几类高维非线性系统的分岔与混沌问题研究
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摘要
在物理、化学、生物及工程等领域中,许多问题的动力学方程都可以约化为高维非线性系统。由于高维非线性系统具有更复杂、更丰富的动力学性质,其研究难度比线性系统以及低维的非线性系统要大得多,因此受到许多学者和专家的关注,也一直是研究的热点和前沿问题,特别是高维非线性系统的分岔与混沌动力学的研究。
本文主要利用规范型理论、全局摄动法、能量相位法、Silnikov方法等解析方法和数值方法研究几类高维非线性动力系统的分岔与混沌问题。
首先,利用规范型理论和数值方法,研究了一类3:1内共振和组合参激共振条件下大范围直线运动梁的稳定性与局部分岔问题,分别对第一阶、第二阶模态主参激共振以及第一、二阶模态间组合参激共振条件下的梁系统进行稳定性与分岔分析,给出了系统在各种特征值情况下的临界分岔曲线、分岔解及其稳定性,得到了这类模型的丰富的动力学性质,并用数值方法验证了理论分析所得结果的正确性。
其次,研究了双耦合参数激励下的非线性Van der Pol振子的全局分岔与混沌问题。利用全局摄动法研究了双耦合参数激励下的非线性Van der Pol振子的Silnikov型单脉冲轨道的存在性及其导致的Silnikov型混沌运动,并利用能量相位法研究了该系统的多脉冲跳跃同宿轨道的存在性及其导致的Smale马蹄意义下的混沌,得到了一些新的有意义的动力学现象。
然后,对两类非线性悬索系统,研究了其单脉冲跳跃轨道、多脉冲跳跃轨道及它们所导致的混沌问题。利用全局摄动法研究了这两类非线性悬索系统的单脉冲轨道的存在性及其导致的Silnikov型混沌运动,利用能量相位法研究了这两类非线性悬索系统的Silnikov型多脉冲轨道的存在性及其导致的Smale马蹄意义下的混沌,得到了一些新的动力学行为。
最后,利用第一Lyapunov系数和Silnikov方法详细研究了Rucklidge系统和一个具有四翼混沌吸引子的三维系统的Hopf分岔与混沌,用待定系数法证明了这些系统的同宿轨道和异宿轨道的存在性,从而存在Smale马蹄意义下的混沌。
In physics, chemistry, biology and engineering, the governing equations of motion for a number of problems can be described by high-dimensional nonlinear systems. Comparing with linear systems and low-dimensional nonlinear systems, the study on the dynamical behavior of high-dimensional nonlinear systems were more difficult, so there are more complexities and richer in the dynamical phenomena of high-dimensional nonlinear systems. High-dimensional nonlinear systems have received considerable attention and play an important part in this area, especially, the study of bifurcations and chaos of high-dimensional nonlinear systems.
By using normal form method, global perturbation method, energy-phase method, Silnikov method and numerical methods, the present dissertation is devoted to the bifurcations and chaos of some high-dimensional nonlinear dynamical systems.
Firstly, both normal form method and numerical approaches are employed to consider the stability and local bifurcation for the flexible beam system undergoing a large linear motion with combination parametric resonance and 3:1 internal resonance. We discuss the stability and local bifurcation for the flexible beam to a principal parametric excitation of either the first or the second mode or a combination parametric resonance of both modes. The critical bifurcation curves, bifurcation solutions and their stabilities are obtained. Some significant dynamical phenomena are obtained. Numerical simulations agree with the analytic prediction.
Secondly,with the global perturbation method and the energy-phase method, we discuss the global bifurcations and chaotic dynamics of two non-linearly coupled parametrically excited van der Pol oscillators. Using the global perturbation method, the existence of Silnikov-type single-pulse orbits and chaotic motions of Silnikov-type single-pulse of two non-linearly coupled parametrically excited van der Pol oscillators are studied in detail. And using the energy-phase method, the existence of Silnikov-type multi-pulse orbits of these systems are studied in detail. These results show that the chaotic motions of Silnikov-type multi-pulse can occur. Some new significant dynamical phenomena are obtained.
Thirdly, Silnikov-type single-pulse orbit, Silnikov-type multi-pulse orbits and chaotic motions of two types of nonlinear suspended cables systems are studed in detail in this dissertation. The global perturbation method is applied to study the existence of Silnikov-type single-pulse orbits and chaotic motions of Silnikov-type single-pulse of two types of nonlinear suspended cables systems. The energy-phase method is utilized to analyze the existence of Silnikov-type multi-pulse orbits of these systems. These results show that the chaotic motions of Silnikov-type multi-pulse can occur.
