可积Hamilton系统和具有不变代数曲面的三维系统的动力学
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摘要
可积Hamilton系统是非线性科学研究的一个重要分支，她广泛地出现在力学、声学，光学，生命科学以及社会科学等各个领域，特别是天体力学、等离子物理、航天科技以及生物工程中很多模型都以可积Hamilton系统或者其扰动系统的形式出现的。另外，很多描述混沌现象、非线性振动和生物数学等模型都是三维的系统，如Lorenz系统，Rabinovich系统、Chen系统、Rikitake系统和Lotka-Volterra系统等等。这些系统尽管形式上非常简单，但其动力学极其复杂。至今都没有完全弄清楚他们的动力学性质。为了简化问题，往往考虑其具有首次积分或不变代数曲面的情况。
在研究微分方程和动力系统的过程中，人们十分关心其是否存在不变量。不变量的存在性问题从Poincare和Hilbert时代起就一直是人们普遍关心的问题。如果一个微分系统存在一个首次积分，他的动力学的研究就可以降低一维。如果可积，则有可能了解系统的整体动力学性质。如果一个系统具有不变代数曲面，则通过不变代数曲面上系统的动力学的研究，可以有助于整个空间中系统动力学的研究。因此可积Hamilton系统的存在性的判定，以及他们的拓扑、几何、代数性质的研究，以及具有不变代数曲面的微分系统的动力学的研究有着很大的实际应用价值。但正如Poincare所指出：不变代数曲面和首次积分的寻找是十分困难的问题。
本文研究一些特殊Riemann流形上Hamilton系统的正交分离可积以及他们的拓扑熵，和一些著名三维系统当具有不变代数曲面时该系统轨道的全局拓扑结构。具体阐述如下。
本文第一部分介绍可积Hamilton系统的应用背景及意义，系统全面的介绍了可积Hamilton系统及其拓扑熵的研究的发展历程、国内外的研究现状和具有不变代数曲面的三维系统的研究意义、历史和发展。
第二部分在两维环面T~2上由三维空间自然诱导的度量下，研究具有两个自由度的自然Hamilton系统的正交分离可积(这里的可积是在Liouville意义下)，得到这类系统所有可能的分类。并证明这类可积流如果是解析的，则在任何紧的正则能量面上的拓扑熵为零。进一步地，我们通过例子显示T~2上的可积Hamilton统可以有复杂的动力学现象。例如，它们有多族不变环面，每一族都由同宿环状柱面和异宿环状柱面所包围。据我们所知，这是第一个具体的例子来表现很多族环面在一个复杂的的方式下同时出现。
第三部分里我们首次在Riemann流形T~2×[0，1]上给出了C~∞。光滑的正交分离的带有势能的自然Hamilton系统的特征。利用这些T~2×[0，1]上的Hamilton系统，我们得到在Riemann流形M_A=T~2×[0，1]／～上C~∞光滑可积的Hamilton系统。进而，我们证明对于任何一个总能量不小于e_H的可积Hamilton系统，存在一个零Lebesgue测度集合Ω(?)D：={e∈R；e≥e_H}，使得对任意的e∈D＼Ω，约束在能量面{H=e}上的Hamilton流有正的拓扑熵。据我们所知，这是首个在Riemann流形上具有正拓扑熵的C~∞光滑Liouville可积的自然Hamilton系统的例子，而Bolsinov和Taimanov的例子是对于Riemann流形M_A上的测地流得到的。
第四部分中，我们在Riemann流形T~n×[0，1]给出所有C~∞光滑正交分离的自然Hamilton系统，即既有动能又有势能，的特征。然后，在这些系统中找出在Anosov映射诱导的n维环面双曲自同构，即Riemann流形M_A：=T~n×[0，1]／～上C~∞光滑可积的系统。特别地，我们讨论了Anosov映射诱导的环面双曲自同构的谱有复特征值的情况。进而，我们证明了限制在正则能量面上的Hamilton系统有正的拓扑熵。这里给出了一个任意有限维空间上的带有正拓扑熵的C~∞Liouvill可积的自然Hamilton系统的例子。
第五部分将研究具有不变代数曲面的Rabinovich系统(?)=hy-v_1x+yz，(?)=hx-v_2y-xz，(?)=-v_3z+xy和Chen系统(?)=a(y-x)，(?)=(c-a)x-xz+cy，(?)=xy-bz，的轨线的全局拓扑结构。我们完全解决了这两类系统的动力学研究。
Integrable Hamiltonian system is an important branch of nonlinear science. They are applied widely in mechanics, acoustics, optics, biology, life sciences, society sciences, etc. Specially, in the fields of biology, astrodynamics, spaceflight engineering technology, lots of models are constructed in integrable Hamiltonian systems or their perturbations. In addition, large numbers of chaotic phenomena, nonlinear oscillations, biology mathematics models are described in 3-dimensional systems such as Lorenz systems, Rabinovich systems, Chen systems, Rikitake systems and Lotka-Volterra systems, etc. Although these systems look simple, but in fact their dynamics are extremely complex. Up to now, their dynamics cannot be understood clearly. To simplify the questions, ones usually study the system having invariants, for instance first integrals or invariant algebraic surfaces.
