更改

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions.  The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions.  The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.

在数学上，<font color="#ff8000">稳定性理论 stability theory</font>研究<font color="#ff8000">微分方程 differential equation</font>解的稳定性和<font color="#ff8000">动力系统 dynamical system</font>在初始条件的小扰动下的轨迹的稳定性。例如，<font color="#ff8000">热传导方程 heat equation</font>是一个稳定的偏微分方程，因为初始数据的微小扰动会导致温度随之产生微小的变化，这是<font color="#ff8000">极大值原理 maximum principle</font>的结果。在偏微分方程中，人们可以使用 Lp 范数或 sup 范数来度量函数之间的距离，而在微分几何中，人们可以使用 Gromov–Hausdorff 距离来度量空间之间的距离。

在数学上，<font color="#ff8000">稳定性理论 Stability theory</font>被用于研究<font color="#ff8000">微分方程Differential equation</font>解的稳定性和动力系统<font color="#ff8000">Dynamical system</font>在初始条件的微小扰动下轨迹的稳定性。例如，热传导方程<font color="#ff8000"> Heat equation</font>是一个稳定的偏微分方程，因为初始数据的微小扰动会导致温度随之产生微小的变化，这是极大值原理<font color="#ff8000"> Maximum principle</font>的结果。在偏微分方程中，人们可以使用 <math>Lp</math> 范数或 <math>sup</math> 范数来度量函数之间的距离，而在微分几何中，人们可以使用 Gromov–Hausdorff 距离来度量空间之间的距离。

 

[[File:Stability_Diagram.png|thumb|550px|Stability diagram classifying [[Poincaré map#Poincaré maps and stability analysis|Poincaré maps]] as stable or unstable according to their features.  Stability generally increases to the left of the diagram.<ref>[http://www.egwald.ca/linearalgebra/lineardifferentialequationsstabilityanalysis.php Egwald Mathematics - Linear Algebra: Systems of Linear Differential Equations: Linear Stability Analysis] Accessed 10 October 2019.</ref>|链接=Special:FilePath/Stability_Diagram.png]]

[[File:Stability_Diagram.png|thumb|550px|Stability diagram classifying [[Poincaré map#Poincaré maps and stability analysis|Poincaré maps]] as stable or unstable according to their features.  Stability generally increases to the left of the diagram.<ref>[http://www.egwald.ca/linearalgebra/lineardifferentialequationsstabilityanalysis.php Egwald Mathematics - Linear Algebra: Systems of Linear Differential Equations: Linear Stability Analysis] Accessed 10 October 2019.</ref>]]

Poincaré maps as stable or unstable according to their features.  Stability generally increases to the left of the diagram.]]

Poincaré maps as stable or unstable according to their features.  Stability generally increases to the left of the diagram.]]

 

==Overview in dynamical systems 动力系统概述==

== Overview in dynamical systems 动力系统概述==

Many parts of the [[qualitative theory of differential equations]] and dynamical systems deal with asymptotic properties of solutions and the trajectories—what happens with the system after a long period of time. The simplest kind of behavior is exhibited by [[equilibrium point]]s, or fixed points, and by [[periodic orbit]]s. If a particular orbit is well understood, it is natural to ask next whether a small change in the initial condition will lead to similar behavior. Stability theory addresses the following questions: Will a nearby orbit indefinitely stay close to a given orbit? Will it converge to the given orbit? In the former case, the orbit is called '''stable'''; in the latter case, it is called '''asymptotically stable''' and the given orbit is said to be '''attracting'''.

Many parts of the [[qualitative theory of differential equations]] and dynamical systems deal with asymptotic properties of solutions and the trajectories—what happens with the system after a long period of time. The simplest kind of behavior is exhibited by [[equilibrium point]]s, or fixed points, and by [[periodic orbit]]s. If a particular orbit is well understood, it is natural to ask next whether a small change in the initial condition will lead to similar behavior. Stability theory addresses the following questions: Will a nearby orbit indefinitely stay close to a given orbit? Will it converge to the given orbit? In the former case, the orbit is called '''stable'''; in the latter case, it is called '''asymptotically stable''' and the given orbit is said to be '''attracting'''.

