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第5行： + − In [[mathematics]], '''stability theory''' addresses the stability of solutions of [[differential equation]]s and of trajectories of [[dynamical system]]s 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 space|Lp norms]] or the sup norm, while in differential geometry one may measure the distance between spaces using the [[Gromov–Hausdorff convergence|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>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。 − − − − In dynamical systems, an [[orbit (dynamics)|orbit]] is called ''[[Lyapunov stability|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 [[eigenvalue]]s of [[matrix (mathematics)|matrices]]. A more general method involves [[Lyapunov function]]s. In practice, any one of a number of different [[stability criterion|stability criteria]] are applied. − + 第50行：
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{{short description|Part of mathematics that addresses the stability of solutions}}
{{short description|Part of mathematics that addresses the stability of solutions}}
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。
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.
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。
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。
*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.
===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''}}:
对于具有一个不动点{{Math|''a''}}的连续可微映射{{Math|''f'': '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>}}，存在一个类似的判据，由{{Math|''a''}}的雅可比矩阵{{Math|''J''<sub>''a''</sub>(''f'')}}表示。如果{{Math|''J''}}的所有特征值都是绝对值严格小于1的实数或复数，则是稳定不动点; 如果其中至少有一个特征值的绝对值严格大于1，则它是不稳定的。就像对于{{Math|''n''}}=1，最大本征值绝对值为1的情况也需要进一步研究ーー雅可比矩阵检验是不确定的。同样的准则对光滑流形的微分同胚也有更广泛的适用性。
对于具有一个不动点{{Math|''a''}}的连续可微映射{{Math|''f'': '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>}}，存在一个类似的判据，由{{Math|''a''}}的雅可比矩阵{{Math|''J''<sub>''a''</sub>(''f'')}}表示。如果{{Math|''J''}}的所有特征值都是绝对值严格小于1的实数或复数，则是稳定不动点; 如果其中至少有一个特征值的绝对值严格大于1，则它是不稳定的。就像对于{{Math|''n''}}=1，最大本征值绝对值为1的情况也需要进一步研究ーー雅可比矩阵检验是不确定的。同样的准则对光滑流形的微分同胚也有更广泛的适用性。
=== 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 equation]]s of first order can be analyzed using the [[eigenvalue]]s of the corresponding matrix.
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.
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}}
==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]].