高维非线性系统的全局分叉和多脉冲混沌动力学的研究及应用
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摘要
力学、航空航天和机械工程等领域中,许多问题的力学模型可用高维非线性系统来描述。当这些系统受到外部激励,并且在系统内部非线性耦合的情况下,系统将表现出复杂的非线性动力学行为,如模态作用、能量转移、跳跃现象和多脉冲混沌动力学等。无论从非线性动力学理论角度还是从实际工程应用方面,寻找这些复杂行为出现的机理并对系统施加合理的控制是一个非常有意义的研究课题。近年来研究高维非线性系统的动力学是国际非线性动力学领域的前沿课题和重要课题,同时也是科研难题。
目前研究高维非线性系统全局动力学的解析方法还不多,主要有研究共振条件下单脉冲混沌动力学的Melnikov方法,研究多脉冲混沌动力学的广义Melnikov方法和能量-相位法,以及非共振条件下的标准Melnikov方法和指数二分法。这些方法为工程问题中的高维非线性系统全局动力学的分析提供了强有力的工具,但是这些方法都具有一定的局限性。例如能量-相位法分析的系统的未扰动部分一般要求是四维的Hamilton系统,广义Melnikov方法分析的系统未扰动作用-角变量部分可以不是Hamilton系统,但是其它的未扰动部分要求是完全可积的高维非线性系统,指数二分法可以分析高维非线性系统的全局动力学,但分析过程比较复杂并且缺乏几何的直观性。
本文首先推广了研究一类高维非线性系统的全局分叉和多脉冲混沌动力学的广义Melnikov方法,然后推导出了该类四维子系统N-脉冲跳跃轨道的能量-相位函数,改进了研究一类高维非自治慢变振子的全局分叉和混沌动力学的Melnikov方法。应用这些全局解析方法分析一些工程系统的全局动力学,揭示系统多脉冲跳跃和多脉冲混沌运动等复杂非线性动力学行为及不稳定运动产生的机理。论文的研究内容主要有以下几个方面。
(1)改进Melnikov方法对高维非线性系统完全可积性条件的限制,利用同宿流形变分方程解的结构、不变流形纤维丛理论和几何奇异摄动理论,进一步发展了研究一类高维非线性系统的全局分叉和多脉冲混沌动力学的广义Melnikov方法。
(2)研究了一类同时具有Hamilton扰动和耗散扰动的四维非线性系统,首次利用广义Melnikov函数推导出了该类系统的N-脉冲跳跃轨道的能量-相位函数。当未扰动系统是四维Hamilton系统时,所得到的能量-相位函数和Haller发展的能量-相位法中定义的能量-相位函数是一致的,进一步分析了广义Melnikov方法和能量-相位法之间的区别和联系。
(3)利用快流形方法和同宿流形变分方程解的结构,改进了研究一类高维非自治慢变振子的全局分叉和混沌动力学的Melnikov方法,推广后的方法更适合分析一些复杂工程系统的全局动力学问题。
(4)基于非线性非平面运动悬臂梁的模型,利用广义Melnikov方法分析了1:1内共振条件下非平面运动悬臂梁的全局分叉和多脉冲混沌运动,首次利用能量-相位法研究了1:2内共振条件下非平面运动悬臂梁的全局分叉和多脉冲混沌动力学。分析结果揭示了非平面运动悬臂梁产生模态作用、能量转移、多脉冲跳跃和多脉冲混沌运动的机理。数值模拟的结果表明非线性非平面运动悬臂梁的确存在多脉冲跳跃和多脉冲混沌运动等复杂的非线性现象。
(5)利用指数二分法和广义平均法,首次研究了屈曲矩形薄板的混沌运动。研究结果表明,当外激励频率和非屈曲线性系统的固有频率发生主共振,当参数激励的频率和非屈曲线性系统的固有频率发生基本参数共振时,矩形薄板的非双曲子系统不会影响整个系统混沌运动发生的临界条件。数值模拟研究了参数激励和外激励对屈曲矩形薄板共振混沌动力学的影响,数值结果验证了双曲子系统的混沌运动决定着整个系统也是混沌运动的结论。
(6)利用广义Melnikov方法分析了1:1内共振条件下功能梯度材料矩形板的全局分叉和多脉冲混沌动力学。为了克服运动控制方程中的平方和立方非线性项为系统动力学分析带来的困难,采用了高阶多尺度方法推导出控制系统的振幅和相位的平均方程。全局摄动分析揭示了功能梯度材料矩形板产生模态作用、能量转移、多脉冲跳跃和多脉冲混沌运动的机理。数值模拟研究了参数激励和外激励对功能梯度材料矩形板动力学的影响,其结果表明了功能梯度材料矩形板的确存在多脉冲混沌运动等复杂的非线性现象。
(7)利用能量-相位法分析了1:1内共振条件下非线性振动减震器的全局分叉和多脉冲混沌动力学。全局摄动分析揭示了非线性减震器产生跳跃现象、模态作用、能量转移和多脉冲混沌运动的机理。数值模拟研究了参数激励、阻尼参数和调谐参数对系统动力学的影响,结果表明非线性振动减震器的确存在多脉冲跳跃和多脉冲混沌运动等复杂的非线性现象。
在结束语中,对全文进行了总结,提出了可能存在的问题和进一步的研究方向。
The mechanical models for a variety of problems in the field of mechanics, aircraft, aerospace, mechanical engineering, can be described by high-dimensional nonlinear systems. When these systems are subjected to external excitations and internal coupling of nonlinearity, complex dynamical behaviors of the high-dimensional nonlinear systems will occur, such as modal interactions, the transfer of energy, the phenomena of jumping, multi-pulse chaotic dynamics. It is a meaningful topic to find the mechanism of these complex dynamical behaviors and improve reasonable control on these systems. Recently, studies on complex dynamics in high-dimensional nonlinear systems have become the leading, significant and difficult topics in the field of nonlinear dynamics.
