最简规范形及机电耦合非线性系统的分岔、混沌研究
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摘要
规范形理论是化简非线性振动系统的重要手段,对于研究分岔和混沌等复杂动力学问题具有深远的影响。最简规范形是在传统规范形理论和非线性变换理论的基础上,最大程度地化简微分方程所得到的一种规范形。最简规范形的研究与实际应用正朝着高维的方向发展,其求解过程非常复杂而繁琐。但是,非线性动力系统经最简规范形理论加以简化后,可以简捷地获取其平衡点附近的动力学特性。针对最简规范形和机电耦合非线性动力系统的动力学特性,论文的研究内容及取得的创新性成果有以下几个方面
(1)利用共轭算子法,研究了高维Hopf分岔和退化Hopf分岔最简规范形的系数。引进特殊的非线性变换和内积,进一步简化了中心流形上的方程。获得了几个关于Hopf分岔和退化Hopf分岔的最简规范形系数与传统规范形系数之间具体关系的定理。借助Mathematica语言,编制了计算高维Hopf分岔和退化Hopf分岔最简规范形系数的程序。通过该程序,只需输入原动力系统方程,可得到系统具体的最简规范形。最后利用编制的程序分别计算了一个6维退化Hopf分岔系统的最简规范形和一个5维Hopf分岔系统的最简规范形。结果表明6维退化Hopf分岔系统的非线性项只包含5阶项和9阶项,5维Hopf分岔系统的非线性项只包含3阶项和5阶项。
(2)研究了高维Neimark-Sacker分岔和退化Neimark-Sacker分岔的最简规范形的系数。根据最简规范形理论,通过引进特殊的非线性变换、直接计算法和第二数学归纳法,对中心流形上的方程进一步化简,计算出Neimark-Sacker分岔和退化Neimark-Sacker分岔最简规范形的非线性项中分别只包含两项。获得了几个关于Neimark-Sacker分岔和退化Neimark-Sacker分岔的最简规范形系数与传统规范形系数之间关系的定理。
(3)运用可逆线性变换和近恒同变换,研究了不经计算传统规范形,直接计算高维非线性动力系统的最简规范形。引进可逆线性变换,将非线性动力系统的线性矩阵拓扑等价于符合实际研究需求的分块对角线矩阵:相伴矩阵分布在对角线上,其余元素均为0。利用低阶项来化简高阶项,得到了高维非线性动力系统的最简规范形。在该最简规范形中,对应于每一个相伴矩阵的非线性系数矩阵,只有最后一行含有非0元素,其余各行元素均为0。借助Mathematica语言,编制了计算高维非线性动力系统最简规范形的通用程序。运行该程序,分别计算了2维、3维、4维、6维和7维非线性动力系统直到4阶的最简规范形。
(4)运用含有参数的可逆线性变换和含有参数的近恒同变换,提出了不计算传统规范形,直接计算含参非线性动力系统最简规范形的一种计算方法。借助Mathematica语言,编制了计算含参非线性动力系统的最简规范形的通用程序。
(5)利用含有参数的可逆线性变换和含有参数的近恒同非线性变换,得到一类机电耦合非线性系统的最简规范形。进一步得到了该系统的普适开折以及开折参数与原系统参数之间的关系。讨论了该系统的余维2分岔,揭示了各参数对机电耦合系统动力学行为的影响,对系统的参数设计、稳定运行和故障诊断提供了理论依据。给出了该机电耦合系统的数值仿真结果,验证了理论分析结果。
(6)利用Silnikov定理,讨论了具有自动频率跟踪功能电磁振动机械系统的混沌特性。借助卡尔达诺公式和微分方程组级数解分别讨论了该系统的特征根问题和同宿轨道的存在性,进而比较严密地证明了该系统Silnikov型Smale混沌的存在性,并给出发生Silnikov型Smale混沌所需条件。利用数值模拟得到该类机电耦合系统的相轨迹图、Lyaponov指数谱和Lyaponov维数,进一步验证了该非线性系统存在奇怪吸引子。
Normal form theory is one of the useful tools in the study of nonlinear dynamics and stresses a profound influence on complex dynamic theory such as bifurcation and chaos dynamics. The simplest normal form is the normal form that can be used to deeply simplify the original differential equations based on conventional normal form and nonlinear transformation theory. The study and the application of the simplest normal form are developing to high-dimension. The computation of the simplest normal form is very complicated. But, nonlinear dynamical systems can be simplified by the simplest normal form method and the nonlinear dynamical behavior of these systems near critical equilibrium can be obtained more easily. Aiming at the simplest normal form and the dynamical behavior of the electromechanical coupled nonlinear dynamical system, the research contents and the innovative contributions of this dissertation are as follows
(1) The coefficients of the simplest normal forms of high-dimensional generalized Hopf and high-dimensional Hopf bifurcations systems are discussed using the adjoint operator method. A particular nonlinear scaling and an inner product are introduced and the central manifold equations are simplified. Theorems are established for the explicit expression of the simplest normal forms in terms of the coefficients of the conventional normal forms of Hopf and generalized Hopf bifurcations systems. Symbolic program is designed to perform the calculation of the coefficients of the simplest normal forms using Mathematica. The original ordinary differential equation is required in the input and the simplest normal form can be obtained as the output. Finally, the simplest normal form of 6-dimensional generalized Hopf and 5-dimensional Hopf bifurcation system are discussed by executing the program. The outputs show that the 5th-order and 9th-order terms remain in 6-dimensional generalized Hopf and the 3rd-order and 5th-order terms remain in 5-dimensional Hopf bifurcation system.
