# 结构稳定性

此词条暂由Henry翻译

In mathematics, **structural stability** is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact *C*^{1}-small perturbations).

In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact C^{1}-small perturbations).

在数学中， 结构稳定性Structural stability是动力系统的一个基本性质，这意味着轨道的定性行为不受小扰动的影响(确切地说是C1-小扰动)。

Examples of such qualitative properties are numbers of fixed points and periodic orbits (but not their periods). Unlike Lyapunov stability, which considers perturbations of initial conditions for a fixed system, structural stability deals with perturbations of the system itself. Variants of this notion apply to systems of ordinary differential equations, vector fields on smooth manifolds and flows generated by them, and diffeomorphisms.

Examples of such qualitative properties are numbers of fixed points and periodic orbits (but not their periods). Unlike Lyapunov stability, which considers perturbations of initial conditions for a fixed system, structural stability deals with perturbations of the system itself. Variants of this notion apply to systems of ordinary differential equations, vector fields on smooth manifolds and flows generated by them, and diffeomorphisms.

这种定性性质的例子有许多不动点和周期轨道(但不包括它们的周期)。与考虑固定系统初始条件扰动的李雅普诺夫稳定性不同，结构稳定性考虑的是系统本身的扰动。这一概念的变体适用于常微分方程组、光滑流形上的向量场及其产生的流，以及微分同胚。

Structurally stable systems were introduced by Aleksandr Andronov and Lev Pontryagin in 1937 under the name "systèmes grossiers", or **rough systems**. They announced a characterization of rough systems in the plane, the Andronov–Pontryagin criterion. In this case, structurally stable systems are *typical*, they form an open dense set in the space of all systems endowed with appropriate topology. In higher dimensions, this is no longer true, indicating that typical dynamics can be very complex (cf strange attractor). An important class of structurally stable systems in arbitrary dimensions is given by Anosov diffeomorphisms and flows.

Structurally stable systems were introduced by Aleksandr Andronov and Lev Pontryagin in 1937 under the name "systèmes grossiers", or rough systems. They announced a characterization of rough systems in the plane, the Andronov–Pontryagin criterion. In this case, structurally stable systems are typical, they form an open dense set in the space of all systems endowed with appropriate topology. In higher dimensions, this is no longer true, indicating that typical dynamics can be very complex (cf strange attractor). An important class of structurally stable systems in arbitrary dimensions is given by Anosov diffeomorphisms and flows.

结构稳定的系统是由Aleksandr Andronov和Lev Pontryagin于1937年提出的，其名称是“系统-总体系统”（Systemèmes grossiers），即粗糙系统。他们宣布了飞机上粗糙系统的一个特征，Andronov-Pontryagin准则。在这种情况下，结构稳定系统是典型的，它们在所有具有适当拓扑结构的系统空间中形成一个开放的稠密集。在更高维中，这不再正确，这表明典型的动力学可能非常复杂（cf奇怪吸引子）。利用Anosov微分同胚和流可以得出一类重要的任意维结构稳定系统。

## Definition定义

Let *G* be an open domain in **R**^{n} with compact closure and smooth (*n*−1)-dimensional boundary. Consider the space *X*^{1}(*G*) consisting of restrictions to *G* of *C*^{1} vector fields on **R**^{n} that are transversal to the boundary of *G* and are inward oriented. This space is endowed with the *C*^{1} metric in the usual fashion. A vector field *F* ∈ *X*^{1}(*G*) is **weakly structurally stable** if for any sufficiently small perturbation *F*_{1}, the corresponding flows are topologically equivalent on *G*: there exists a homeomorphism *h*: *G* → *G* which transforms the oriented trajectories of *F* into the oriented trajectories of *F*_{1}. If, moreover, for any *ε* > 0 the homeomorphism *h* may be chosen to be *C*^{0} *ε*-close to the identity map when *F*_{1} belongs to a suitable neighborhood of *F* depending on *ε*, then *F* is called (strongly) **structurally stable**. These definitions extend in a straightforward way to the case of *n*-dimensional compact smooth manifolds with boundary. Andronov and Pontryagin originally considered the strong property. Analogous definitions can be given for diffeomorphisms in place of vector fields and flows: in this setting, the homeomorphism *h* must be a topological conjugacy.

Let G be an open domain in R^{n} with compact closure and smooth (n−1)-dimensional boundary. Consider the space X^{1}(G) consisting of restrictions to G of C^{1} vector fields on R^{n} that are transversal to the boundary of G and are inward oriented. This space is endowed with the C^{1} metric in the usual fashion. A vector field F ∈ X^{1}(G) is weakly structurally stable if for any sufficiently small perturbation F_{1}, the corresponding flows are topologically equivalent on G: there exists a homeomorphism h: G → G which transforms the oriented trajectories of F into the oriented trajectories of F_{1}. If, moreover, for any ε > 0 the homeomorphism h may be chosen to be C^{0} ε-close to the identity map when F_{1} belongs to a suitable neighborhood of F depending on ε, then F is called (strongly) structurally stable. These definitions extend in a straightforward way to the case of n-dimensional compact smooth manifolds with boundary. Andronov and Pontryagin originally considered the strong property. Analogous definitions can be given for diffeomorphisms in place of vector fields and flows: in this setting, the homeomorphism h must be a topological conjugacy.