Up to now, there is very few analytical method which can be used to study the global dynamics of high-dimensional nonlinear systems. These methods mainly include the Melnikov method, the global perturbation method for investigating the single-pulse chaotic dynamics, the extended Melnikov method and energy-phase method for studying the multi-pulse chaotic dynamics in the case of resonance, and the method of exponential dichotomies for studying the chaotic dynamics in the case of non-resonance. These methods are powerful tools to investigate the global bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering. However, these methods have many limitations for applying them to analyze the global bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering. For example, the unperturbed parts of high-dimensional nonlinear systems, which can be analyzed by the energy-phase method, are generally requested to be four-dimensional Hamiltonian systems. The extended Melnikov is usually employed to analyze the global bifurcations and chaotic dynamics of high-dimensional nonlinear systems with the action-angle variable. The unperturbed part of the action-angle variable may not be a Hamiltonian system. However, other parts of the unperturbed systems must be Hamiltonian with completely integrability. The method of exponential dichotomies can be utilized to analyze the chaotic dynamics of high-dimensional nonlinear systems. However, the process of analysis is too complex and lack of the geometric intuition for engineering researchers.
In this dissertation, firstly, the extended Melnikov method is further generalized to study the global bifurcations and multi-pulse chaotic dynamics for a class of high-dimensional nonlinear systems. The energy-phase function is derived for the four-dimensional subsystem of a class of high-dimensional nonlinear systems. The Melnikov method is also improved to study the global bifurcations and chaotic dynamics for a class of high-dimensional non-autonomous slowly varying oscillators. Then, the aforementioned global perturbation methods are utilized to investigate the global bifurcations and multi-pulse chaotic dynamics of several engineering systems. The results obtained here reveal the mechanism on complex nonlinear dynamical behaviors and the instability of motions, such as the multi-pulse jumping and multi-pulse chaotic motions of high-dimensional nonlinear systems in engineering.
The major research contents of this dissertation are briefly summarized as follows.
