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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，稳定性理论研究微分方程解的稳定性和动力系统在初始条件的小扰动下的轨迹的稳定性。例如，热传导方程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>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。

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，如果一个轨道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，则称其为李雅普诺夫稳定。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特征值的问题，而这已经有很多研究。更一般的方法涉及李雅普诺夫函数。在实践中，很多稳定性标准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。

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.

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平衡点或不动点，以及周期轨道。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，轨道被称为是稳定的；在后一种情况下，轨道是渐近稳定的，给定的轨道称为吸引子。

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。

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.

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 维相空间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(Hartman-Grobman 定理)。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，参考李雅普诺夫稳定性和指数稳定性。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==

An autonomous system

An autonomous system

Let  be the  Jacobian matrix of the vector field  at the point . If all eigenvalues of  have strictly negative real part then the solution is asymptotically stable. This condition can be tested using the Routh–Hurwitz criterion.

Let  be the  Jacobian matrix of the vector field  at the point . If all eigenvalues of  have strictly negative real part then the solution is asymptotically stable. This condition can be tested using the Routh–Hurwitz criterion.

设{{Math|''J''_''p''(''v'')}}为向量场 {{Math|''v''}}在点{{Math|''p''}}的{{Math|''n''×''n''}}雅可比矩阵。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==