设G是Rn中具有紧闭包和光滑（n−1）维边界的开域。考虑空间X1（G），该空间由位于Rn上的C1向量场的G的约束组成，这些约束与G的边界是横向的，并且是向内的。这个空间通常被赋予C1度量。向量场F∈X1（G）是弱结构稳定的，如果对任何足够小的扰动F1，相应的流在G上拓扑等价：存在一个同胚h:G→G，它将F的定向轨迹转化为F1的定向轨迹。此外，如果对于任何ε>0，当F1属于依赖于ε的F的合适邻域时，同胚h可以选择为C0ε-接近恒等式映射，则F被称为（强）结构稳定的。这些定义以一种直接的方式推广到n维紧光滑流形的情形。安德罗诺夫和庞特里亚金最初认为这是很强的性质。对于不同的同胚可以给出类似的定义来代替向量场和流：在这种情况下，同胚h必须是拓扑共轭的。

It is important to note that topological equivalence is realized with a loss of smoothness: the map *h* cannot, in general, be a diffeomorphism. Moreover, although topological equivalence respects the oriented trajectories, unlike topological conjugacy, it is not time-compatible. Thus the relevant notion of topological equivalence is a considerable weakening of the naïve *C*^{1} conjugacy of vector fields. Without these restrictions, no continuous time system with fixed points or periodic orbits could have been structurally stable. Weakly structurally stable systems form an open set in *X*^{1}(*G*), but it is unknown whether the same property holds in the strong case.

It is important to note that topological equivalence is realized with a loss of smoothness: the map h cannot, in general, be a diffeomorphism. Moreover, although topological equivalence respects the oriented trajectories, unlike topological conjugacy, it is not time-compatible. Thus the relevant notion of topological equivalence is a considerable weakening of the naïve C^{1} conjugacy of vector fields. Without these restrictions, no continuous time system with fixed points or periodic orbits could have been structurally stable. Weakly structurally stable systems form an open set in X^{1}(G), but it is unknown whether the same property holds in the strong case.

值得注意的是，拓扑等价是在光滑性损失的情况下实现的：映射h一般不能是微分同胚。此外，虽然拓扑等价尊重有向轨迹，但与拓扑共轭不同，它不是时间相容的。因此，拓扑等价的相关概念大大削弱了向量场的C1共轭性。没有这些限制，任何具有不动点或周期轨道的连续时间系统都不可能结构稳定。弱结构稳定系统在X1（G）中形成了一个开集，但在强情形下是否成立是未知的。

## Examples 例子

Necessary and sufficient conditions for the structural stability of *C*^{1} vector fields on the unit disk *D* that are transversal to the boundary and on the two-sphere *S*^{2} have been determined in the foundational paper of Andronov and Pontryagin. According to the Andronov–Pontryagin criterion, such fields are structurally stable if and only if they have only finitely many singular points (equilibrium states) and periodic trajectories (limit cycles), which are all non-degenerate (hyperbolic), and do not have saddle-to-saddle connections. Furthermore, the non-wandering set of the system is precisely the union of singular points and periodic orbits. In particular, structurally stable vector fields in two dimensions cannot have homoclinic trajectories, which enormously complicate the dynamics, as discovered by Henri Poincaré.

Necessary and sufficient conditions for the structural stability of C^{1} vector fields on the unit disk D that are transversal to the boundary and on the two-sphere S^{2} have been determined in the foundational paper of Andronov and Pontryagin. According to the Andronov–Pontryagin criterion, such fields are structurally stable if and only if they have only finitely many singular points (equilibrium states) and periodic trajectories (limit cycles), which are all non-degenerate (hyperbolic), and do not have saddle-to-saddle connections. Furthermore, the non-wandering set of the system is precisely the union of singular points and periodic orbits. In particular, structurally stable vector fields in two dimensions cannot have homoclinic trajectories, which enormously complicate the dynamics, as discovered by Henri Poincaré.