For studying dynamical systems, ones are very interested in whether the systems have invariants. The study on the existence of invariants of a system can be go back to Poincare's and Hilbert's era. If a system has a first integral, then its study can be reduced in one dimension. If a system is integrable, it is possible to characterize its dynamics globally. If a system has an invariant algebraic surface, it is helpful to study the dynamics in whole space by studying the dynamics on the invariant algebraic surface. It is valuable to study the Liouvillian integrability of Hamiltonian systems and their topology, geometry and algebra and to study 3-dimensional systems having an invariant algebraic surface. But as Poincare had realized, it is difficult to find invariant algebraic surfaces and first integrals for a given system.
In this paper, we study some integrable Hamiltonian systems and their topological entropy on special Riemannian manifolds, and the global topological structure of orbits of some famous 3-dimension systems having an invariant algebraic surface.
In chapter 1, we introduce Hamiltonian systems associated with their integrability and topological entropy, and 3-dimensional systems having an invariant algebraic surface.
In chapter 2, we characterize the Liouvillian integrable orthogonal separable Hamiltonian systems on T~2 for a given metric, and prove that the Hamiltonian flow restricted to any compact level hypersurface has zero topological entropy. Furthermore, by examples we show that the integrable Hamiltonian systems on T~2 can have complicated dynamical phenomena. For instance they can have several families of invariant tori, each family is bounded by the homoclinic-loop-like cylinders and heteroclinic-loop-like cylinders. As we know, it is the first concrete example to present families of invariant tori at the same time appearing in such a complicated way.
In chapter 3, we will first characterize C~∞smoothly orthogonally separable Hamiltonian Systems with a potential energy on T~2×[0,1]. Then using these integrable Hamiltonian systems on T~2×[0,1] we obtain a class of integrable Hamiltonian systems on the Riemannian manifold M_A.Moreover, we prove that for the integrable natural Hamiltonian H with total energy no less than e_H there exists a subsetΩ(?)D:= {e∈R; e≥e_H} of Lebesgue measure zero such that the Hamiltonian flow restricted to each energy surface {H = e} with e∈D\Ωhas a positive topological entropy. As a result, we obtain the first example, as our knowledge, of C~∞Liouvillian integrable natural Hamiltonian flows on a Riemannian manifold which has a positive topological entropy.
In chapter 4, we give the characterization of the integrable natural Hamiltonian systems, which orthogonal separable on T~n×[0,1], with the configuration space a three dimensional quotient manifold induced by the Anosov map. Moreover, we prove that the Hamiltonian flow restricted to suitable regular energy surfaces has a positive topological entropy.
