六维非自治非线性系统的复杂动力学研究及应用
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摘要
随着科学技术的进步，研究低维非线性系统的动力学问题已经不能满足一些实际工程的需要。为了得到非线性系统中更加详细的复杂动力学行为，我们需要研究更高维数的非线性动力学系统。但是与研究低维系统的非线性动力学行为相比较，研究高维系统的非线性动力学问题的理论是很困难的。特别是为了得到与原始系统较为接近的理论分析结果，我们需要直接研究高维非自治非线性系统的复杂动力学问题。这与以前的研究方法相比，减少了中间化简的过程，保留了更多原始系统的特性，因此研究高维非自治非线性系统的复杂动力学行为具有重要的科学意义和工程应用价值。
本文在研究四维非自治非线性动力学系统的基础上，将一些方法推广应用到六维非自治非线性系统，并研究一些六维非自治非线性系统的复杂动力学问题，如下：
(1)将研究单脉冲混沌动力学的多自由度Melnikov方法推广应用到六维非自治非线性动力学系统。并利用此方法研究面内激励作用下四边简支屈曲矩形薄板的单脉冲混沌动力学。首先对面内激励作用下四边简支矩形薄板的运动方程进行三阶Galerkin离散，得到一个三自由度非自治非线性动力学方程。利用规范形理论对系统进行分析，将一些对系统影响较小的项视为扰动项。最后，通过理论分析和数值模拟发现面内激励作用下四边简支屈曲矩形薄板存在单脉冲混沌运动。此外，利用此方法研究了面内激励与横向激励联合作用下四边简支屈曲矩形薄板的单脉冲混沌动力学。
(2)将研究多脉冲混沌动力学的广义Melnikov方法推广到混合坐标系下六维非自治非线性动力学系统。在研究过程中，引进了横截面，这样就将混合坐标系下六维非自治非线性动力学系统化成七维自治非线性动力学系统。并应用归纳法证明了Melnikov函数的计算过程。
利用改进后的广义Melnikov方法研究三种板的多脉冲混沌动力学，例如面内激励作用下四边简支屈曲矩形薄板、面内激励与横向激励联合作用下四边简支屈曲矩形薄板和面内激励与横向激励联合作用下压电复合材料层合矩形板。在计算过程中，由于这三个系统比较复杂，无法直接利用改进后的方法进行研究，我们首先利用三阶规范形程序对三自由度非线性动力学方程进行计算。然后，通过坐标变换把这三个系统化成后四维是极坐标并且前两维与后四维是解耦的Hamilton系统。分析解耦系统的前两维方程发现这三个系统存在同宿分叉。研究解耦系统的后四维方程发现这三个系统存在Shilnikov型多脉冲混沌运动。在计算Melnikov函数时，由于一些积分比较复杂，我们利用Taylor级数进行展开和留数理论得到了这三个系统的k-脉冲Melnikov函数。最后通过数值模拟发现这三个系统存在多脉冲混沌运动，进一步验证了理论分析的结果。
(3)将广义Melnikov方法推广到直角坐标系下六维非自治非线性动力学系统并在理论上做了证明。在研究过程中，引进了横截面，这样就将直角坐标系下六维非自治非线性动力学系统化成七维自治非线性动力学系统。
利用此方法研究了横向激励作用下四边简支复合材料层合矩形板和面内激励与横向激励联合作用下四边简支蜂窝夹层矩形板的多脉冲混沌动力学。在研究过程中，首先应用三阶规范形程序对这两个系统进行分析。经过规范形分析后发现系统中的一些项不存在，根据规范形理论的本质，说明这些项在系统中起次要的作用，因此我们把这些项放在扰动项中进行研究。然后，通过坐标变换将这两个系统的未扰动系统化为六维直角坐标系下的Hamilton系统并且系统的前两维与后四维是解耦的系统。接下来，考虑解耦系统的前两维方程分别发现横向激励作用下四边简支层合矩形板存在异宿轨道，而面内激励与横向激励联合作用下四边简支蜂窝夹层矩形板存在同宿轨道。在共振的情况下，研究这两个系统的后四维方程，发现这两个系统存在Shilnikov型多脉冲混沌运动。最后数值计算也发现了这两个系统存在多脉冲混沌运动，进一步验证了理论分析的结果。
With the development of science and technology, study of the dynamical problem forthe low-dimensional nonlinear dynamical system cannot satisfy the requirement in someof practical engineering. In order to accurately analyze the complicated dynamicalproblem of nonlinear system, the higher-dimensional nonlinear dynamical system needs tobe considered. However, compared with the research of the dynamical problem for thelow-dimensional nonlinear system, it is difficult to study the theory of high-dimensionalnonlinear system. Especially, In order to obtain the theoretical results which closer to theoriginal system, the complicated dynamics of the high-dimensional non-autonomousnonlinear system is investigated directly. The process of simplification is reducedcomparing with the previous methods. Then more characteristics of the original system areretained. Therefore, study of the complicated dynamics for the high-dimensionalnon-autonomous nonlinear system is very important subject in science and engineeringapplications.