在Andronov和Pontryagin的基础论文中，给出了单位圆盘D上横截于边界的C1向量场结构稳定的充分必要条件。根据Andronov–Pontryagin准则，这类场在结构上是稳定的当且仅当它们只有有限多个奇点（平衡态）和周期轨迹（极限环），它们都是非退化（双曲型），并且没有鞍-鞍连接。此外，系统的非游荡集正是奇点与周期轨道的并。特别是二维结构稳定的向量场不能有同宿轨迹，这使得动力学变得非常复杂，正如Henri Poincaré发现的那样。

Structural stability of non-singular smooth vector fields on the torus can be investigated using the theory developed by Poincaré and Arnaud Denjoy. Using the Poincaré recurrence map, the question is reduced to determining structural stability of diffeomorphisms of the circle. As a consequence of the Denjoy theorem, an orientation preserving *C*^{2} diffeomorphism *ƒ* of the circle is structurally stable if and only if its rotation number is rational, *ρ*(*ƒ*) = *p*/*q*, and the periodic trajectories, which all have period *q*, are non-degenerate: the Jacobian of *ƒ*^{q} at the periodic points is different from 1, see circle map.

Structural stability of non-singular smooth vector fields on the torus can be investigated using the theory developed by Poincaré and Arnaud Denjoy. Using the Poincaré recurrence map, the question is reduced to determining structural stability of diffeomorphisms of the circle. As a consequence of the Denjoy theorem, an orientation preserving C^{2} diffeomorphism ƒ of the circle is structurally stable if and only if its rotation number is rational, ρ(ƒ) = p/q, and the periodic trajectories, which all have period q, are non-degenerate: the Jacobian of ƒ^{q} at the periodic points is different from 1, see circle map.

环面上非奇异光滑向量场的结构稳定性可以用Poincaré和Arnaud Denjoy提出的理论进行研究。利用Poincaré递推映射，将问题归结为确定圆的微分同胚的结构稳定性。作为Denjoy定理的一个结果，一个保持方向的C2微分同胚ƒ在结构上是稳定的当且仅当它的旋转数是有理的，ρ（ƒ）=p/q，并且周期轨迹都是非退化的：周期点上ƒq的雅可比与1不同，参见圆图。

Dmitri Anosov discovered that hyperbolic automorphisms of the torus, such as the Arnold's cat map, are structurally stable. He then generalized this statement to a wider class of systems, which have since been called Anosov diffeomorphisms and Anosov flows. One celebrated example of Anosov flow is given by the geodesic flow on a surface of constant negative curvature, cf Hadamard billiards.

Dmitri Anosov discovered that hyperbolic automorphisms of the torus, such as the Arnold's cat map, are structurally stable. He then generalized this statement to a wider class of systems, which have since been called Anosov diffeomorphisms and Anosov flows. One celebrated example of Anosov flow is given by the geodesic flow on a surface of constant negative curvature, cf Hadamard billiards.

Dmitri Anosov发现环面的双曲自同构，如阿诺德猫图，在结构上是稳定的。然后，他将这一说法推广到更广泛的一类系统中，这类系统后来被称为Anosov微分同胚和Anosov流。阿诺索夫流的一个著名例子是常负曲率曲面上的测地线流，如哈达玛台球。

## History and significance历史与重要性

Structural stability of the system provides a justification for applying the qualitative theory of dynamical systems to analysis of concrete physical systems. The idea of such qualitative analysis goes back to the work of Henri Poincaré on the three-body problem in celestial mechanics. Around the same time, Aleksandr Lyapunov rigorously investigated stability of small perturbations of an individual system. In practice, the evolution law of the system (i.e. the differential equations) is never known exactly, due to the presence of various small interactions. It is, therefore, crucial to know that basic features of the dynamics are the same for any small perturbation of the "model" system, whose evolution is governed by a certain known physical law. Qualitative analysis was further developed by George Birkhoff in the 1920s, but was first formalized with introduction of the concept of rough system by Andronov and Pontryagin in 1937. This was immediately applied to analysis of physical systems with oscillations by Andronov, Witt, and Khaikin. The term "structural stability" is due to Solomon Lefschetz, who oversaw translation of their monograph into English. Ideas of structural stability were taken up by Stephen Smale and his school in the 1960s in the context of hyperbolic dynamics. Earlier, Marston Morse and Hassler Whitney initiated and René Thom developed a parallel theory of stability for differentiable maps, which forms a key part of singularity theory. Thom envisaged applications of this theory to biological systems. Both Smale and Thom worked in direct contact with Maurício Peixoto, who developed Peixoto's theorem in the late 1950s.

