore prediction and control of this sort of nonlinear system can be more effectively recognized, understood and achieved.
Equation (1) is the dynamic equation that contains quadratic nonlinear term and cubic non-linear term. When =0, this equation becomes
Equation (2) is the Duffing equation, whose characteristic is that the nonlinear terms on the left side of the equation are all cubic power. Many studies have been carried out on the system described by Duffing equation (2), but very few on equation (1) by using the analytical method. The quadratic nonlinear term and the cubic nonlinear term that coexist in equation (1), thus causes difficulty in studying this sort of system through the analytical method and correspondingly makes the solution of analytical expression of homoclinic orbit or heteroclinic orbit in Hamilton system extremely difficult. By using Melinkov Method, the chaotic movement of equation (1) is thoroughly studied in this paper. The main work and results are as follows:
1. Nature of the singular point of plane Poincare mapping that is established by equation (1) is analyzed. Relationship between the three parameters and homoclinic orbit and heteroclinic orbit of the Hamilton system corresponding to this sort of equation is discussed. The adequate and essential condition of the existing homoclinic orbit or heteroclinic orbit for the Hamilton system is presented.
2. An analytical expression of homoclinic orbit or heteroclinic orbit is worked out. By using Melnikov method, the Melnikov function of homoclinic orbit or heteroclinic orbit is calculated and established. The critical value when Poincare mapping taken the form of Smale horseshoes chaos is presented
3. Within the homoclinic orbit or heteroclinic orbit, the analytical expression of one set of periodical track surrounding the center-type singular point is worked out. The Melnikov function of subharmonic orbits is calculated and established and the criterion of appearing periodical m point of Poincare mapping is presented.
4. The specific route of the system going into Smale horse-hoof chaos through subharmonic bifurcation is discussed.
Every result in the paper is presented through specific analytic expression, including the analytic expression of homoclinic orbit or heteroclinic orbit and its Melnikov function, analytical expression of periodical track surrounding the center-type singular point within the homoclinic orbit or heteroclinic orbit and its Melinkov function, the critical value when periodical m point appears, the critical value when Smale horseshoes chaos appears, etc.. These results are of great significance for studying and analyzing the Smale horseshoes chaos of equation (1).
Influence of the three parameters in the equation is persistently considered in the research work conducted in this paper. Because the determination of the dynamic behavior of the system is decided by the parameters , every result in this paper has general and universal meaning. At this time, the criterion concerning the Smale horse-hoes chaos of equation (1) and its related problems are basically solved in this paper.
对方程(1)的动力学性质的研究，是当前固体力学的研究领域中前沿的研究内容。对方程(1)的研究的解析结果，能有效的分析、计算和掌握梁、板、壳、拱等非线性弹性动力系统的发展演化规律，更好的认识、理解和实现对这类非线性系统的预测和控制。
方程(1)是含有二次非线性项和三次非线性项的动力学方程。当
9：0时，方程 (1) 成为
~+ax+Tx’’占g(x，i,t) (2)
方程(2)是Duffing方程，其特点是方程等式左边中的非线性项为三次幂。前人对Duffing方程(2)描述的系统进行了许多研究，但很少见到用解析方法研究方程(1)。二次非线性项和三次非线性项共同存在于方程(1)中，使得用解析方法研究这类系统的难度增大，对应Hamilton系统中的同宿轨道或异宿轨道的解析表达式的求解相当困难。本文用Melinkov方法对方程(1)的混沌运动进行了全面和详细地研究。主要工作和结果如下：
1．分析了方程(1)建立的平面Poincare映射的奇点性质，讨论了此类方程对应的Hamilton系统的同宿轨道和异宿轨道与三个参数O、 夕、尸的关系，给出了Hamilton系统存在同宿轨道或异宿轨道的充分必要条件。
2．得出了同宿轨道或异宿轨道的解析表达式。应用Melnikov方法，计算并建立了同宿轨道或异宿轨道的Melnikov函数。给出了Poincare映射出现Smale马蹄混沌的临界值。
3．得到了同宿轨道或异宿轨道内的，围绕中心型奇点的一族周期轨道的解析表达式。计算并建立了次谐周期轨道的Melnikov函数，给出了Poincare映射出现周期m点的判据。
4．讨论了系统经过次谐分叉进入Smale马蹄混沌的具体途径。
文中的各个结果均以具体的解析形式给出，其中包括同宿轨道或异宿轨道的解析表达式及其Melnikov函数；同(异)宿轨道内围绕中心型奇点的周期轨道的解析表达式及其Melnikov函数；出现周期m点的临界值；出现Smale马蹄混沌的临界值等。这些结果对于分析和研究方程(1)的Smale马蹄
