高维非线性系统全局分岔与混沌若干问题的研究
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摘要
工程中的真实动力系统几乎都含有各种各样的非线性因素。由于线性系统对非线性系统的逼近并非总是可靠的,且线性系统的解的适定性对非线性系统不再成立,因而非线性系统具有更丰富的动力学行为。非线性系统的分岔与混沌行为研究是目前国际上非线性动力学领域中的热点和前沿课题,研究非线性系统的模态分岔及相互作用具有潜在的应用价值和实际意义。本文主要研究几类四维非线性系统的全局分岔与混沌问题,一共分为如下六部分。
第一章,主要介绍了非线性动力学中动力系统分岔、混沌等与本文相关的一些基本概念、性质、方法和定理等。
第二章,研究了谐波激励下简支矩形金属板1:1内共振下的全局分岔行为和混沌运动。首先,在非Hamilton共振下,运用Melnikov方法,得到了无阻尼时系统的混沌行为。其次,在Hamilton共振下,运用Kovacic-Wiggins全局摄动法和能量-相位法,研究了系统同宿于慢流形的单脉冲和多脉冲同宿轨的存在性及其导致的Smale马蹄意义下的混沌。这与Melnikov方法得到的结果有本质上的区别。最后通过数值模拟,得到了一些新的动力学现象。
第三章,研究了粘弹性运动梁1:2内共振和主共振条件下的全局动力学行为。在多尺度法得到平均方程的基础上,运用规范型理论得到对应一对非半简零特征根和一对共轭纯虚特征根的规范型。运用Kovacic-Wiggins全局摄动法和能量-相位法,研究了系统在Hamilton共振情形下同宿于慢流形的单脉冲和多脉冲同宿轨的存在性。本节结果从理论上解释了粘弹性运动梁跳跃行为。
第四章,研究了周期激励浅拱的全局分岔,得到了系统在Hamilton共振带上连结鞍点和汇点的同宿轨和异宿轨。与Kovacic-Wiggins全局摄动法和能量-相位法得到同宿于慢流形的轨道不同,这些轨道是交替地位于快慢流形之上的。
第五章,利用Silnikov方法讨论了两类四维系统的混沌行为。利用待定系数法证明了这些系统同宿轨和异宿轨的存在性,并给出了系统发生混沌运动的判据。
第六章是全文总结与展望。
There are various nonlinear factors in the real engineering systems. The certainty for the solutions of linear systems is not satisfied for that of nonlinear systems, so there are more complexities in nonlinear systems. The study of bifurcations and chaos of nonlinear systems are cutting-edge topics. The investigations of mode bifurcations and interactions play an important part in this area. The present dissertation is devoted to the global bifurcations and chaos of some nonlinear four-dimensional mechanical systems. The paper is mainly divided into six chapters.
In the first charter, some important concepts, properties, methods and theories of bifurcation and chaos are introduced, which are quoted here for earlier references until they are applied in later chapters.
In the second charter, the global bifurcations in mode interactions of a simply supported rectangular metallic plate subjected to a transverse harmonic excitation are investigated with the case of the 1:1 internal resonance. Firstly, for the“non-resonant”case, Melnikov method has been used to study the chaotic behaviors. Secondly, for the“resonant”case, the Kovacic-Wiggins global perturbation method and energy-phase method are utilized to analyze the global bifurcations for the rectangular metallic plate. The existence of Silnikov-type single-pulse and multi-pulse homoclinic orbits is obtained, which imply that chaotic motions may occur for this class of rectangular metallic plates. This is quite different from the earlier case based on Melnikov method. Finally, numerical results are presented and some new dynamical phenomena are obtained.
In the third chapter, the global bifurcations and chaotic dynamics of an axially moving viscoelastic beam are investigated with the case of 1:2 internal resonance. On the basis of the modulation equations derived by the method of multiple scales, the theory of normal form is utilized to find the explicit formulas of normal form associated with a double zero eigenvalues and a pair of pure imaginary eigenvalues. The Kovacic-Wiggins perturbation method and energy-phase method are employed to analyze the global bifurcations for the axially moving viscoelastic beam. The results obtained here indicate that there exist the Silnikov-type single-pulse and multi-pulse orbits homoclinic to certain invariant sets for the“resonant”case, leading to chaos in the system. The results give the explanation for the jumping behaviors observed in this class of axially moving viscoelastic beams.
In the fourth chapter, the existence of homoclinic orbits and heteroclinic orbits for a shallow arch subjected to periodic excitation with internal resonance is obtained. Rather than possess finitely many fast pieces that follow one after the other, orbits in this class are composed of alternating slow and fast pieces. In this sense, the results obtained here are different from the Kovacic-Wiggins global perturbation method and energy-phase method.