Structural stability of the system provides a justification for applying the qualitative theory of dynamical systems to analysis of concrete physical systems. The idea of such qualitative analysis goes back to the work of Henri Poincaré on the three-body problem in celestial mechanics. Around the same time, Aleksandr Lyapunov rigorously investigated stability of small perturbations of an individual system. In practice, the evolution law of the system (i.e. the differential equations) is never known exactly, due to the presence of various small interactions. It is, therefore, crucial to know that basic features of the dynamics are the same for any small perturbation of the "model" system, whose evolution is governed by a certain known physical law. Qualitative analysis was further developed by George Birkhoff in the 1920s, but was first formalized with introduction of the concept of rough system by Andronov and Pontryagin in 1937. This was immediately applied to analysis of physical systems with oscillations by Andronov, Witt, and Khaikin. The term "structural stability" is due to Solomon Lefschetz, who oversaw translation of their monograph into English. Ideas of structural stability were taken up by Stephen Smale and his school in the 1960s in the context of hyperbolic dynamics. Earlier, Marston Morse and Hassler Whitney initiated and René Thom developed a parallel theory of stability for differentiable maps, which forms a key part of singularity theory. Thom envisaged applications of this theory to biological systems. Both Smale and Thom worked in direct contact with Maurício Peixoto, who developed Peixoto's theorem in the late 1950s.

系统的结构稳定性为将动力系统的定性理论应用于具体物理系统的分析提供了依据。这种定性分析的思想可以追溯到Henri Poincaré关于天体力学中三体问题的工作。同时，Aleksandr-Lyapunov严格地研究了单个系统小扰动的稳定性。在实践中，由于各种小的相互作用的存在，系统的演化规律（即微分方程）永远无法准确地知道。因此，重要的是要知道，对于“模型”系统的任何小扰动，动力学的基本特征都是相同的，其演化受某个已知物理定律的支配。20世纪20年代，George Birkhoff进一步发展了定性分析，但随着Andronov和Pontryagin在1937年引入粗糙系统的概念，定性分析首先被形式化。 Andronov, Witt,和Khaikin立即将这一理论应用于具有振荡的物理系统的分析。“结构稳定性”一词源于Solomon Lefschetz，他负责将他们的专著翻译成英语。结构稳定性的思想是由斯蒂芬·斯梅尔和他的学派在20世纪60年代在双曲动力学的背景下提出的。早些时候，Marston Morse和 Hassler Whitney提出并发展了可微映射稳定性的并行理论，这是奇点理论的一个重要组成部分。汤姆设想了这一理论在生物系统中的应用。Smale和Thom都与Maurício Peixoto直接接触，后者在20世纪50年代末发展了Peixoto定理。

When Smale started to develop the theory of hyperbolic dynamical systems, he hoped that structurally stable systems would be "typical". This would have been consistent with the situation in low dimensions: dimension two for flows and dimension one for diffeomorphisms. However, he soon found examples of vector fields on higher-dimensional manifolds that cannot be made structurally stable by an arbitrarily small perturbation (such examples have been later constructed on manifolds of dimension three). This means that in higher dimensions, structurally stable systems are not dense. In addition, a structurally stable system may have transversal homoclinic trajectories of hyperbolic saddle closed orbits and infinitely many periodic orbits, even though the phase space is compact. The closest higher-dimensional analogue of structurally stable systems considered by Andronov and Pontryagin is given by the Morse–Smale systems.

When Smale started to develop the theory of hyperbolic dynamical systems, he hoped that structurally stable systems would be "typical". This would have been consistent with the situation in low dimensions: dimension two for flows and dimension one for diffeomorphisms. However, he soon found examples of vector fields on higher-dimensional manifolds that cannot be made structurally stable by an arbitrarily small perturbation (such examples have been later constructed on manifolds of dimension three). This means that in higher dimensions, structurally stable systems are not dense. In addition, a structurally stable system may have transversal homoclinic trajectories of hyperbolic saddle closed orbits and infinitely many periodic orbits, even though the phase space is compact. The closest higher-dimensional analogue of structurally stable systems considered by Andronov and Pontryagin is given by the Morse–Smale systems.

当斯梅尔开始发展双曲动力系统理论时，他希望结构稳定的系统是“典型的”。这与低维的情况是一致的: 流的维数为二，微分同胚的维数为一。然而，他很快发现了高维流形上的向量场的例子，这些例子不能通过任意小的扰动使结构稳定(这样的例子后来被构造在三维流形上)。这意味着在更高的维度中，结构稳定的系统并不密集。此外，即使相空间紧凑，结构稳定的系统也可能具有横向同宿轨迹，即双曲型鞍闭轨和无穷多个周期轨。结构稳定系统的最接近的高维模拟由Andronov和Pontryagin的Morse–Smale系统给出。

## See also参见

平衡

自我稳定 超稳定

稳定理论

## References参考

- Andronov, Aleksandr A.; Lev S. Pontryagin (1988) [1937]. V. I. Arnold (ed.). "Грубые системы" [Coarse systems].
*Geometric Methods in the Theory of Differential Equations*. Grundlehren der Mathematischen Wissenschaften, 250. Springer-Verlag, New York. ISBN 0-387-96649-8.

Category:Differential equations

类别: 微分方程

Category:Dynamical systems

类别: 动力系统

Category:Stability theory

范畴: 稳定性理论

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