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第5行： − + − In dynamical systems, an orbit is called Lyapunov stable if the forward orbit of any point is in a small enough neighborhood or it stays in a small (but perhaps, larger) neighborhood. Various criteria have been developed to prove stability or instability of an orbit. Under favorable circumstances, the question may be reduced to a well-studied problem involving eigenvalues of matrices. A more general method involves Lyapunov functions. In practice, any one of a number of different stability criteria are applied.+ − 在动力系统中，如果一个<font color="#ff8000">轨道 Orbit</font>上任意点的前向轨道处于一个足够小的邻域内，或者这个轨道处于一个较小的邻域(但可能是较大的邻域)内，则称其为<font color="#ff8000">李雅普诺夫稳定 Lyapunov stable</font>。有各种标准来证明轨道的稳定性或不稳定性。在有利的条件下，这个问题可以简化为一个涉及矩阵<font color="#ff8000">特征值 Eigenvalue</font>的问题，而这已经有很多研究。更一般的方法涉及<font color="#ff8000">李雅普诺夫函数 Lyapunov function</font>。在实践中，很多<font color="#ff8000">稳定性判据 Stability criterion</font>中的任何一个都是适用的。 + − [[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]] − + − 根据它们的特征，庞加莱映射是稳定的还是不稳定的。稳定性通常增加到图的左边。] + − ==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 points, or fixed points, and by periodic orbits. 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.+ − 微分方程和动力系统定性理论的许多部分关心解的渐近性质和轨迹——系统经过很长时间后会发生什么。最简单的行为表现为<font color="#ff8000">平衡点 equilibrium points</font>或不动点，以及<font color="#ff8000">周期轨道 periodic orbit</font>。如果一个特定的轨道被很好地理解，那么很自然地会问下一个问题：初始条件的一个微小变化是否会导致类似的行为。稳定性理论解决了以下问题: 附近的轨道是否会无限靠近给定的轨道？它会收敛到给定的轨道吗？在前一种情况下，轨道被称为是<font color="#ff8000">稳定 stable</font>的；在后一种情况下，轨道是<font color="#ff8000">渐近稳定 asymptotically stable </font>的，给定的轨道称为吸引子。 − − − An equilibrium solution <math>f_e</math> to an autonomous system of first order ordinary differential equations is called: − − An equilibrium solution <math>f_e</math> to an autonomous system of first order ordinary differential equations is called: 第40行：
第31行： − + − + − Stability means that the trajectories do not change too much under small perturbations. The opposite situation, where a nearby orbit is getting repelled from the given orbit, is also of interest. In general, perturbing the initial state in some directions results in the trajectory asymptotically approaching the given one and in other directions to the trajectory getting away from it. There may also be directions for which the behavior of the perturbed orbit is more complicated (neither converging nor escaping completely), and then stability theory does not give sufficient information about the dynamics. 第51行：
第41行： − − One of the key ideas in stability theory is that the qualitative behavior of an orbit under perturbations can be analyzed using the [[linearization]] of the system near the orbit. In particular, at each equilibrium of a smooth dynamical system with an ''n''-dimensional [[phase space]], there is a certain [[square matrix|''n''×''n'' matrix]] ''A'' whose [[eigenvalue]]s characterize the behavior of the nearby points ([[Hartman–Grobman theorem]]). More precisely, if all eigenvalues are negative [[real number]]s or [[complex number]]s with negative real parts then the point is a stable attracting fixed point, and the nearby points converge to it at an [[exponential decay|exponential]] rate, cf [[Lyapunov stability]] and [[exponential stability]]. If none of the eigenvalues are purely imaginary (or zero) then the attracting and repelling directions are related to the eigenspaces of the matrix ''A'' with eigenvalues whose real part is negative and, respectively, positive. Analogous statements are known for perturbations of more complicated orbits. 第60行：