(1) After improving the limitations of completely integrability for the Melnikov method in high-dimensional nonlinear systems and using the structure of solutions for variational equation along homoclinic manifolds, the theory of fibers for invariant manifolds and geometric singular perturbation theory, the extended Melnikov method is further improved to study the global bifurcations and multi-pulse chaotic dynamics for a class of high-dimensional nonlinear systems.
(2) A class of four dimensional nonlinear systems subjected to both Hamiltonian and dissipative perturbations are investigated. The energy-phase function for N-pulse jumping orbits is derived using the extended Melnikov function for the first time. The energy-phase function is the same as the one given by Haller in energy-phase method when the unperturbed system is Hamiltonian. Moreover, the differences and connections between the energy-phase method and the extended Melnikov method are also analyzed.
(3) Applying the technique of fast manifold and the structure of solutions for variational equation along homoclinic manifolds, the Melnikov method is improved to investigate the global bifurcations and chaotic dynamics for a class of high-dimensional non-autonomous slowly varying oscillators. The generalized methods are more suitable to analyze the global dynamics of complex engineering systems.
(4) Based on the governing equations of nonlinear nonplanar motion for a cantilever beam, the global bifurcations and multi-pulse chaotic motions for the cantilever beam in the case of one-to-one internal resonance are analyzed using the extended Melnikov method. The global bifurcations and multi-pulse chaotic motions for the cantilever beam in the case of one-to-two internal resonance are investigated using the energy-phase method for the first time. These results obtained here can reveal the mechanism of complex nonlinear motions for the cantilever beam, such as modal interactions, the transfer of energy, the multi-pulse jumping and the multi-pulse chaotic motions. Numerical simulations are implemented to demonstrate the existence of the complex nonlinear phenomena, such as multi-pulse jumping and the multi-pulse chaotic motions for the cantilever beam.
(5) The exponential dichotomies and the generalized averaged procedure are employed to study the chaotic motions for a buckled rectangular thin plate for the first time. In the case of primary resonance and principle parametric resonance, it is found that non-hyperbolic subsystem of the rectangular thin plate does not affect the critical condition on the occurrence of chaotic motions for the full system of the rectangular thin plate. Numerical simulations are implemented to explain the influence of external and parametric excitations on the chaotic motions for the rectangular thin plate. Numerical results show that the chaotic motions of the hyperbolic subsystem are shadowed by the chaotic motions for the full system of the rectangular thin plate.
(6) The global bifurcations and multi-pulse chaotic dynamics for a functionally graded material rectangular plate are investigated in the case of one-to-one internal resonance using the extended Melnikov method. In order to overcome the difficulty for dynamical analysis brought by the square and cubic nonlinear terms in the governing equations of motion, the method of high-order multiple scales is applied to derive the averaged equation governing the amplitude and phase. The extended Melnikov method is employed to detect the mechanism of complex motions for the FGM rectangular plate, such as modal interactions, the transfer of energy, the multi-pulse jumping and the multi-pulse chaotic motions. Numerical simulations are finished to study the influence of the external and parametric excitations on the nonlinear dynamics for the FGM rectangular thin plate. Numerical results indicate the existence of multi-pulse chaotic motions and other complex nonlinear phenomena for the FGM rectangular plate.
(7) The global bifurcations and multi-pulse chaotic dynamics for a nonlinear vibration absorber are studied in the case of one-to-one internal resonance using the energy-phase method. The mechanism of complex motions for the nonlinear vibration absorber is detected, such as modal interactions, the transfer of energy, the multi-pulse jumping and the multi-pulse chaotic motions. Numerical simulations are given to study the effect of the parametric excitation, the damping parameter and the detuning parameters on the nonlinear dynamic responses for the nonlinear vibration absorber. Numerical results illustrate the existence of multi-pulse jumping, multi-pulse chaotic motions and other complex nonlinear phenomena for the nonlinear vibration absorber.
In the last section, the dissertation is summarized. Moreover, the further studies on global bifurcations and multi-pulse chaotic dynamics for high-dimensional nonlinear systems are discussed.