Based on the method of study four-dimensional non-autonomous nonlineardynamical system, some of methods are improved and applied to study thesix-dimensional non-autonomous nonlinear dynamical system. The complicated dynamicsof some six-dimensional non-autonomous nonlinear systems is investigated using thesemethods as follows.
(1) The multi-degree-of-freedom Melnikov method which studied the single-pulsechaotic dynamics is improved to investigate the six-dimensional non-autonomousnonlinear dynamical system. The single-pulse chaotic dynamics of a four-edge simplysupported buckled rectangular thin plate under in-plane excitation are investigated usingthe improved method. Firstly, the Galerkin method is employed to discretize the motionequations of the four-edge simply supported buckled rectangular thin plate under in-planeexcitation. A non-autonomous nonlinear dynamics equation with three-degree-of-freedomis derived. Then, the three-order normal form is used to analyze this system. Somenonlinear terms in this system have less effect than other terms. So these terms areconsidered as perturbation terms. In the end, the single-pulse chaotic motions of thefour-edge simply supported buckled rectangular thin plate under in-plane excitation arefound from theoretical analysis and numerical simulation. Furthermore, the single-pulsechaotic dynamics of a four-edge simply supported buckled rectangular thin plate under thecombination of in-plane and transversal excitations is investigated using this method.
(2) The extended Melnikov method which studied the multi-pulse chaotic dynamicsis improved to investigate the six-dimensional non-autonomous nonlinear dynamicalsystem in mixed coordinate. In the process of investigation, when a cross-section is appled to the six-dimensional non-autonomous nonlinear dynamical system, a seven-dimensionalautonomous nonlinear dynamical system is obtained. The finite inductive approach is usedto prove the Melnikov function.
The multi-pulse chaotic dynamics of three kinds of plate are investigated using theimproved extended Melnikov method. Such as the four-edge simply supported buckledrectangular thin plate under in-plane excitation, the four-edge simply supported buckledrectangular thin plate under the combination of in-plane and transversal excitations and afour-edge simply supported composite laminated piezoelectric rectangular plate underin-plane and transversal excitations. In the process of computation, because these systemsare too complex to study using the improved method, the three-order normal form is usedto simplify the three-degree-of-freedom nonlinear dynamical equations. Then the lastfour-dimensional equations are transformed into the polar form, and the first two equationsare decoupled with the last four equations by coordinate transformation. A homoclinicbifurcation is found from the first two equations and a multi-pulse Shilnikov type chaoticmotion is found from the last four equations when studing these decoupled systems. Whencalculating the Melnikov functions, some of integrals are too complex to calculate.The k-pulse Melnikov function of these systems are obtained using the Taylor series andthe residue theory. In the end, the multi-pulse chaotic motions of these systems are foundfrom the numerical simulations which further verify the result of theoretical analysis.
(3) The extended Melnikov method is improved to investigate the six-dimensionalnon-autonomous nonlinear dynamical system in Cartesian coordinate and correctness ofsuch a method is theoretically proved. In the process of investigation, when a cross-sectionis appled to six-dimensional non-autonomous nonlinear dynamical system, aseven-dimensional autonomous nonlinear dynamical system is obtained.
The multi-pulse chaotic dynamics of a four-edge simply supported compositelaminated rectangular plate subjected to transversal excitation and a four-edge simplysupported honeycomb sandwich rectangular plate under in-plane and transversalexcitations are investigated using the improved extended Melnikov method. In the processof computation, the three-order normal form is used to analyze these systems. Somenonlinear terms of these systems disappear after the normal form calculation. By virtue ofthe theory of the normal form, these nonlinear terms in these systems have less effect thanother terms. So these terms are considered as perturbation terms in these systems. Theunperturbed systems of these systems are transformed into Hamiltonian systems and thefirst two equations are decoupled with the last four equations by coordinatetransformation. When studing the first two equations of these decoupled systems, aheteroclinic orbit is found from the four-edge simply supported laminated rectangularplate subjected to transversal excitation, and a homoclinic orbit is found from thefour-edge simply supported honeycomb sandwich rectangular plate under in-plane and transversal excitations. Multi-pulse Shilnikov type chaotic motions of these systems arefound in the condition of resonance. In the end, multi-pulse chaotic motions of thesesystems are found from numerical simulations which further verify the result of theoreticalanalysis.