第48行： − 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. 第67行：
第54行： − − There are useful tests of stability for the case of a linear system. Stability of a nonlinear system can often be inferred from the stability of its [[linearization]]. 第75行：
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第114行： − − There is an analogous criterion for a continuously differentiable map {{Math|''f'': '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>}} with a fixed point {{Math|''a''}}, expressed in terms of its [[Jacobian matrix]] at {{Math|''a''}}, {{Math|''J''<sub>''a''</sub>(''f'')}}. If all [[eigenvalues]] of {{Math|''J''}} are real or complex numbers with absolute value strictly less than 1 then {{Math|''a''}} is a stable fixed point; if at least one of them has absolute value strictly greater than 1 then {{Math|''a''}} is unstable. Just as for {{Math|''n''}}=1, the case of the largest absolute value being 1 needs to be investigated further — the Jacobian matrix test is inconclusive. The same criterion holds more generally for [[diffeomorphism]]s of a [[smooth manifold]]. 第138行：
第121行： − The stability of fixed points of a system of constant coefficient [[linear differential equation]]s of first order can be analyzed using the [[eigenvalue]]s of the corresponding matrix.+ 第248行：
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无编辑摘要
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<font color="#ff8000">动力系统 Dynamical system</font>在初始条件的微小扰动下轨迹的稳定性问题。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<font color="#ff8000">极大值原理 Maximum principle</font>的存在，初始数据的微小扰动会导致温度随之产生微小的变化。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 <math>Gromov–Hausdorff</math> 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<font color="#ff8000">动力系统 Dynamical system</font>在初始条件的微小扰动下轨迹的稳定性问题。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<font color="#ff8000">极大值原理 Maximum principle</font>的存在，初始数据的微小扰动会导致温度随之产生微小的变化。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。
在动力系统中，如果一条<font color="#ff8000">轨道 Orbit</font>上任意点的前向轨道都处于一个足够小的邻域内，或者这条轨道整体停留在一个邻域（一般是较小的邻域，也有可能是较大的邻域）内，则称该轨道的状态为<font color="#ff8000">李雅普诺夫稳定 Lyapunov stable</font>。有各种标准来证明轨道的稳定性或不稳定性。在适当的条件下，这个问题可以简化为一个涉及矩阵<font color="#ff8000">特征值 Eigenvalue</font>的问题，关于这类矩阵特征值的问题已被大量研究并且该领域已经比较成熟。一种更一般的方法涉及<font color="#ff8000">李雅普诺夫函数 Lyapunov function</font>。在实践中，很多<font color="#ff8000">稳定性判据 Stability criterion</font>都可以使用，我们可以使用其中的任何一个作为判断系统稳定性的准则。
[[File:Stability_Diagram.png|thumb|550px|稳定性图将<font color="#ff8000">庞加莱映射 Poincaré map</font> 根据其特征划分为稳定或不稳定区间。如图可见，图中下半部分区域中系统的稳定性增加。<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]]
Poincaré maps as stable or unstable according to their features. Stability generally increases to the left of the diagram.]]