In chapter 5, we characterize the global topological structure of orbits of Rabinovich system (?)= hy -v_1x + yz,(?) = hx - v_2y - xz,(?)= -v_3z + xy and the Chen system (?) = a(y - x), (?) = (c - a)x - xz + cy, (?) = xy - bz having an invariant algebraic surface. We complete the classification of dynamics of these two systems.
在研究微分方程和动力系统的过程中，人们十分关心其是否存在不变量。不变量的存在性问题从Poincare和Hilbert时代起就一直是人们普遍关心的问题。如果一个微分系统存在一个首次积分，他的动力学的研究就可以降低一维。如果可积，则有可能了解系统的整体动力学性质。如果一个系统具有不变代数曲面，则通过不变代数曲面上系统的动力学的研究，可以有助于整个空间中系统动力学的研究。因此可积Hamilton系统的存在性的判定，以及他们的拓扑、几何、代数性质的研究，以及具有不变代数曲面的微分系统的动力学的研究有着很大的实际应用价值。但正如Poincare所指出：不变代数曲面和首次积分的寻找是十分困难的问题。
本文研究一些特殊Riemann流形上Hamilton系统的正交分离可积以及他们的拓扑熵，和一些著名三维系统当具有不变代数曲面时该系统轨道的全局拓扑结构。具体阐述如下。
本文第一部分介绍可积Hamilton系统的应用背景及意义，系统全面的介绍了可积Hamilton系统及其拓扑熵的研究的发展历程、国内外的研究现状和具有不变代数曲面的三维系统的研究意义、历史和发展。
第二部分在两维环面T~2上由三维空间自然诱导的度量下，研究具有两个自由度的自然Hamilton系统的正交分离可积(这里的可积是在Liouville意义下)，得到这类系统所有可能的分类。并证明这类可积流如果是解析的，则在任何紧的正则能量面上的拓扑熵为零。进一步地，我们通过例子显示T~2上的可积Hamilton统可以有复杂的动力学现象。例如，它们有多族不变环面，每一族都由同宿环状柱面和异宿环状柱面所包围。据我们所知，这是第一个具体的例子来表现很多族环面在一个复杂的的方式下同时出现。
第三部分里我们首次在Riemann流形T~2×[0，1]上给出了C~∞。光滑的正交分离的带有势能的自然Hamilton系统的特征。利用这些T~2×[0，1]上的Hamilton系统，我们得到在Riemann流形M_A=T~2×[0，1]／～上C~∞光滑可积的Hamilton系统。进而，我们证明对于任何一个总能量不小于e_H的可积Hamilton系统，存在一个零Lebesgue测度集合Ω(?)D：={e∈R；e≥e_H}，使得对任意的e∈D＼Ω，约束在能量面{H=e}上的Hamilton流有正的拓扑熵。据我们所知，这是首个在Riemann流形上具有正拓扑熵的C~∞光滑Liouville可积的自然Hamilton系统的例子，而Bolsinov和Taimanov的例子是对于Riemann流形M_A上的测地流得到的。
第四部分中，我们在Riemann流形T~n×[0，1]给出所有C~∞光滑正交分离的自然Hamilton系统，即既有动能又有势能，的特征。然后，在这些系统中找出在Anosov映射诱导的n维环面双曲自同构，即Riemann流形M_A：=T~n×[0，1]／～上C~∞光滑可积的系统。特别地，我们讨论了Anosov映射诱导的环面双曲自同构的谱有复特征值的情况。进而，我们证明了限制在正则能量面上的Hamilton系统有正的拓扑熵。这里给出了一个任意有限维空间上的带有正拓扑熵的C~∞Liouvill可积的自然Hamilton系统的例子。
第五部分将研究具有不变代数曲面的Rabinovich系统(?)=hy-v_1x+yz，(?)=hx-v_2y-xz，(?)=-v_3z+xy和Chen系统(?)=a(y-x)，(?)=(c-a)x-xz+cy，(?)=xy-bz，的轨线的全局拓扑结构。我们完全解决了这两类系统的动力学研究。
Integrable Hamiltonian system is an important branch of nonlinear science. They are applied widely in mechanics, acoustics, optics, biology, life sciences, society sciences, etc. Specially, in the fields of biology, astrodynamics, spaceflight engineering technology, lots of models are constructed in integrable Hamiltonian systems or their perturbations. In addition, large numbers of chaotic phenomena, nonlinear oscillations, biology mathematics models are described in 3-dimensional systems such as Lorenz systems, Rabinovich systems, Chen systems, Rikitake systems and Lotka-Volterra systems, etc. Although these systems look simple, but in fact their dynamics are extremely complex. Up to now, their dynamics cannot be understood clearly. To simplify the questions, ones usually study the system having invariants, for instance first integrals or invariant algebraic surfaces.