In the fifth chapter, Silnikov chaos is discussed in detail for two four-dimensional dynamical systems with Silnikov method. By applying the undetermined coefficient method, the Silnikov-type homoclinic and heteroclinic orbits in these systems are found analytically and the uniform convergence of the corresponding series expansions of these orbits is proved. The criterions for chaos are obtained.
The last chapter is the summary and outlook of this paper.
第一章,主要介绍了非线性动力学中动力系统分岔、混沌等与本文相关的一些基本概念、性质、方法和定理等。
第二章,研究了谐波激励下简支矩形金属板1:1内共振下的全局分岔行为和混沌运动。首先,在非Hamilton共振下,运用Melnikov方法,得到了无阻尼时系统的混沌行为。其次,在Hamilton共振下,运用Kovacic-Wiggins全局摄动法和能量-相位法,研究了系统同宿于慢流形的单脉冲和多脉冲同宿轨的存在性及其导致的Smale马蹄意义下的混沌。这与Melnikov方法得到的结果有本质上的区别。最后通过数值模拟,得到了一些新的动力学现象。
第三章,研究了粘弹性运动梁1:2内共振和主共振条件下的全局动力学行为。在多尺度法得到平均方程的基础上,运用规范型理论得到对应一对非半简零特征根和一对共轭纯虚特征根的规范型。运用Kovacic-Wiggins全局摄动法和能量-相位法,研究了系统在Hamilton共振情形下同宿于慢流形的单脉冲和多脉冲同宿轨的存在性。本节结果从理论上解释了粘弹性运动梁跳跃行为。
第四章,研究了周期激励浅拱的全局分岔,得到了系统在Hamilton共振带上连结鞍点和汇点的同宿轨和异宿轨。与Kovacic-Wiggins全局摄动法和能量-相位法得到同宿于慢流形的轨道不同,这些轨道是交替地位于快慢流形之上的。
第五章,利用Silnikov方法讨论了两类四维系统的混沌行为。利用待定系数法证明了这些系统同宿轨和异宿轨的存在性,并给出了系统发生混沌运动的判据。
第六章是全文总结与展望。
There are various nonlinear factors in the real engineering systems. The certainty for the solutions of linear systems is not satisfied for that of nonlinear systems, so there are more complexities in nonlinear systems. The study of bifurcations and chaos of nonlinear systems are cutting-edge topics. The investigations of mode bifurcations and interactions play an important part in this area. The present dissertation is devoted to the global bifurcations and chaos of some nonlinear four-dimensional mechanical systems. The paper is mainly divided into six chapters.
In the first charter, some important concepts, properties, methods and theories of bifurcation and chaos are introduced, which are quoted here for earlier references until they are applied in later chapters.
In the second charter, the global bifurcations in mode interactions of a simply supported rectangular metallic plate subjected to a transverse harmonic excitation are investigated with the case of the 1:1 internal resonance. Firstly, for the“non-resonant”case, Melnikov method has been used to study the chaotic behaviors. Secondly, for the“resonant”case, the Kovacic-Wiggins global perturbation method and energy-phase method are utilized to analyze the global bifurcations for the rectangular metallic plate. The existence of Silnikov-type single-pulse and multi-pulse homoclinic orbits is obtained, which imply that chaotic motions may occur for this class of rectangular metallic plates. This is quite different from the earlier case based on Melnikov method. Finally, numerical results are presented and some new dynamical phenomena are obtained.
In the third chapter, the global bifurcations and chaotic dynamics of an axially moving viscoelastic beam are investigated with the case of 1:2 internal resonance. On the basis of the modulation equations derived by the method of multiple scales, the theory of normal form is utilized to find the explicit formulas of normal form associated with a double zero eigenvalues and a pair of pure imaginary eigenvalues. The Kovacic-Wiggins perturbation method and energy-phase method are employed to analyze the global bifurcations for the axially moving viscoelastic beam. The results obtained here indicate that there exist the Silnikov-type single-pulse and multi-pulse orbits homoclinic to certain invariant sets for the“resonant”case, leading to chaos in the system. The results give the explanation for the jumping behaviors observed in this class of axially moving viscoelastic beams.
In the fourth chapter, the existence of homoclinic orbits and heteroclinic orbits for a shallow arch subjected to periodic excitation with internal resonance is obtained. Rather than possess finitely many fast pieces that follow one after the other, orbits in this class are composed of alternating slow and fast pieces. In this sense, the results obtained here are different from the Kovacic-Wiggins global perturbation method and energy-phase method.
In the fifth chapter, Silnikov chaos is discussed in detail for two four-dimensional dynamical systems with Silnikov method. By applying the undetermined coefficient method, the Silnikov-type homoclinic and heteroclinic orbits in these systems are found analytically and the uniform convergence of the corresponding series expansions of these orbits is proved. The criterions for chaos are obtained.
The last chapter is the summary and outlook of this paper.
引文
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