常微分方程在经历了长期的求精确解的努力后逐渐停滞，庞加莱在分析的基础上引入几何方法，开创了常微分方程定性理论，同时在分析中引入几何方法，搭建起分析与几何之间的沟通桥梁，带来了微分方程研究的新突破。李雅普诺夫则在庞加莱定性分析的基础上 ,转而进入了新的稳定性研究。如今 ,李雅普诺夫稳定性理论被普遍认为是微分方程定性理论的基本成就之一。不仅有精确的定义 ,更有严格的分析证明 ,将微分方程及稳定性理论的研究推向了新的高度。庞加莱被公认是19世纪后四分之一和二十世纪初的领袖数学家，是对于数学和它的应用具有全面知识的最后一个人，他在数学方面的杰出工作对20世纪和当今的数学造成极其深远的影响。<font color="#ff8000">庞加莱映射 Poincaré map</font>是由相空间中轨道运动定义的一种映射，是当轨道反复穿越同一截面时，反映后继点对先行点依赖关系的映射。一个连续非线性动力系统的求解是非常困难的，庞加莱给出了相图分析法。在相图中虽然不能定量地知道物理量随时间的变化，但可以定性地得到轨线的形态类型及其拓扑结构，从而了解动力系统运动的全局图像。为了更清楚了解高维相空间运动的形态，在连续运动的轨线上用一个截面（称庞加莱截面）将其横截，轨线在截面上穿过的情况就可以简捷地判断运动的形态。对于庞加莱映射是稳定的还是不稳定的判断则取决于其特征，如图所示，在相空间区间中向下的方向上稳定性增加。
==Overview in dynamical systems 动力系统概述 ==
微分方程和动力系统定性理论的许多部分关心系统或者方程解的渐近性质及其轨迹，这也意味着系统经过很长时间后会发生什么。系统最简单的行为表现为<font color="#ff8000">平衡点 Equilibrium points</font>或不动点，以及<font color="#ff8000">周期轨道 Periodic orbit</font>。如果我们已经很好地理解了一个特定的轨道，那么很自然地就会问下一个问题：初始条件的一个微小变化对于系统来说是否仍会保持类似的行为。稳定性理论解决了以下问题：附近的轨道是否会无限靠近给定的轨道？已知的轨道会收敛到给定的轨道吗？在前一种情况下，轨道被称为是<font color="#ff8000">稳定 Stable</font>的；在后一种情况下，轨道是<font color="#ff8000">渐近稳定 Asymptotically stable </font>的，并且收敛到给定的轨道称为<font color="#ff8000">吸引子 Attractor</font>。
对于一个一阶常微分方程自治系统的平衡解<math>f_e</math>：
对于一个一阶常微分方程自治系统的平衡解<math>f_e</math>：
*asymptotically stable if it is stable and, in addition, there exists <math>\delta_0 > 0</math> such that whenever <math>\| f(t_0) - f_e \| < \delta_0 </math> then <math>f(t) \rightarrow f_e </math>as <math>t \rightarrow \infty </math>.
*asymptotically stable if it is stable and, in addition, there exists <math>\delta_0 > 0</math> such that whenever <math>\| f(t_0) - f_e \| < \delta_0 </math> then <math>f(t) \rightarrow f_e </math>as <math>t \rightarrow \infty </math>.
*如果对于任意（小的）<math>\epsilon > 0</math>，存在<math>\delta > 0 </math>，使得只要初始条件与平衡点的距离在<math> \delta </math>范围内，例如<math> \| f(t_0) - f_e \| < \delta</math>，就有，对任何<math> t \ge t_0 </math>满足解 <math>f(t) </math> 与平衡点的距离在 <math> \epsilon </math> 范围内，例如<math>\| f(t) - f_e \| < \epsilon</math>，那么该平衡点称为稳定的。
* 如果对于任意（小的）<math>\epsilon > 0</math>，存在<math>\delta > 0 </math>，使得只要初始条件与平衡点的距离在<math> \delta </math>范围内，例如<math> \| f(t_0) - f_e \| < \delta</math>，就有，对任何<math> t \ge t_0 </math>满足解 <math>f(t) </math> 与平衡点的距离在 <math> \epsilon </math> 范围内，例如<math>\| f(t) - f_e \| < \epsilon</math>，那么该平衡点称为稳定的。
* 如果该平衡点是稳定的，并且存在 <math>\delta_0 > 0</math>，使得对于任何<math>\| f(t_0) - f_e \| < \delta_0 </math>，当<math>t \rightarrow \infty </math>时都有<math>f(t) \rightarrow f_e </math>，那么该平衡点时渐近稳定的。
*如果该平衡点是稳定的，并且存在 <math>\delta_0 > 0</math>，使得对于任何<math>\| f(t_0) - f_e \| < \delta_0 </math>，当<math>t \rightarrow \infty </math>时都有<math>f(t) \rightarrow f_e </math>，那么该平衡点时渐近稳定的。
Stability means that the trajectories do not change too much under small perturbations. The opposite situation, where a nearby orbit is getting repelled from the given orbit, is also of interest. In general, perturbing the initial state in some directions results in the trajectory asymptotically approaching the given one and in other directions to the trajectory getting away from it. There may also be directions for which the behavior of the perturbed orbit is more complicated (neither converging nor escaping completely), and then stability theory does not give sufficient information about the dynamics.