Lastly, for Rucklidge systems and a new 3-D quadratic autonomous system with a four-wing chaotic attractor, Hopf bifurcation and Silnikov chaos are discussed in detail with the first Lyapunov coefficient and Silnikov method. Using undetermined coefficient method, the existence of heteroclinic and homoclinic orbits of these systems are proved, and the Silnikov criterion guarantees that there exists the Smale horseshoe chaos motion.
本文主要利用规范型理论、全局摄动法、能量相位法、Silnikov方法等解析方法和数值方法研究几类高维非线性动力系统的分岔与混沌问题。
首先,利用规范型理论和数值方法,研究了一类3:1内共振和组合参激共振条件下大范围直线运动梁的稳定性与局部分岔问题,分别对第一阶、第二阶模态主参激共振以及第一、二阶模态间组合参激共振条件下的梁系统进行稳定性与分岔分析,给出了系统在各种特征值情况下的临界分岔曲线、分岔解及其稳定性,得到了这类模型的丰富的动力学性质,并用数值方法验证了理论分析所得结果的正确性。
其次,研究了双耦合参数激励下的非线性Van der Pol振子的全局分岔与混沌问题。利用全局摄动法研究了双耦合参数激励下的非线性Van der Pol振子的Silnikov型单脉冲轨道的存在性及其导致的Silnikov型混沌运动,并利用能量相位法研究了该系统的多脉冲跳跃同宿轨道的存在性及其导致的Smale马蹄意义下的混沌,得到了一些新的有意义的动力学现象。
然后,对两类非线性悬索系统,研究了其单脉冲跳跃轨道、多脉冲跳跃轨道及它们所导致的混沌问题。利用全局摄动法研究了这两类非线性悬索系统的单脉冲轨道的存在性及其导致的Silnikov型混沌运动,利用能量相位法研究了这两类非线性悬索系统的Silnikov型多脉冲轨道的存在性及其导致的Smale马蹄意义下的混沌,得到了一些新的动力学行为。
最后,利用第一Lyapunov系数和Silnikov方法详细研究了Rucklidge系统和一个具有四翼混沌吸引子的三维系统的Hopf分岔与混沌,用待定系数法证明了这些系统的同宿轨道和异宿轨道的存在性,从而存在Smale马蹄意义下的混沌。
In physics, chemistry, biology and engineering, the governing equations of motion for a number of problems can be described by high-dimensional nonlinear systems. Comparing with linear systems and low-dimensional nonlinear systems, the study on the dynamical behavior of high-dimensional nonlinear systems were more difficult, so there are more complexities and richer in the dynamical phenomena of high-dimensional nonlinear systems. High-dimensional nonlinear systems have received considerable attention and play an important part in this area, especially, the study of bifurcations and chaos of high-dimensional nonlinear systems.
By using normal form method, global perturbation method, energy-phase method, Silnikov method and numerical methods, the present dissertation is devoted to the bifurcations and chaos of some high-dimensional nonlinear dynamical systems.
Firstly, both normal form method and numerical approaches are employed to consider the stability and local bifurcation for the flexible beam system undergoing a large linear motion with combination parametric resonance and 3:1 internal resonance. We discuss the stability and local bifurcation for the flexible beam to a principal parametric excitation of either the first or the second mode or a combination parametric resonance of both modes. The critical bifurcation curves, bifurcation solutions and their stabilities are obtained. Some significant dynamical phenomena are obtained. Numerical simulations agree with the analytic prediction.
Secondly,with the global perturbation method and the energy-phase method, we discuss the global bifurcations and chaotic dynamics of two non-linearly coupled parametrically excited van der Pol oscillators. Using the global perturbation method, the existence of Silnikov-type single-pulse orbits and chaotic motions of Silnikov-type single-pulse of two non-linearly coupled parametrically excited van der Pol oscillators are studied in detail. And using the energy-phase method, the existence of Silnikov-type multi-pulse orbits of these systems are studied in detail. These results show that the chaotic motions of Silnikov-type multi-pulse can occur. Some new significant dynamical phenomena are obtained.
Thirdly, Silnikov-type single-pulse orbit, Silnikov-type multi-pulse orbits and chaotic motions of two types of nonlinear suspended cables systems are studed in detail in this dissertation. The global perturbation method is applied to study the existence of Silnikov-type single-pulse orbits and chaotic motions of Silnikov-type single-pulse of two types of nonlinear suspended cables systems. The energy-phase method is utilized to analyze the existence of Silnikov-type multi-pulse orbits of these systems. These results show that the chaotic motions of Silnikov-type multi-pulse can occur.
Lastly, for Rucklidge systems and a new 3-D quadratic autonomous system with a four-wing chaotic attractor, Hopf bifurcation and Silnikov chaos are discussed in detail with the first Lyapunov coefficient and Silnikov method. Using undetermined coefficient method, the existence of heteroclinic and homoclinic orbits of these systems are proved, and the Silnikov criterion guarantees that there exists the Smale horseshoe chaos motion.
引文
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