稳定性理论的关键思想之一是利用轨道附近系统的线性化，来分析轨道在扰动下的定性行为。特别地，在 n 维<font color="#ff8000">相空间 phase space</font>的光滑动力系统的每个平衡点上，都存在一个 n×n 的矩阵 A，其特征值刻画了邻近点的行为(<font color="#ff8000">Hartman-Grobman 定理 Hartman–Grobman theorem</font>)。更确切地说，如果所有的特征值都是负实数或实部为负的复数，那么这个平衡点就是一个稳定的吸引子，并且附近的点以指数速率收敛到它，参考<font color="#ff8000">李雅普诺夫稳定性 Lyapunov stability</font>和<font color="#ff8000">指数稳定性 exponential stability</font>。如果所有的特征值都不是纯虚数(或零) ，那么吸引方向和排斥方向都与矩阵 A 的特征空间有关，其特征值的实部分别为负和正。对于更复杂的轨道的扰动，也有类似的陈述。

稳定性理论的关键思想之一是利用轨道附近系统的线性化，来分析轨道在扰动下的定性行为。特别地，在 n 维<font color="#ff8000">相空间 phase space</font>的光滑动力系统的每个平衡点上，都存在一个 n×n 的矩阵 A，其特征值刻画了邻近点的行为(<font color="#ff8000">Hartman-Grobman 定理 Hartman–Grobman theorem</font>)。更确切地说，如果所有的特征值都是负实数或实部为负的复数，那么这个平衡点就是一个稳定的吸引子，并且附近的点以指数速率收敛到它，参考<font color="#ff8000">李雅普诺夫稳定性 Lyapunov stability</font>和<font color="#ff8000">指数稳定性 exponential stability</font>。如果所有的特征值都不是纯虚数(或零) ，那么吸引方向和排斥方向都与矩阵 A 的特征空间有关，其特征值的实部分别为负和正。对于更复杂的轨道的扰动，也有类似的陈述。

== Stability of fixed points 不动点稳定性==

==Stability of fixed points 不动点稳定性==

The simplest kind of an orbit is a fixed point, or an equilibrium. If a mechanical system is in a stable equilibrium state then a small push will result in a localized motion, for example, small [[oscillation]]s as in the case of a [[pendulum]]. In a system with [[damping]], a stable equilibrium state is moreover asymptotically stable. On the other hand, for an unstable equilibrium, such as a ball resting on a top of a hill, certain small pushes will result in a motion with a large amplitude that may or may not converge to the original state.

The simplest kind of an orbit is a fixed point, or an equilibrium. If a mechanical system is in a stable equilibrium state then a small push will result in a localized motion, for example, small [[oscillation]]s as in the case of a [[pendulum]]. In a system with [[damping]], a stable equilibrium state is moreover asymptotically stable. On the other hand, for an unstable equilibrium, such as a ball resting on a top of a hill, certain small pushes will result in a motion with a large amplitude that may or may not converge to the original state.

 

===Maps 映射===

=== Maps 映射===

Let {{Math|''f'': '''R''' → '''R'''}} be a [[continuously differentiable function]] with a fixed point {{Math|''a''}}, {{Math|1=''f''(''a'') = ''a''}}. Consider the dynamical system obtained by iterating the function {{Math|''f''}}:

Let {{Math|''f'': '''R''' → '''R'''}} be a [[continuously differentiable function]] with a fixed point {{Math|''a''}}, {{Math|1=''f''(''a'') = ''a''}}. Consider the dynamical system obtained by iterating the function {{Math|''f''}}:

 

:<math> x_{n+1}=f(x_n), \quad n=0,1,2,\ldots.</math>

: <math> x_{n+1}=f(x_n), \quad n=0,1,2,\ldots.</math>

  <math> x_{n+1}=f(x_n), \quad n=0,1,2,\ldots.</math>

  <math> x_{n+1}=f(x_n), \quad n=0,1,2,\ldots.</math>

 

:<math> f(x) \approx f(a)+f'(a)(x-a). </math>

: <math> f(x) \approx f(a)+f'(a)(x-a). </math>

  <math> f(x) \approx f(a)+f'(a)(x-a). </math>

  <math> f(x) \approx f(a)+f'(a)(x-a). </math>

 

:<math>x_{n+1}-x_{n} = f(x_n)-x_n \simeq f(a) + f'(a)(x_n-a)-x_n = a + f'(a)(x_n-a)-x_n = (f'(a)-1)(x_n-a) \to \frac{x_{n+1}-x_{n}}{x_n-a}=f'(a)-1</math>

: <math>x_{n+1}-x_{n} = f(x_n)-x_n \simeq f(a) + f'(a)(x_n-a)-x_n = a + f'(a)(x_n-a)-x_n = (f'(a)-1)(x_n-a) \to \frac{x_{n+1}-x_{n}}{x_n-a}=f'(a)-1</math>

  <math>x_{n+1}-x_{n} = f(x_n)-x_n \simeq f(a) + f'(a)(x_n-a)-x_n = a + f'(a)(x_n-a)-x_n = (f'(a)-1)(x_n-a) \to \frac{x_{n+1}-x_{n}}{x_n-a}=f'(a)-1</math>

  <math>x_{n+1}-x_{n} = f(x_n)-x_n \simeq f(a) + f'(a)(x_n-a)-x_n = a + f'(a)(x_n-a)-x_n = (f'(a)-1)(x_n-a) \to \frac{x_{n+1}-x_{n}}{x_n-a}=f'(a)-1</math>