Stability means that the trajectories do not change too much under small perturbations. The opposite situation, where a nearby orbit is getting repelled from the given orbit, is also of interest. In general, perturbing the initial state in some directions results in the trajectory asymptotically approaching the given one and in other directions to the trajectory getting away from it. There may also be directions for which the behavior of the perturbed orbit is more complicated (neither converging nor escaping completely), and then stability theory does not give sufficient information about the dynamics.
One of the key ideas in stability theory is that the qualitative behavior of an orbit under perturbations can be analyzed using the linearization of the system near the orbit. In particular, at each equilibrium of a smooth dynamical system with an n-dimensional phase space, there is a certain n×n matrix A whose eigenvalues characterize the behavior of the nearby points (Hartman–Grobman theorem). More precisely, if all eigenvalues are negative real numbers or complex numbers with negative real parts then the point is a stable attracting fixed point, and the nearby points converge to it at an exponential rate, cf Lyapunov stability and exponential stability. If none of the eigenvalues are purely imaginary (or zero) then the attracting and repelling directions are related to the eigenspaces of the matrix A with eigenvalues whose real part is negative and, respectively, positive. Analogous statements are known for perturbations of more complicated orbits.
One of the key ideas in stability theory is that the qualitative behavior of an orbit under perturbations can be analyzed using the linearization of the system near the orbit. In particular, at each equilibrium of a smooth dynamical system with an n-dimensional phase space, there is a certain n×n matrix A whose eigenvalues characterize the behavior of the nearby points (Hartman–Grobman theorem). More precisely, if all eigenvalues are negative real numbers or complex numbers with negative real parts then the point is a stable attracting fixed point, and the nearby points converge to it at an exponential rate, cf Lyapunov stability and exponential stability. If none of the eigenvalues are purely imaginary (or zero) then the attracting and repelling directions are related to the eigenspaces of the matrix A with eigenvalues whose real part is negative and, respectively, positive. Analogous statements are known for perturbations of more complicated orbits.
==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 oscillations 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 oscillations 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.
There are useful tests of stability for the case of a linear system. Stability of a nonlinear system can often be inferred from the stability of its linearization.
There are useful tests of stability for the case of a linear system. Stability of a nonlinear system can often be inferred from the stability of its linearization.
===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''}}:
There is an analogous criterion for a continuously differentiable map with a fixed point , expressed in terms of its Jacobian matrix at , . If all eigenvalues of are real or complex numbers with absolute value strictly less than 1 then is a stable fixed point; if at least one of them has absolute value strictly greater than 1 then is unstable. Just as for =1, the case of the largest absolute value being 1 needs to be investigated further — the Jacobian matrix test is inconclusive. The same criterion holds more generally for diffeomorphisms of a smooth manifold.
There is an analogous criterion for a continuously differentiable map with a fixed point , expressed in terms of its Jacobian matrix at , . If all eigenvalues of are real or complex numbers with absolute value strictly less than 1 then is a stable fixed point; if at least one of them has absolute value strictly greater than 1 then is unstable. Just as for =1, the case of the largest absolute value being 1 needs to be investigated further — the Jacobian matrix test is inconclusive. The same criterion holds more generally for diffeomorphisms of a smooth manifold.
===Linear autonomous systems 线性自治系统===
===Linear autonomous systems 线性自治系统===
The stability of fixed points of a system of constant coefficient linear differential equations of first order can be analyzed using the eigenvalues of the corresponding matrix.
The stability of fixed points of a system of constant coefficient linear differential equations of first order can be analyzed using the eigenvalues of the corresponding matrix.
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]]
*[[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}}
== External links==
==External links==
*[http://demonstrations.wolfram.com/StableEquilibria/ Stable Equilibria] by Michael Schreiber, [[The Wolfram Demonstrations Project]].
*[http://demonstrations.wolfram.com/StableEquilibria/ Stable Equilibria] by Michael Schreiber, [[The Wolfram Demonstrations Project]].