本文在研究四维非自治非线性动力学系统的基础上，将一些方法推广应用到六维非自治非线性系统，并研究一些六维非自治非线性系统的复杂动力学问题，如下：
(1)将研究单脉冲混沌动力学的多自由度Melnikov方法推广应用到六维非自治非线性动力学系统。并利用此方法研究面内激励作用下四边简支屈曲矩形薄板的单脉冲混沌动力学。首先对面内激励作用下四边简支矩形薄板的运动方程进行三阶Galerkin离散，得到一个三自由度非自治非线性动力学方程。利用规范形理论对系统进行分析，将一些对系统影响较小的项视为扰动项。最后，通过理论分析和数值模拟发现面内激励作用下四边简支屈曲矩形薄板存在单脉冲混沌运动。此外，利用此方法研究了面内激励与横向激励联合作用下四边简支屈曲矩形薄板的单脉冲混沌动力学。
(2)将研究多脉冲混沌动力学的广义Melnikov方法推广到混合坐标系下六维非自治非线性动力学系统。在研究过程中，引进了横截面，这样就将混合坐标系下六维非自治非线性动力学系统化成七维自治非线性动力学系统。并应用归纳法证明了Melnikov函数的计算过程。
利用改进后的广义Melnikov方法研究三种板的多脉冲混沌动力学，例如面内激励作用下四边简支屈曲矩形薄板、面内激励与横向激励联合作用下四边简支屈曲矩形薄板和面内激励与横向激励联合作用下压电复合材料层合矩形板。在计算过程中，由于这三个系统比较复杂，无法直接利用改进后的方法进行研究，我们首先利用三阶规范形程序对三自由度非线性动力学方程进行计算。然后，通过坐标变换把这三个系统化成后四维是极坐标并且前两维与后四维是解耦的Hamilton系统。分析解耦系统的前两维方程发现这三个系统存在同宿分叉。研究解耦系统的后四维方程发现这三个系统存在Shilnikov型多脉冲混沌运动。在计算Melnikov函数时，由于一些积分比较复杂，我们利用Taylor级数进行展开和留数理论得到了这三个系统的k-脉冲Melnikov函数。最后通过数值模拟发现这三个系统存在多脉冲混沌运动，进一步验证了理论分析的结果。
(3)将广义Melnikov方法推广到直角坐标系下六维非自治非线性动力学系统并在理论上做了证明。在研究过程中，引进了横截面，这样就将直角坐标系下六维非自治非线性动力学系统化成七维自治非线性动力学系统。
利用此方法研究了横向激励作用下四边简支复合材料层合矩形板和面内激励与横向激励联合作用下四边简支蜂窝夹层矩形板的多脉冲混沌动力学。在研究过程中，首先应用三阶规范形程序对这两个系统进行分析。经过规范形分析后发现系统中的一些项不存在，根据规范形理论的本质，说明这些项在系统中起次要的作用，因此我们把这些项放在扰动项中进行研究。然后，通过坐标变换将这两个系统的未扰动系统化为六维直角坐标系下的Hamilton系统并且系统的前两维与后四维是解耦的系统。接下来，考虑解耦系统的前两维方程分别发现横向激励作用下四边简支层合矩形板存在异宿轨道，而面内激励与横向激励联合作用下四边简支蜂窝夹层矩形板存在同宿轨道。在共振的情况下，研究这两个系统的后四维方程，发现这两个系统存在Shilnikov型多脉冲混沌运动。最后数值计算也发现了这两个系统存在多脉冲混沌运动，进一步验证了理论分析的结果。
With the development of science and technology, study of the dynamical problem forthe low-dimensional nonlinear dynamical system cannot satisfy the requirement in someof practical engineering. In order to accurately analyze the complicated dynamicalproblem of nonlinear system, the higher-dimensional nonlinear dynamical system needs tobe considered. However, compared with the research of the dynamical problem for thelow-dimensional nonlinear system, it is difficult to study the theory of high-dimensionalnonlinear system. Especially, In order to obtain the theoretical results which closer to theoriginal system, the complicated dynamics of the high-dimensional non-autonomousnonlinear system is investigated directly. The process of simplification is reducedcomparing with the previous methods. Then more characteristics of the original system areretained. Therefore, study of the complicated dynamics for the high-dimensionalnon-autonomous nonlinear system is very important subject in science and engineeringapplications.