西南交通大学博士研究生学位论文 第11页
混浊运动具有重要意义。
本文进行的研究讨论工作始终考虑方程中三个参数a、p、厂对系统的
影响，出于参数。、g、厂决定着确定系统的动力学行为，因而文中所得’
到的各个结果具有一般性和普适性。至此，本文基本解诀了方程 门）的关
于Smale马蹄混炖的判别及相关问题。
Converted by Galerkin principle, problems of forced vibration of nonlinear structure such as beam, arch, slab and shell can be converted into the following nonlinear dynamic equation
Researches on dynamic properties of equation (1) are the leading work of present research activities of Solid Mechanics. Through the analytical research result of equation (1), evolution rules of nonlinear elastic dynamic system such as beam, slab, shell and arch can be effectively analyzed, calculated and mastered. Therefore prediction and control of this sort of nonlinear system can be more effectively recognized, understood and achieved.
Equation (1) is the dynamic equation that contains quadratic nonlinear term and cubic non-linear term. When =0, this equation becomes
Equation (2) is the Duffing equation, whose characteristic is that the nonlinear terms on the left side of the equation are all cubic power. Many studies have been carried out on the system described by Duffing equation (2), but very few on equation (1) by using the analytical method. The quadratic nonlinear term and the cubic nonlinear term that coexist in equation (1), thus causes difficulty in studying this sort of system through the analytical method and correspondingly makes the solution of analytical expression of homoclinic orbit or heteroclinic orbit in Hamilton system extremely difficult. By using Melinkov Method, the chaotic movement of equation (1) is thoroughly studied in this paper. The main work and results are as follows:
1. Nature of the singular point of plane Poincare mapping that is established by equation (1) is analyzed. Relationship between the three parameters and homoclinic orbit and heteroclinic orbit of the Hamilton system corresponding to this sort of equation is discussed. The adequate and essential condition of the existing homoclinic orbit or heteroclinic orbit for the Hamilton system is presented.
2. An analytical expression of homoclinic orbit or heteroclinic orbit is worked out. By using Melnikov method, the Melnikov function of homoclinic orbit or heteroclinic orbit is calculated and established. The critical value when Poincare mapping taken the form of Smale horseshoes chaos is presented
3. Within the homoclinic orbit or heteroclinic orbit, the analytical expression of one set of periodical track surrounding the center-type singular point is worked out. The Melnikov function of subharmonic orbits is calculated and established and the criterion of appearing periodical m point of Poincare mapping is presented.
4. The specific route of the system going into Smale horse-hoof chaos through subharmonic bifurcation is discussed.
Every result in the paper is presented through specific analytic expression, including the analytic expression of homoclinic orbit or heteroclinic orbit and its Melnikov function, analytical expression of periodical track surrounding the center-type singular point within the homoclinic orbit or heteroclinic orbit and its Melinkov function, the critical value when periodical m point appears, the critical value when Smale horseshoes chaos appears, etc.. These results are of great significance for studying and analyzing the Smale horseshoes chaos of equation (1).
Influence of the three parameters in the equation is persistently considered in the research work conducted in this paper. Because the determination of the dynamic behavior of the system is decided by the parameters , every result in this paper has general and universal meaning. At this time, the criterion concerning the Smale horse-hoes chaos of equation (1) and its related problems are basically solved in this paper.
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