待定固有频率法与非线性动力系统的复杂动力学
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摘要
非线性动力系统蕴含着复杂的动力学行为,如分岔、混沌等。正是这些复杂动力学行为的存在并伴以新现象的不断涌现,成为促进近代非线性动力学理论研究方法的产生、发展并日臻完善的源动力。待定固有频率法是在规范形理论全面发展以及强非线性振动问题深入研究的基础上提出的,最初的研究内容主要涉及非线性动力系统由Hopf分岔而产生的稳态响应,本文是这一方法在非线性动力系统复杂问题领域的进一步推广,针对:○1强非线性振动系统的静态与动态动力学行为研究;○2提高Melnikov方法分析非线性动力系统同(异)宿分岔问题的求解精度;○3三维系统的Shilnikov同宿轨道与倍周期分岔等问题,结合非线性动力学理论,开展研究工作,提出有效的解决方法。
本文的研究内容与主要研究成果体现在如下几个方面
(1).提出了以待定固有频率为基础,计算强非线性系统同(异)分岔问题的解析判据,克服了弱非线性系统分析方法在该领域的应用局限性。通过在复数形式规范形求解过程中引入待定固有频率,获得了强非线性系统周期响应,以该待定频率趋于零和相平面上极限环趋于鞍点为途径,确定了发生分岔的临界参数值。
(2).研究了一类强非线性转摆系统的高余维静态分岔问题。提出利用约束分岔理论和奇异性理论,充分考察强非线性振动系统的余维3静态分岔特性,全面建立了系统转迁集与分岔图之间的对应关系,并且讨论了平凡解的稳定性问题。
(3).研究了参数激励和强迫激励联合作用下强非线性系统的动态分岔问题。当控制方程中的激励形式比较复杂时,得到的平均系统会展现出丰富的动力学行为,形成的动态分岔问题无法用Melnikov方法对原有系统直接积分获取。因此,本文首先利用待定固有频率法得到了强非线性振动系统的平均方程,再通过Melnikov函数,考察系统中由于复杂激励形式存在而出现的鞍点状态与同(异)宿分岔。
(4).提出了利用待定固有频率法提高Melnikov函数分析非线性动力系统混沌门槛值计算精度的简单方法。对于复杂激励形式作用下的三阱势能系统,通过引入由待定固有频率形成的时间尺度变换,从同宿及异宿分岔两个角度获得了系统的混沌临界值,使得非线性扰动量对于基频的影响有效地体现于Melnikov函数表达式中,进而结合相应的分析过程提高了所得结果的计算精度。
(5).研究了三维PID控制系统与简化WINDMI模型Shilnikov意义下的Samle马蹄混沌。在常规的同宿轨道存在性证明基础上,进一步得到了系统Shilnikov类型同宿轨道的解析表达式。文中所述的动力系统均具有比较复杂的非线性项形式,利用Shilnikov定理对系统的混沌行为进行了充分的研究,将计算Lorenz类系统不变流形的相关方法拓展至建立三维系统鞍-焦平衡点处Shilnikov类型同宿轨道。此外,针对三维系统的周期倍化分岔问题,尝试采用待定固有频率法,提高倍周期分岔值的计算精度,有效地避免了由于寻求高阶近似解而引发的计算复杂性问题。
General nonlinear dynamical systems contain the complicated dynamical phenomenon mostly, such as bifurcation and chaos. That makes it the fundamental motility for the emerging, developing, and gradually optimizing of the modern nonlinear dynamical research because the complicated problems exciting and unfamiliar phenomenon rush out in the physical world all the time. The emergency of the undetermined fundamental frequency just accorded with the circumstance, where the research of the normal form theory has come to its mature stage and the strongly nonlinear problems bring us the enough temptation to practise our research. In this article, the initial targets of the method is extend, from pursuiting the steady asymptotic responses to the broader areas. It includes the static and dynamic bifurcation behaviors of the strongly nonlinear oscillation system (SNOS), improving the computational precision of the Homoclinic and Heteroclinic bifurcation in terms of the Melnikov method, and Shilnikov sense Homoclinic orbit and flip (periodic doubling) bifurcation in the three dimensional nonlinear system. So we pay our attention to these fields with the application of modern nonlinear dynamical methodology, to find the appropriate resolvents.
The research contents and obtained major results of this dissertation are as follows.
