六维非线性系统的复杂动力学研究
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摘要
非线性科学是自上世纪60年代以来,在各门以非线性为特征的分支学科的基础上逐步发展起来的一门研究非线性共性现象的基础学科,被誉为上世纪自然科学的“第三次大革命”。非线性动力学系统理论把丰富的工程和物理内容与近代数学的抽象方法有机地结合在一起,是古典力学的现代数学表示形式。工程非线性系统中往往存在混沌运动,一些被认为是安全的实际工程系统中存在着平衡稳态的混沌运动、混沌运动的瞬变过程以及不稳定混沌运动的可能性。
机械系统中许多问题的数学模型和动力学方程往往都可以用高维非线性系统来描述。对于高维非线性动力学系统的研究来说,不仅理论方法上有困难,几何描述和数值计算都有困难,因此,其系统动力学特性的研究难度比低维非线性动力学系统要大得多。目前,国际和国内关于高维非线性系统全局分叉和混沌动力学的理论方法研究的还不是很多,仍处于发展阶段。如何全面系统地了解和掌握高维非线性系统的全局分叉和混沌运动学特性,是分析高维非线性系统动力学特性的难题,也是国际非线性动力学领域的前沿研究课题。
全局摄动方法和能量-相位法自从20世纪90年代被提出来以后,主要用于研究四维非线性系统的全局分叉和单脉冲及多脉冲混沌动力学。国际和国内在四维非线性系统的全局分叉和单脉冲及多脉冲混沌动力学的研究方面已经取得了一些成果,而对于六维非线性系统的全局分叉和单脉冲及多脉冲混沌动力学的研究还不多见。本文以四维非线性系统全局分叉和混沌动力学的研究结果为基础,在六维相空间中改进和推广了全局摄动方法和能量-相位法,首次利用这两种解析方法研究了具有实际工程应用背景的几个六维非线性系统,探索了这些系统中所存在的全局分叉和单脉冲及多脉冲混沌动力学特性,利用数值方法进行模拟验证了理论分析的结果。
本文主要的研究内容包括以下几个方面。
1.研究了1:2:3内共振情况下六维粘弹性传动带系统的同宿分叉现象和混沌动力学
我们首先利用规范形理论和内积法对粘弹性传动带系统的六维平均方程进行化简,得到与原系统拓扑等价的较为简单的规范形。在此基础上,应用全局摄动方法对六维非线性系统进行摄动分析。研究结果发现,在小扰动情况下,系统存在连接同宿轨道的Shilnikov型单脉冲混沌运动。我们应用扩展的能量-相位法同样分析了这个六维非线性系统的复杂动力学特性,研究结果发现系统存在着从不变流形的不动点出发,经过n个跳跃以后,落点仍然位于该不动点收敛域内的Shilnikov型多脉冲轨道。数值模拟结果表明系统出现具有跳跃轨道的混沌运动,激励f 1和f 2对粘弹性传动带系统的非线性动力学行为具有显著的影响。
2.研究了1:3:3内共振情况下的复合材料层合矩形板的混沌动力学
应用规范形理论对1:3:3内共振情况下复合材料层合矩形板的六维平均方程进行化简,得到较为简单的规范形。在此基础上,应用全局摄动方法研究了这个系统的同宿分叉和Shilnikov型单脉冲混沌运动;应用能量-相位法分析了系统的异宿分叉和Shilnikov型多脉冲轨道。应用数值模拟进一步验证了理论分析结果的正确性,数值结果表明系统存在着单脉冲和多脉冲混沌运动,同时,我们还发现在一定的参数条件下,激励f 3对系统的非线性动力学行为有显著的影响,系统响应的变化过程呈现为混沌→周期→混沌→周期。
3.利用能量-相位法对1:2:4内共振情况下压电复合材料层合矩形板的混沌动力学进行研究
应用规范形理论对1:2:4内共振情况下压电复合材料层合矩形板的六维平均方程进行化简,得到与原系统拓扑等价的较为简单的规范形。应用扩展的能量-相位法研究了这个系统的复杂动力学特性。理论研究结果表明,在一定的条件下,系统可以发生同宿分叉,在小扰动情况下,系统存在着Shilnikov型多脉冲轨道和混沌运动。通过数值模拟研究,我们发现当取某些参数时,压电复合材料层合矩形板可以出现具有跳跃轨道的混沌运动,进一步验证了理论分析结果的正确性。
Since the 1960s, the basic research of nonlinear phenomenon was successfully developed from different branches in nonlinear sciences. Henceforth, it is regarded as“The Third Revolution”in natural sciences in the 20th century. The theories of nonlinear science are established from the multidisciplinary subjects in engineering, physics and mathematics. There frequently exist the chaotic motions in nonlinear dynamical systems. In reality, it is possible to find the steady state response, transient process and instable state of chaotic motions in practical problems.