(2) The coefficients of the simplest normal forms of both high-dimensional Neimark-Sacker and generalized Neimark-Sacker bifurcation systems are discussed. On the basis of the simplest normal form theory, using appropriate nonlinear transformations, direct computation and the second complete induction, the central manifold equations are further reduced to the simplest normal forms which only contain two nonlinear terms. Theorems are established for the explicit expression of the simplest normal forms in terms of the coefficients of the conventional normal forms of Neimark-Sacker and generalized Neimark-Sacker bifurcations systems.
(3) Applying a reversible linear transformation and a near-identity transformation, the simplest normal forms for high-dimensional nonlinear dynamical system is studied without calculating its traditional normal form. Using a reversible linear transformation, the matrix of the linear part for the nonlinear dynamical system is topologically equivalent to the block diagonal matrix that adapts to the demand of the practical research: companion matrixes distribute on the diagonal line and the remaining elements are zero. In order to obtain the simplest normal form, we use lower order nonlinear terms in the normal form for the simplifications of higher order terms. In the simplest normal form, the nonlinear coefficient matrix contains non-zero elements only in the row corresponding to the last row of each companion matrix and zero elements in the remaining rows. The general program with the Mathematica language is provided, which can compute the simplest normal form of an arbitrary nonlinear dynamical system easily. For nonlinear dynamical systems of 2-dimensional, 3-dimensional, 4-dimensional, 6-dimensional and 7-dimension, the simplest normal forms up to order 4 are discussed by executing the program.
(4) Applying a reversible linear transformation and a near-identity nonlinear transformation with small parameters, the calculating method according to the simplest normal forms of nonlinear dynamical systems with perturbation parameters is obtained. The general program with the Mathematica language is provided, which can compute the simplest normal form of the mentioned above nonlinear dynamical system easily.
(5) Applying a reversible linear transformation and a near-identity nonlinear transformation with parameters, the simplest normal form of an electromechanical coupled nonlinear dynamical system is obtained. Furthermore, the universal unfolding and the relationship between its parameters and the parameters of the original nonlinear dynamical system are obtained. The codimension-two bifurcation is analyzed and the effects of various parameters to the dynamical behavior of the system mentioned above are revealed, which lay a theoretical foundation for the parameter design, stable operation and fault diagnosis of a real system. The numerical simulating results of the electromechanical coupled nonlinear dynamical system are obtained, which verify the corresponding theoretical analysis result.