(1). The analytical criterions of the Homoclinic (Heteroclinic) bifurcation of the strongly nonlinear oscillator are presented. We consider the approximate periodic solution of the system subject to the quintic nonlinearity by introducing the undetermined fundamental frequency into the complex normal form operation. For the occurrence of Homoclinicity (Heteroclinicity), the bifurcation criterions are accomplished. They depend on the contact of the limit cycle with the saddle equilibrium and the vanishing undermined fundamental frequency. So the available ranges of the former criterions are extended from the weakly nonlinear oscillation system to the SNOS.
(2). The static bifurcation of the parametrically excited rotating arm with strongly nonlinearity is researched. For the discussion of static bifurcation, the bifurcation problem is described as the 3-codimension unfolding with Z2 symmetry on the basis of the singularity theory. The transition set and bifurcation diagrams, for the singularity are wholly presented with constraints, while the stability of the zero solution is researched by the eigenvalues in various parameter regions.
(3). The dynamic bifurcations of strongly nonlinear oscillator induced by parametric and external excitation are researched. It is known that the parametric and external excitation may induce additional saddle states, and result chaos in the phase space, which can not be detected by applying the Melnikov method directly. A feasible solution for this problem is the combination of the averaged equations and Melnikov method. So we consider the averaged equations of the system subject to Duffing-Van der Pol type strong nonlinearity by introducing the undetermined fundamental frequency. Then the saddle states and bifurcation values of Homoclinic structure formation are detected through the combined application of the new averaged equations with Melnikov method.
(4). The simple approach to improve the computational precision of Melnikov method is presented by using the undetermined fundamental frequency and normal form method. We construct the improved Melnikov expression for a triple well nonlinear oscillator subject to principal parametric resonance and external excitation. For the occurrence of chaos, the threshold value approximations of chaotic motion are obtained in the Homoclinicity and Heteroclinicity points of view. It depends on the introduction of undetermined fundamental frequency, and adopting new time transformation for fulfilling the Homoclinic and Heteroclinic orbits, so that the effect of disturbing parameter can be easily detected and embodied in the Melnikov operation.
(5). The strategy of predicting the Shilnikov sense horseshoe and Homoclinic orbit of the new PID controller and reduced solar wind driven magnetosphere ionosphere model (WINDMI) is presented by using the Shilnikov theorem and the invariant manifolds of the saddle-focus equilibrium point. It provides the quadratic and cubic nonlinearity to the controller systems, and extends the canonical forms of continuous time quadratic autonomous chaotic systems in three dimensions. The Shilnikov type Homoclinic orbit is concerned not only about its existence but also the analytical series expression, which distinguishes the present work from the general description of the Shilnikov sense chaos. For the discussion of flip bifurcation in the three dimensional system, the introduction of the undetermined fundemental frequency improve the computational precise of normal form method and efficiently avoid the difficulties of seeking higher-order asymptotic solutions during the course of operation.
本文的研究内容与主要研究成果体现在如下几个方面
(1).提出了以待定固有频率为基础,计算强非线性系统同(异)分岔问题的解析判据,克服了弱非线性系统分析方法在该领域的应用局限性。通过在复数形式规范形求解过程中引入待定固有频率,获得了强非线性系统周期响应,以该待定频率趋于零和相平面上极限环趋于鞍点为途径,确定了发生分岔的临界参数值。
(2).研究了一类强非线性转摆系统的高余维静态分岔问题。提出利用约束分岔理论和奇异性理论,充分考察强非线性振动系统的余维3静态分岔特性,全面建立了系统转迁集与分岔图之间的对应关系,并且讨论了平凡解的稳定性问题。
(3).研究了参数激励和强迫激励联合作用下强非线性系统的动态分岔问题。当控制方程中的激励形式比较复杂时,得到的平均系统会展现出丰富的动力学行为,形成的动态分岔问题无法用Melnikov方法对原有系统直接积分获取。因此,本文首先利用待定固有频率法得到了强非线性振动系统的平均方程,再通过Melnikov函数,考察系统中由于复杂激励形式存在而出现的鞍点状态与同(异)宿分岔。
(4).提出了利用待定固有频率法提高Melnikov函数分析非线性动力系统混沌门槛值计算精度的简单方法。对于复杂激励形式作用下的三阱势能系统,通过引入由待定固有频率形成的时间尺度变换,从同宿及异宿分岔两个角度获得了系统的混沌临界值,使得非线性扰动量对于基频的影响有效地体现于Melnikov函数表达式中,进而结合相应的分析过程提高了所得结果的计算精度。
(5).研究了三维PID控制系统与简化WINDMI模型Shilnikov意义下的Samle马蹄混沌。在常规的同宿轨道存在性证明基础上,进一步得到了系统Shilnikov类型同宿轨道的解析表达式。文中所述的动力系统均具有比较复杂的非线性项形式,利用Shilnikov定理对系统的混沌行为进行了充分的研究,将计算Lorenz类系统不变流形的相关方法拓展至建立三维系统鞍-焦平衡点处Shilnikov类型同宿轨道。此外,针对三维系统的周期倍化分岔问题,尝试采用待定固有频率法,提高倍周期分岔值的计算精度,有效地避免了由于寻求高阶近似解而引发的计算复杂性问题。