一个<font color="#ff8000">自治系统 autonomous system</font>

一个<font color="#ff8000">自治系统 autonomous system</font>

 

:<math>x(t)=0.</math>

: <math>x(t)=0.</math>

  <math>x(t)=0.</math>

  <math>x(t)=0.</math>

为了判定线性系统原点的稳定性，劳斯-赫尔维茨稳定性判据推动了这一结果在实践中的应用。矩阵的特征值是其特征多项式的根。如果所有根的实部都是严格负的，那么一个具有实系数的单变量多项式称为赫尔维茨多项式。劳斯-赫尔维茨定理通过避免计算根的算法暗示了赫尔维茨多项式的特征。

为了判定线性系统原点的稳定性，劳斯-赫尔维茨稳定性判据推动了这一结果在实践中的应用。矩阵的特征值是其特征多项式的根。如果所有根的实部都是严格负的，那么一个具有实系数的单变量多项式称为赫尔维茨多项式。劳斯-赫尔维茨定理通过避免计算根的算法暗示了赫尔维茨多项式的特征。

=== Non-linear autonomous systems 非线性自治系统===

===Non-linear autonomous systems 非线性自治系统===

Asymptotic stability of fixed points of a non-linear system can often be established using the [[Hartman–Grobman theorem]].

Asymptotic stability of fixed points of a non-linear system can often be established using the [[Hartman–Grobman theorem]].

假设{{Math|''v''}}是{{Math|'''R'''^''n''}}上的一个{{Math|''C''¹}}-向量场，并且下降至某一点{{Math|''p''}}有{{Math|1=''v''(''p'') = 0}}。那么相应的自治系统

假设{{Math|''v''}}是{{Math|'''R'''^''n''}}上的一个{{Math|''C''¹}}-向量场，并且下降至某一点{{Math|''p''}}有{{Math|1=''v''(''p'') = 0}}。那么相应的自治系统

 

:<math> x(t)=p.</math>

: <math> x(t)=p.</math>

  <math> x(t)=p.</math>

  <math> x(t)=p.</math>

设{{Math|''J''_''p''(''v'')}}为向量场 {{Math|''v''}}在点{{Math|''p''}}的{{Math|''n''×''n''}}<font color="#ff8000">雅可比矩阵 Jacobian matrix</font>。如果{{Math|''J''}}的所有特征值都是严格负实部，则解是渐近稳定的。这个条件可以用劳斯-赫尔维茨准则来检验。

设{{Math|''J''_''p''(''v'')}}为向量场 {{Math|''v''}}在点{{Math|''p''}}的{{Math|''n''×''n''}}<font color="#ff8000">雅可比矩阵 Jacobian matrix</font>。如果{{Math|''J''}}的所有特征值都是严格负实部，则解是渐近稳定的。这个条件可以用劳斯-赫尔维茨准则来检验。

== Lyapunov function for general dynamical systems 一般动力系统的李雅普诺夫函数==

==Lyapunov function for general dynamical systems 一般动力系统的李雅普诺夫函数==

{{main|Lyapunov function}}

{{main|Lyapunov function}}

建立动力系统的李雅普诺夫稳定性或渐近稳定的一般方法是利用李亚普诺夫函数。

建立动力系统的李雅普诺夫稳定性或渐近稳定的一般方法是利用李亚普诺夫函数。

== See also 参见==

==See also 参见==

* [[Asymptotic stability 渐近稳定性]]

* [[Asymptotic stability 渐近稳定性]]

* [[Hyperstability 超稳定性]]

*[[Hyperstability 超稳定性]]

* [[Linear stability 线性稳定性]]

* [[Linear stability 线性稳定性]]

* [[Orbital stability 轨道稳定性]]

*[[Orbital stability 轨道稳定性]]

* [[Stability criterion 稳定性判据]]

*[[Stability criterion 稳定性判据]]

* [[Stability radius 稳定半径]]

* [[Stability radius 稳定半径]]

* [[Structural stability 结构稳定性]]

*[[Structural stability 结构稳定性]]

* [[von Neumann stability analysis 冯诺依曼稳定性分析]]

*[[von Neumann stability analysis 冯诺依曼稳定性分析]]

== References ==

==References==

{{Reflist}}

{{Reflist}}

* {{Scholarpedia|title=Stability|urlname=Stability|curator=Philip Holmes and Eric T. Shea-Brown}}

*{{Scholarpedia|title=Stability|urlname=Stability|curator=Philip Holmes and Eric T. Shea-Brown}}

 

 

 

[[Category:Stability theory| ]]

[[分类:Stability theory| ]]

[[Category:Limit sets]]

[[Category:Limit sets]]