(6) Based on the Silnikov criterion, the chaotic characters of mechanically and electrically coupled nonlinear dynamical systems are discussed. Using Cardano formula and series solution of differential equation, eigenvalue problem and the existence of homoclinic orbit are studied respectively. Furthermore, a rigorous proof for the existence of Silnikov-sense Smale horseshoes chaos is presented and some conditions which lead to the chaos are obtained. The space trajectory, the Lyapunov exponent and the Lyapunov dimension are investigated via numerical simulation, which show chaotic attractor existed in the non-linear dynamical systems.
(1)利用共轭算子法,研究了高维Hopf分岔和退化Hopf分岔最简规范形的系数。引进特殊的非线性变换和内积,进一步简化了中心流形上的方程。获得了几个关于Hopf分岔和退化Hopf分岔的最简规范形系数与传统规范形系数之间具体关系的定理。借助Mathematica语言,编制了计算高维Hopf分岔和退化Hopf分岔最简规范形系数的程序。通过该程序,只需输入原动力系统方程,可得到系统具体的最简规范形。最后利用编制的程序分别计算了一个6维退化Hopf分岔系统的最简规范形和一个5维Hopf分岔系统的最简规范形。结果表明6维退化Hopf分岔系统的非线性项只包含5阶项和9阶项,5维Hopf分岔系统的非线性项只包含3阶项和5阶项。
(2)研究了高维Neimark-Sacker分岔和退化Neimark-Sacker分岔的最简规范形的系数。根据最简规范形理论,通过引进特殊的非线性变换、直接计算法和第二数学归纳法,对中心流形上的方程进一步化简,计算出Neimark-Sacker分岔和退化Neimark-Sacker分岔最简规范形的非线性项中分别只包含两项。获得了几个关于Neimark-Sacker分岔和退化Neimark-Sacker分岔的最简规范形系数与传统规范形系数之间关系的定理。
(3)运用可逆线性变换和近恒同变换,研究了不经计算传统规范形,直接计算高维非线性动力系统的最简规范形。引进可逆线性变换,将非线性动力系统的线性矩阵拓扑等价于符合实际研究需求的分块对角线矩阵:相伴矩阵分布在对角线上,其余元素均为0。利用低阶项来化简高阶项,得到了高维非线性动力系统的最简规范形。在该最简规范形中,对应于每一个相伴矩阵的非线性系数矩阵,只有最后一行含有非0元素,其余各行元素均为0。借助Mathematica语言,编制了计算高维非线性动力系统最简规范形的通用程序。运行该程序,分别计算了2维、3维、4维、6维和7维非线性动力系统直到4阶的最简规范形。
(4)运用含有参数的可逆线性变换和含有参数的近恒同变换,提出了不计算传统规范形,直接计算含参非线性动力系统最简规范形的一种计算方法。借助Mathematica语言,编制了计算含参非线性动力系统的最简规范形的通用程序。
(5)利用含有参数的可逆线性变换和含有参数的近恒同非线性变换,得到一类机电耦合非线性系统的最简规范形。进一步得到了该系统的普适开折以及开折参数与原系统参数之间的关系。讨论了该系统的余维2分岔,揭示了各参数对机电耦合系统动力学行为的影响,对系统的参数设计、稳定运行和故障诊断提供了理论依据。给出了该机电耦合系统的数值仿真结果,验证了理论分析结果。
(6)利用Silnikov定理,讨论了具有自动频率跟踪功能电磁振动机械系统的混沌特性。借助卡尔达诺公式和微分方程组级数解分别讨论了该系统的特征根问题和同宿轨道的存在性,进而比较严密地证明了该系统Silnikov型Smale混沌的存在性,并给出发生Silnikov型Smale混沌所需条件。利用数值模拟得到该类机电耦合系统的相轨迹图、Lyaponov指数谱和Lyaponov维数,进一步验证了该非线性系统存在奇怪吸引子。
Normal form theory is one of the useful tools in the study of nonlinear dynamics and stresses a profound influence on complex dynamic theory such as bifurcation and chaos dynamics. The simplest normal form is the normal form that can be used to deeply simplify the original differential equations based on conventional normal form and nonlinear transformation theory. The study and the application of the simplest normal form are developing to high-dimension. The computation of the simplest normal form is very complicated. But, nonlinear dynamical systems can be simplified by the simplest normal form method and the nonlinear dynamical behavior of these systems near critical equilibrium can be obtained more easily. Aiming at the simplest normal form and the dynamical behavior of the electromechanical coupled nonlinear dynamical system, the research contents and the innovative contributions of this dissertation are as follows
(1) The coefficients of the simplest normal forms of high-dimensional generalized Hopf and high-dimensional Hopf bifurcations systems are discussed using the adjoint operator method. A particular nonlinear scaling and an inner product are introduced and the central manifold equations are simplified. Theorems are established for the explicit expression of the simplest normal forms in terms of the coefficients of the conventional normal forms of Hopf and generalized Hopf bifurcations systems. Symbolic program is designed to perform the calculation of the coefficients of the simplest normal forms using Mathematica. The original ordinary differential equation is required in the input and the simplest normal form can be obtained as the output. Finally, the simplest normal form of 6-dimensional generalized Hopf and 5-dimensional Hopf bifurcation system are discussed by executing the program. The outputs show that the 5th-order and 9th-order terms remain in 6-dimensional generalized Hopf and the 3rd-order and 5th-order terms remain in 5-dimensional Hopf bifurcation system.