General nonlinear dynamical systems contain the complicated dynamical phenomenon mostly, such as bifurcation and chaos. That makes it the fundamental motility for the emerging, developing, and gradually optimizing of the modern nonlinear dynamical research because the complicated problems exciting and unfamiliar phenomenon rush out in the physical world all the time. The emergency of the undetermined fundamental frequency just accorded with the circumstance, where the research of the normal form theory has come to its mature stage and the strongly nonlinear problems bring us the enough temptation to practise our research. In this article, the initial targets of the method is extend, from pursuiting the steady asymptotic responses to the broader areas. It includes the static and dynamic bifurcation behaviors of the strongly nonlinear oscillation system (SNOS), improving the computational precision of the Homoclinic and Heteroclinic bifurcation in terms of the Melnikov method, and Shilnikov sense Homoclinic orbit and flip (periodic doubling) bifurcation in the three dimensional nonlinear system. So we pay our attention to these fields with the application of modern nonlinear dynamical methodology, to find the appropriate resolvents.
The research contents and obtained major results of this dissertation are as follows.
(1). The analytical criterions of the Homoclinic (Heteroclinic) bifurcation of the strongly nonlinear oscillator are presented. We consider the approximate periodic solution of the system subject to the quintic nonlinearity by introducing the undetermined fundamental frequency into the complex normal form operation. For the occurrence of Homoclinicity (Heteroclinicity), the bifurcation criterions are accomplished. They depend on the contact of the limit cycle with the saddle equilibrium and the vanishing undermined fundamental frequency. So the available ranges of the former criterions are extended from the weakly nonlinear oscillation system to the SNOS.
(2). The static bifurcation of the parametrically excited rotating arm with strongly nonlinearity is researched. For the discussion of static bifurcation, the bifurcation problem is described as the 3-codimension unfolding with Z2 symmetry on the basis of the singularity theory. The transition set and bifurcation diagrams, for the singularity are wholly presented with constraints, while the stability of the zero solution is researched by the eigenvalues in various parameter regions.
(3). The dynamic bifurcations of strongly nonlinear oscillator induced by parametric and external excitation are researched. It is known that the parametric and external excitation may induce additional saddle states, and result chaos in the phase space, which can not be detected by applying the Melnikov method directly. A feasible solution for this problem is the combination of the averaged equations and Melnikov method. So we consider the averaged equations of the system subject to Duffing-Van der Pol type strong nonlinearity by introducing the undetermined fundamental frequency. Then the saddle states and bifurcation values of Homoclinic structure formation are detected through the combined application of the new averaged equations with Melnikov method.
(4). The simple approach to improve the computational precision of Melnikov method is presented by using the undetermined fundamental frequency and normal form method. We construct the improved Melnikov expression for a triple well nonlinear oscillator subject to principal parametric resonance and external excitation. For the occurrence of chaos, the threshold value approximations of chaotic motion are obtained in the Homoclinicity and Heteroclinicity points of view. It depends on the introduction of undetermined fundamental frequency, and adopting new time transformation for fulfilling the Homoclinic and Heteroclinic orbits, so that the effect of disturbing parameter can be easily detected and embodied in the Melnikov operation.
(5). The strategy of predicting the Shilnikov sense horseshoe and Homoclinic orbit of the new PID controller and reduced solar wind driven magnetosphere ionosphere model (WINDMI) is presented by using the Shilnikov theorem and the invariant manifolds of the saddle-focus equilibrium point. It provides the quadratic and cubic nonlinearity to the controller systems, and extends the canonical forms of continuous time quadratic autonomous chaotic systems in three dimensions. The Shilnikov type Homoclinic orbit is concerned not only about its existence but also the analytical series expression, which distinguishes the present work from the general description of the Shilnikov sense chaos. For the discussion of flip bifurcation in the three dimensional system, the introduction of the undetermined fundemental frequency improve the computational precise of normal form method and efficiently avoid the difficulties of seeking higher-order asymptotic solutions during the course of operation.
引文
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