目前研究高维非线性系统全局动力学的解析方法还不多,主要有研究共振条件下单脉冲混沌动力学的Melnikov方法,研究多脉冲混沌动力学的广义Melnikov方法和能量-相位法,以及非共振条件下的标准Melnikov方法和指数二分法。这些方法为工程问题中的高维非线性系统全局动力学的分析提供了强有力的工具,但是这些方法都具有一定的局限性。例如能量-相位法分析的系统的未扰动部分一般要求是四维的Hamilton系统,广义Melnikov方法分析的系统未扰动作用-角变量部分可以不是Hamilton系统,但是其它的未扰动部分要求是完全可积的高维非线性系统,指数二分法可以分析高维非线性系统的全局动力学,但分析过程比较复杂并且缺乏几何的直观性。
本文首先推广了研究一类高维非线性系统的全局分叉和多脉冲混沌动力学的广义Melnikov方法,然后推导出了该类四维子系统N-脉冲跳跃轨道的能量-相位函数,改进了研究一类高维非自治慢变振子的全局分叉和混沌动力学的Melnikov方法。应用这些全局解析方法分析一些工程系统的全局动力学,揭示系统多脉冲跳跃和多脉冲混沌运动等复杂非线性动力学行为及不稳定运动产生的机理。论文的研究内容主要有以下几个方面。
(1)改进Melnikov方法对高维非线性系统完全可积性条件的限制,利用同宿流形变分方程解的结构、不变流形纤维丛理论和几何奇异摄动理论,进一步发展了研究一类高维非线性系统的全局分叉和多脉冲混沌动力学的广义Melnikov方法。
(2)研究了一类同时具有Hamilton扰动和耗散扰动的四维非线性系统,首次利用广义Melnikov函数推导出了该类系统的N-脉冲跳跃轨道的能量-相位函数。当未扰动系统是四维Hamilton系统时,所得到的能量-相位函数和Haller发展的能量-相位法中定义的能量-相位函数是一致的,进一步分析了广义Melnikov方法和能量-相位法之间的区别和联系。
(3)利用快流形方法和同宿流形变分方程解的结构,改进了研究一类高维非自治慢变振子的全局分叉和混沌动力学的Melnikov方法,推广后的方法更适合分析一些复杂工程系统的全局动力学问题。
(4)基于非线性非平面运动悬臂梁的模型,利用广义Melnikov方法分析了1:1内共振条件下非平面运动悬臂梁的全局分叉和多脉冲混沌运动,首次利用能量-相位法研究了1:2内共振条件下非平面运动悬臂梁的全局分叉和多脉冲混沌动力学。分析结果揭示了非平面运动悬臂梁产生模态作用、能量转移、多脉冲跳跃和多脉冲混沌运动的机理。数值模拟的结果表明非线性非平面运动悬臂梁的确存在多脉冲跳跃和多脉冲混沌运动等复杂的非线性现象。
(5)利用指数二分法和广义平均法,首次研究了屈曲矩形薄板的混沌运动。研究结果表明,当外激励频率和非屈曲线性系统的固有频率发生主共振,当参数激励的频率和非屈曲线性系统的固有频率发生基本参数共振时,矩形薄板的非双曲子系统不会影响整个系统混沌运动发生的临界条件。数值模拟研究了参数激励和外激励对屈曲矩形薄板共振混沌动力学的影响,数值结果验证了双曲子系统的混沌运动决定着整个系统也是混沌运动的结论。
(6)利用广义Melnikov方法分析了1:1内共振条件下功能梯度材料矩形板的全局分叉和多脉冲混沌动力学。为了克服运动控制方程中的平方和立方非线性项为系统动力学分析带来的困难,采用了高阶多尺度方法推导出控制系统的振幅和相位的平均方程。全局摄动分析揭示了功能梯度材料矩形板产生模态作用、能量转移、多脉冲跳跃和多脉冲混沌运动的机理。数值模拟研究了参数激励和外激励对功能梯度材料矩形板动力学的影响,其结果表明了功能梯度材料矩形板的确存在多脉冲混沌运动等复杂的非线性现象。
(7)利用能量-相位法分析了1:1内共振条件下非线性振动减震器的全局分叉和多脉冲混沌动力学。全局摄动分析揭示了非线性减震器产生跳跃现象、模态作用、能量转移和多脉冲混沌运动的机理。数值模拟研究了参数激励、阻尼参数和调谐参数对系统动力学的影响,结果表明非线性振动减震器的确存在多脉冲跳跃和多脉冲混沌运动等复杂的非线性现象。
在结束语中,对全文进行了总结,提出了可能存在的问题和进一步的研究方向。
The mechanical models for a variety of problems in the field of mechanics, aircraft, aerospace, mechanical engineering, can be described by high-dimensional nonlinear systems. When these systems are subjected to external excitations and internal coupling of nonlinearity, complex dynamical behaviors of the high-dimensional nonlinear systems will occur, such as modal interactions, the transfer of energy, the phenomena of jumping, multi-pulse chaotic dynamics. It is a meaningful topic to find the mechanism of these complex dynamical behaviors and improve reasonable control on these systems. Recently, studies on complex dynamics in high-dimensional nonlinear systems have become the leading, significant and difficult topics in the field of nonlinear dynamics.