In engineering problems, the mathematical models and its dynamical equations can be governed by the high-dimensional nonlinear systems. Contrary to the low-dimensional nonlinear systems, the theory, geometric description and numerical simulation of high-dimensional ones are much more sophisticated. Heretofore, the analytical methods are often unavailable for studying the global bifurcation and chaotic dynamics of high-dimensional nonlinear systems. The development of theories and the provision of systematic applications to many engineering problems are very appealing and challenging. Hence, it is important to provide an all-embracing understanding of high-dimensional nonlinear systems, including the global bifurcation and the chaotic dynamics. It is a research forefront in nonlinear science as well.
The Melnikov method and the energy-phase method are mainly used to probe the global bifurcation and chaotic dynamics of four-dimensional nonlinear systems. There have been numerous studies on the four-dimensional nonlinear system. In this dissertation, the global perturbation and energy-phase methods are improved to study the global bifurcation and chaotic dynamics of six-dimensional nonlinear mechanical systems. Numerical simulation is also given to verify the analytical solutions. The main research scopes of this dissertation are categorized as follows.
1. The chaotic dynamics and jumping orbits of the six-dimensional nonlinear system are investigated for the first time, which represents the averaged equation of an axially moving viscoelastic belt in the case of 1:2:3 internal resonance.
To simplify the six-dimensional averaged equation to a simpler normal form with the same topological equivalence, the theory of normal form and the method of the inner product are adopted. The global perturbation method is employed to analyze the existence of the homoclinic bifurcation and the single-pulse Shilnikov orbits for the six-dimensional nonlinear system. Using the energy-phase method, it is found that the system contains the multi-pulses jumping. The takeoff point is a fixed point in the invariant manifold and the landing point lies in the domain of the attraction of the fixed point. From the numerical results, the chaotic motions and jumping phenomena of orbits in the nonlinear system are demonstrated. Besides, it is found that the excitations f 1 and f 2 have significant effects on the nonlinear dynamical behavior of the belt.
2. The dynamical characteristics of the six-dimensional nonlinear system are revealed for the first time, which is the nonlinear averaged equation of the composite laminated plate in the case of 1:2:3 internal resonances.
First, the theory of normal form is used to simplify the six-dimensional averaged equation to a simpler normal form. Then, the global perturbation method is also employed to study the homorclinic bifurcation and the single-pulse Shilnikov orbits for the nonlinear system with small perturbation. in addition, The energy-phase method is extended to study the heteroclinic bifurcation and multi-pulse Shilnikov type orbits of the nonlinear system. From the numerical results, the chaotic motions and jumping phenomena of orbits for the composite laminated thin plate are discovered under certain conditions. It is also shown that the excitation f 3 performs significant effect on the nonlinear dynamical behaviors of the composite laminated thin plate. The motions of the composite laminated thin plate are changed to the particular sequence by increasing the forcing excitation f 3 as: the chaotic motion→the periodic motion→the chaotic motion→the periodic motion.
3. By combining the theory of normal form and the energy-phase method, the multi-pulse and dynamic characteristics of the six-dimensional nonlinear system are studied for the first time. This high-dimensional system is used to describe the laminated composite piezoelectric rectangular plate in the case of 1:2:4 internal resonances.