Based on the method of study four-dimensional non-autonomous nonlineardynamical system, some of methods are improved and applied to study thesix-dimensional non-autonomous nonlinear dynamical system. The complicated dynamicsof some six-dimensional non-autonomous nonlinear systems is investigated using thesemethods as follows.
(1) The multi-degree-of-freedom Melnikov method which studied the single-pulsechaotic dynamics is improved to investigate the six-dimensional non-autonomousnonlinear dynamical system. The single-pulse chaotic dynamics of a four-edge simplysupported buckled rectangular thin plate under in-plane excitation are investigated usingthe improved method. Firstly, the Galerkin method is employed to discretize the motionequations of the four-edge simply supported buckled rectangular thin plate under in-planeexcitation. A non-autonomous nonlinear dynamics equation with three-degree-of-freedomis derived. Then, the three-order normal form is used to analyze this system. Somenonlinear terms in this system have less effect than other terms. So these terms areconsidered as perturbation terms. In the end, the single-pulse chaotic motions of thefour-edge simply supported buckled rectangular thin plate under in-plane excitation arefound from theoretical analysis and numerical simulation. Furthermore, the single-pulsechaotic dynamics of a four-edge simply supported buckled rectangular thin plate under thecombination of in-plane and transversal excitations is investigated using this method.
(2) The extended Melnikov method which studied the multi-pulse chaotic dynamicsis improved to investigate the six-dimensional non-autonomous nonlinear dynamicalsystem in mixed coordinate. In the process of investigation, when a cross-section is appled to the six-dimensional non-autonomous nonlinear dynamical system, a seven-dimensionalautonomous nonlinear dynamical system is obtained. The finite inductive approach is usedto prove the Melnikov function.
The multi-pulse chaotic dynamics of three kinds of plate are investigated using theimproved extended Melnikov method. Such as the four-edge simply supported buckledrectangular thin plate under in-plane excitation, the four-edge simply supported buckledrectangular thin plate under the combination of in-plane and transversal excitations and afour-edge simply supported composite laminated piezoelectric rectangular plate underin-plane and transversal excitations. In the process of computation, because these systemsare too complex to study using the improved method, the three-order normal form is usedto simplify the three-degree-of-freedom nonlinear dynamical equations. Then the lastfour-dimensional equations are transformed into the polar form, and the first two equationsare decoupled with the last four equations by coordinate transformation. A homoclinicbifurcation is found from the first two equations and a multi-pulse Shilnikov type chaoticmotion is found from the last four equations when studing these decoupled systems. Whencalculating the Melnikov functions, some of integrals are too complex to calculate.The k-pulse Melnikov function of these systems are obtained using the Taylor series andthe residue theory. In the end, the multi-pulse chaotic motions of these systems are foundfrom the numerical simulations which further verify the result of theoretical analysis.
(3) The extended Melnikov method is improved to investigate the six-dimensionalnon-autonomous nonlinear dynamical system in Cartesian coordinate and correctness ofsuch a method is theoretically proved. In the process of investigation, when a cross-sectionis appled to six-dimensional non-autonomous nonlinear dynamical system, aseven-dimensional autonomous nonlinear dynamical system is obtained.
The multi-pulse chaotic dynamics of a four-edge simply supported compositelaminated rectangular plate subjected to transversal excitation and a four-edge simplysupported honeycomb sandwich rectangular plate under in-plane and transversalexcitations are investigated using the improved extended Melnikov method. In the processof computation, the three-order normal form is used to analyze these systems. Somenonlinear terms of these systems disappear after the normal form calculation. By virtue ofthe theory of the normal form, these nonlinear terms in these systems have less effect thanother terms. So these terms are considered as perturbation terms in these systems. Theunperturbed systems of these systems are transformed into Hamiltonian systems and thefirst two equations are decoupled with the last four equations by coordinatetransformation. When studing the first two equations of these decoupled systems, aheteroclinic orbit is found from the four-edge simply supported laminated rectangularplate subjected to transversal excitation, and a homoclinic orbit is found from thefour-edge simply supported honeycomb sandwich rectangular plate under in-plane and transversal excitations. Multi-pulse Shilnikov type chaotic motions of these systems arefound in the condition of resonance. In the end, multi-pulse chaotic motions of thesesystems are found from numerical simulations which further verify the result of theoreticalanalysis.
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