Up to now, there is very few analytical method which can be used to study the global dynamics of high-dimensional nonlinear systems. These methods mainly include the Melnikov method, the global perturbation method for investigating the single-pulse chaotic dynamics, the extended Melnikov method and energy-phase method for studying the multi-pulse chaotic dynamics in the case of resonance, and the method of exponential dichotomies for studying the chaotic dynamics in the case of non-resonance. These methods are powerful tools to investigate the global bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering. However, these methods have many limitations for applying them to analyze the global bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering. For example, the unperturbed parts of high-dimensional nonlinear systems, which can be analyzed by the energy-phase method, are generally requested to be four-dimensional Hamiltonian systems. The extended Melnikov is usually employed to analyze the global bifurcations and chaotic dynamics of high-dimensional nonlinear systems with the action-angle variable. The unperturbed part of the action-angle variable may not be a Hamiltonian system. However, other parts of the unperturbed systems must be Hamiltonian with completely integrability. The method of exponential dichotomies can be utilized to analyze the chaotic dynamics of high-dimensional nonlinear systems. However, the process of analysis is too complex and lack of the geometric intuition for engineering researchers.
In this dissertation, firstly, the extended Melnikov method is further generalized to study the global bifurcations and multi-pulse chaotic dynamics for a class of high-dimensional nonlinear systems. The energy-phase function is derived for the four-dimensional subsystem of a class of high-dimensional nonlinear systems. The Melnikov method is also improved to study the global bifurcations and chaotic dynamics for a class of high-dimensional non-autonomous slowly varying oscillators. Then, the aforementioned global perturbation methods are utilized to investigate the global bifurcations and multi-pulse chaotic dynamics of several engineering systems. The results obtained here reveal the mechanism on complex nonlinear dynamical behaviors and the instability of motions, such as the multi-pulse jumping and multi-pulse chaotic motions of high-dimensional nonlinear systems in engineering.
The major research contents of this dissertation are briefly summarized as follows.
(1) After improving the limitations of completely integrability for the Melnikov method in high-dimensional nonlinear systems and using the structure of solutions for variational equation along homoclinic manifolds, the theory of fibers for invariant manifolds and geometric singular perturbation theory, the extended Melnikov method is further improved to study the global bifurcations and multi-pulse chaotic dynamics for a class of high-dimensional nonlinear systems.
(2) A class of four dimensional nonlinear systems subjected to both Hamiltonian and dissipative perturbations are investigated. The energy-phase function for N-pulse jumping orbits is derived using the extended Melnikov function for the first time. The energy-phase function is the same as the one given by Haller in energy-phase method when the unperturbed system is Hamiltonian. Moreover, the differences and connections between the energy-phase method and the extended Melnikov method are also analyzed.
(3) Applying the technique of fast manifold and the structure of solutions for variational equation along homoclinic manifolds, the Melnikov method is improved to investigate the global bifurcations and chaotic dynamics for a class of high-dimensional non-autonomous slowly varying oscillators. The generalized methods are more suitable to analyze the global dynamics of complex engineering systems.