The theory of normal form is used to simplify the six-dimensional averaged equation to a simpler normal form with the same topological equivalence. The energy-phase method is extended to study the homorclinic bifurcation and the multi-pulse Shilnikov orbits for the nonlinear system with small perturbation. From the numerical results, it is illustrated that there exist the chaotic motions and jumping phenomena of orbits in the laminated composite piezoelectric rectangular plate under certain conditions.
机械系统中许多问题的数学模型和动力学方程往往都可以用高维非线性系统来描述。对于高维非线性动力学系统的研究来说,不仅理论方法上有困难,几何描述和数值计算都有困难,因此,其系统动力学特性的研究难度比低维非线性动力学系统要大得多。目前,国际和国内关于高维非线性系统全局分叉和混沌动力学的理论方法研究的还不是很多,仍处于发展阶段。如何全面系统地了解和掌握高维非线性系统的全局分叉和混沌运动学特性,是分析高维非线性系统动力学特性的难题,也是国际非线性动力学领域的前沿研究课题。
全局摄动方法和能量-相位法自从20世纪90年代被提出来以后,主要用于研究四维非线性系统的全局分叉和单脉冲及多脉冲混沌动力学。国际和国内在四维非线性系统的全局分叉和单脉冲及多脉冲混沌动力学的研究方面已经取得了一些成果,而对于六维非线性系统的全局分叉和单脉冲及多脉冲混沌动力学的研究还不多见。本文以四维非线性系统全局分叉和混沌动力学的研究结果为基础,在六维相空间中改进和推广了全局摄动方法和能量-相位法,首次利用这两种解析方法研究了具有实际工程应用背景的几个六维非线性系统,探索了这些系统中所存在的全局分叉和单脉冲及多脉冲混沌动力学特性,利用数值方法进行模拟验证了理论分析的结果。
本文主要的研究内容包括以下几个方面。
1.研究了1:2:3内共振情况下六维粘弹性传动带系统的同宿分叉现象和混沌动力学
我们首先利用规范形理论和内积法对粘弹性传动带系统的六维平均方程进行化简,得到与原系统拓扑等价的较为简单的规范形。在此基础上,应用全局摄动方法对六维非线性系统进行摄动分析。研究结果发现,在小扰动情况下,系统存在连接同宿轨道的Shilnikov型单脉冲混沌运动。我们应用扩展的能量-相位法同样分析了这个六维非线性系统的复杂动力学特性,研究结果发现系统存在着从不变流形的不动点出发,经过n个跳跃以后,落点仍然位于该不动点收敛域内的Shilnikov型多脉冲轨道。数值模拟结果表明系统出现具有跳跃轨道的混沌运动,激励f 1和f 2对粘弹性传动带系统的非线性动力学行为具有显著的影响。
2.研究了1:3:3内共振情况下的复合材料层合矩形板的混沌动力学
应用规范形理论对1:3:3内共振情况下复合材料层合矩形板的六维平均方程进行化简,得到较为简单的规范形。在此基础上,应用全局摄动方法研究了这个系统的同宿分叉和Shilnikov型单脉冲混沌运动;应用能量-相位法分析了系统的异宿分叉和Shilnikov型多脉冲轨道。应用数值模拟进一步验证了理论分析结果的正确性,数值结果表明系统存在着单脉冲和多脉冲混沌运动,同时,我们还发现在一定的参数条件下,激励f 3对系统的非线性动力学行为有显著的影响,系统响应的变化过程呈现为混沌→周期→混沌→周期。
3.利用能量-相位法对1:2:4内共振情况下压电复合材料层合矩形板的混沌动力学进行研究
应用规范形理论对1:2:4内共振情况下压电复合材料层合矩形板的六维平均方程进行化简,得到与原系统拓扑等价的较为简单的规范形。应用扩展的能量-相位法研究了这个系统的复杂动力学特性。理论研究结果表明,在一定的条件下,系统可以发生同宿分叉,在小扰动情况下,系统存在着Shilnikov型多脉冲轨道和混沌运动。通过数值模拟研究,我们发现当取某些参数时,压电复合材料层合矩形板可以出现具有跳跃轨道的混沌运动,进一步验证了理论分析结果的正确性。
Since the 1960s, the basic research of nonlinear phenomenon was successfully developed from different branches in nonlinear sciences. Henceforth, it is regarded as“The Third Revolution”in natural sciences in the 20th century. The theories of nonlinear science are established from the multidisciplinary subjects in engineering, physics and mathematics. There frequently exist the chaotic motions in nonlinear dynamical systems. In reality, it is possible to find the steady state response, transient process and instable state of chaotic motions in practical problems.