For studying dynamical systems, ones are very interested in whether the systems have invariants. The study on the existence of invariants of a system can be go back to Poincare's and Hilbert's era. If a system has a first integral, then its study can be reduced in one dimension. If a system is integrable, it is possible to characterize its dynamics globally. If a system has an invariant algebraic surface, it is helpful to study the dynamics in whole space by studying the dynamics on the invariant algebraic surface. It is valuable to study the Liouvillian integrability of Hamiltonian systems and their topology, geometry and algebra and to study 3-dimensional systems having an invariant algebraic surface. But as Poincare had realized, it is difficult to find invariant algebraic surfaces and first integrals for a given system.
In this paper, we study some integrable Hamiltonian systems and their topological entropy on special Riemannian manifolds, and the global topological structure of orbits of some famous 3-dimension systems having an invariant algebraic surface.
In chapter 1, we introduce Hamiltonian systems associated with their integrability and topological entropy, and 3-dimensional systems having an invariant algebraic surface.
In chapter 2, we characterize the Liouvillian integrable orthogonal separable Hamiltonian systems on T~2 for a given metric, and prove that the Hamiltonian flow restricted to any compact level hypersurface has zero topological entropy. Furthermore, by examples we show that the integrable Hamiltonian systems on T~2 can have complicated dynamical phenomena. For instance they can have several families of invariant tori, each family is bounded by the homoclinic-loop-like cylinders and heteroclinic-loop-like cylinders. As we know, it is the first concrete example to present families of invariant tori at the same time appearing in such a complicated way.
In chapter 3, we will first characterize C~∞smoothly orthogonally separable Hamiltonian Systems with a potential energy on T~2×[0,1]. Then using these integrable Hamiltonian systems on T~2×[0,1] we obtain a class of integrable Hamiltonian systems on the Riemannian manifold M_A.Moreover, we prove that for the integrable natural Hamiltonian H with total energy no less than e_H there exists a subsetΩ(?)D:= {e∈R; e≥e_H} of Lebesgue measure zero such that the Hamiltonian flow restricted to each energy surface {H = e} with e∈D\Ωhas a positive topological entropy. As a result, we obtain the first example, as our knowledge, of C~∞Liouvillian integrable natural Hamiltonian flows on a Riemannian manifold which has a positive topological entropy.
In chapter 4, we give the characterization of the integrable natural Hamiltonian systems, which orthogonal separable on T~n×[0,1], with the configuration space a three dimensional quotient manifold induced by the Anosov map. Moreover, we prove that the Hamiltonian flow restricted to suitable regular energy surfaces has a positive topological entropy.
In chapter 5, we characterize the global topological structure of orbits of Rabinovich system (?)= hy -v_1x + yz,(?) = hx - v_2y - xz,(?)= -v_3z + xy and the Chen system (?) = a(y - x), (?) = (c - a)x - xz + cy, (?) = xy - bz having an invariant algebraic surface. We complete the classification of dynamics of these two systems.
引文
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