(2) The coefficients of the simplest normal forms of both high-dimensional Neimark-Sacker and generalized Neimark-Sacker bifurcation systems are discussed. On the basis of the simplest normal form theory, using appropriate nonlinear transformations, direct computation and the second complete induction, the central manifold equations are further reduced to the simplest normal forms which only contain two nonlinear terms. Theorems are established for the explicit expression of the simplest normal forms in terms of the coefficients of the conventional normal forms of Neimark-Sacker and generalized Neimark-Sacker bifurcations systems.
(3) Applying a reversible linear transformation and a near-identity transformation, the simplest normal forms for high-dimensional nonlinear dynamical system is studied without calculating its traditional normal form. Using a reversible linear transformation, the matrix of the linear part for the nonlinear dynamical system is topologically equivalent to the block diagonal matrix that adapts to the demand of the practical research: companion matrixes distribute on the diagonal line and the remaining elements are zero. In order to obtain the simplest normal form, we use lower order nonlinear terms in the normal form for the simplifications of higher order terms. In the simplest normal form, the nonlinear coefficient matrix contains non-zero elements only in the row corresponding to the last row of each companion matrix and zero elements in the remaining rows. The general program with the Mathematica language is provided, which can compute the simplest normal form of an arbitrary nonlinear dynamical system easily. For nonlinear dynamical systems of 2-dimensional, 3-dimensional, 4-dimensional, 6-dimensional and 7-dimension, the simplest normal forms up to order 4 are discussed by executing the program.
(4) Applying a reversible linear transformation and a near-identity nonlinear transformation with small parameters, the calculating method according to the simplest normal forms of nonlinear dynamical systems with perturbation parameters is obtained. The general program with the Mathematica language is provided, which can compute the simplest normal form of the mentioned above nonlinear dynamical system easily.
(5) Applying a reversible linear transformation and a near-identity nonlinear transformation with parameters, the simplest normal form of an electromechanical coupled nonlinear dynamical system is obtained. Furthermore, the universal unfolding and the relationship between its parameters and the parameters of the original nonlinear dynamical system are obtained. The codimension-two bifurcation is analyzed and the effects of various parameters to the dynamical behavior of the system mentioned above are revealed, which lay a theoretical foundation for the parameter design, stable operation and fault diagnosis of a real system. The numerical simulating results of the electromechanical coupled nonlinear dynamical system are obtained, which verify the corresponding theoretical analysis result.
(6) Based on the Silnikov criterion, the chaotic characters of mechanically and electrically coupled nonlinear dynamical systems are discussed. Using Cardano formula and series solution of differential equation, eigenvalue problem and the existence of homoclinic orbit are studied respectively. Furthermore, a rigorous proof for the existence of Silnikov-sense Smale horseshoes chaos is presented and some conditions which lead to the chaos are obtained. The space trajectory, the Lyapunov exponent and the Lyapunov dimension are investigated via numerical simulation, which show chaotic attractor existed in the non-linear dynamical systems.
引文
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