In engineering problems, the mathematical models and its dynamical equations can be governed by the high-dimensional nonlinear systems. Contrary to the low-dimensional nonlinear systems, the theory, geometric description and numerical simulation of high-dimensional ones are much more sophisticated. Heretofore, the analytical methods are often unavailable for studying the global bifurcation and chaotic dynamics of high-dimensional nonlinear systems. The development of theories and the provision of systematic applications to many engineering problems are very appealing and challenging. Hence, it is important to provide an all-embracing understanding of high-dimensional nonlinear systems, including the global bifurcation and the chaotic dynamics. It is a research forefront in nonlinear science as well.
The Melnikov method and the energy-phase method are mainly used to probe the global bifurcation and chaotic dynamics of four-dimensional nonlinear systems. There have been numerous studies on the four-dimensional nonlinear system. In this dissertation, the global perturbation and energy-phase methods are improved to study the global bifurcation and chaotic dynamics of six-dimensional nonlinear mechanical systems. Numerical simulation is also given to verify the analytical solutions. The main research scopes of this dissertation are categorized as follows.
1. The chaotic dynamics and jumping orbits of the six-dimensional nonlinear system are investigated for the first time, which represents the averaged equation of an axially moving viscoelastic belt in the case of 1:2:3 internal resonance.
To simplify the six-dimensional averaged equation to a simpler normal form with the same topological equivalence, the theory of normal form and the method of the inner product are adopted. The global perturbation method is employed to analyze the existence of the homoclinic bifurcation and the single-pulse Shilnikov orbits for the six-dimensional nonlinear system. Using the energy-phase method, it is found that the system contains the multi-pulses jumping. The takeoff point is a fixed point in the invariant manifold and the landing point lies in the domain of the attraction of the fixed point. From the numerical results, the chaotic motions and jumping phenomena of orbits in the nonlinear system are demonstrated. Besides, it is found that the excitations f 1 and f 2 have significant effects on the nonlinear dynamical behavior of the belt.
2. The dynamical characteristics of the six-dimensional nonlinear system are revealed for the first time, which is the nonlinear averaged equation of the composite laminated plate in the case of 1:2:3 internal resonances.
First, the theory of normal form is used to simplify the six-dimensional averaged equation to a simpler normal form. Then, the global perturbation method is also employed to study the homorclinic bifurcation and the single-pulse Shilnikov orbits for the nonlinear system with small perturbation. in addition, The energy-phase method is extended to study the heteroclinic bifurcation and multi-pulse Shilnikov type orbits of the nonlinear system. From the numerical results, the chaotic motions and jumping phenomena of orbits for the composite laminated thin plate are discovered under certain conditions. It is also shown that the excitation f 3 performs significant effect on the nonlinear dynamical behaviors of the composite laminated thin plate. The motions of the composite laminated thin plate are changed to the particular sequence by increasing the forcing excitation f 3 as: the chaotic motion→the periodic motion→the chaotic motion→the periodic motion.
3. By combining the theory of normal form and the energy-phase method, the multi-pulse and dynamic characteristics of the six-dimensional nonlinear system are studied for the first time. This high-dimensional system is used to describe the laminated composite piezoelectric rectangular plate in the case of 1:2:4 internal resonances.
The theory of normal form is used to simplify the six-dimensional averaged equation to a simpler normal form with the same topological equivalence. The energy-phase method is extended to study the homorclinic bifurcation and the multi-pulse Shilnikov orbits for the nonlinear system with small perturbation. From the numerical results, it is illustrated that there exist the chaotic motions and jumping phenomena of orbits in the laminated composite piezoelectric rectangular plate under certain conditions.
引文
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