(4) Based on the governing equations of nonlinear nonplanar motion for a cantilever beam, the global bifurcations and multi-pulse chaotic motions for the cantilever beam in the case of one-to-one internal resonance are analyzed using the extended Melnikov method. The global bifurcations and multi-pulse chaotic motions for the cantilever beam in the case of one-to-two internal resonance are investigated using the energy-phase method for the first time. These results obtained here can reveal the mechanism of complex nonlinear motions for the cantilever beam, such as modal interactions, the transfer of energy, the multi-pulse jumping and the multi-pulse chaotic motions. Numerical simulations are implemented to demonstrate the existence of the complex nonlinear phenomena, such as multi-pulse jumping and the multi-pulse chaotic motions for the cantilever beam.
(5) The exponential dichotomies and the generalized averaged procedure are employed to study the chaotic motions for a buckled rectangular thin plate for the first time. In the case of primary resonance and principle parametric resonance, it is found that non-hyperbolic subsystem of the rectangular thin plate does not affect the critical condition on the occurrence of chaotic motions for the full system of the rectangular thin plate. Numerical simulations are implemented to explain the influence of external and parametric excitations on the chaotic motions for the rectangular thin plate. Numerical results show that the chaotic motions of the hyperbolic subsystem are shadowed by the chaotic motions for the full system of the rectangular thin plate.
(6) The global bifurcations and multi-pulse chaotic dynamics for a functionally graded material rectangular plate are investigated in the case of one-to-one internal resonance using the extended Melnikov method. In order to overcome the difficulty for dynamical analysis brought by the square and cubic nonlinear terms in the governing equations of motion, the method of high-order multiple scales is applied to derive the averaged equation governing the amplitude and phase. The extended Melnikov method is employed to detect the mechanism of complex motions for the FGM rectangular plate, such as modal interactions, the transfer of energy, the multi-pulse jumping and the multi-pulse chaotic motions. Numerical simulations are finished to study the influence of the external and parametric excitations on the nonlinear dynamics for the FGM rectangular thin plate. Numerical results indicate the existence of multi-pulse chaotic motions and other complex nonlinear phenomena for the FGM rectangular plate.
(7) The global bifurcations and multi-pulse chaotic dynamics for a nonlinear vibration absorber are studied in the case of one-to-one internal resonance using the energy-phase method. The mechanism of complex motions for the nonlinear vibration absorber is detected, such as modal interactions, the transfer of energy, the multi-pulse jumping and the multi-pulse chaotic motions. Numerical simulations are given to study the effect of the parametric excitation, the damping parameter and the detuning parameters on the nonlinear dynamic responses for the nonlinear vibration absorber. Numerical results illustrate the existence of multi-pulse jumping, multi-pulse chaotic motions and other complex nonlinear phenomena for the nonlinear vibration absorber.
In the last section, the dissertation is summarized. Moreover, the further studies on global bifurcations and multi-pulse chaotic dynamics for high-dimensional nonlinear systems are discussed.
引文
1. V. K. Melnikov, On the stability of the centre for time periodic perturbations, Transantions of the Moscow Mathematical Society 12, p1-56, 1963.
2. V. I. Arnold, Instability of dynamical systems with many degrees of freedom, Soviet Mathematics Doklady 5, p581-585, 1964.
3. P. Holmes, A nonlinear oscillation with a strange attractor, Philosophical Transactions of the Royal Society A 292, p419-448, 1979.
4. S. Wiggins. Global bifurcations and chaos, New York: Springer-Verlag, 1988.
5. G. Kovacic and S. Wiggins, Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation, Physica D 57, p185-225, 1992.
6. G. Kovacic, Hamiltonian dynamics of orbits homoclinic to a resonance band, Physics Letters A 167, p137-142, 1992.
7. G. Kovacic, Dissipative dynamics of orbits homoclinic to a resonance band, Physics Letters A 167, p143-150, 1992.
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