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第36行： − 微分方程和动力系统定性理论的许多部分涉及解的渐近性质和轨迹ーー系统经过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: 初始条件的一个小的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期地靠近给定的轨道？它会收敛到给定的轨道吗？在前一种情况下，轨道是稳定的; 在后一种情况下，轨道是渐近稳定的，给定的轨道是吸引的。+ 第44行：
第44行： − 一个一阶常微分方程自治系统的平衡解被称为:+ 第50行：
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第58行： − 稳定性意味着在微小的扰动下轨迹不会发生太大的变化。相反的情况，即附近的轨道与给定的轨道相反，这也是有趣的。一般来说，在某些方向扰动初始状态使得轨迹渐近地接近给定轨迹，而在其他方向扰动则使得轨迹远离给定轨迹。也可能存在扰动轨道行为比较复杂(既不会收敛也不会完全逃逸)的方向，因此稳定性理论不能给出关于动力学的充分信息。+ 第65行：
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→Overview in dynamical systems
== Overview in dynamical systems ==
== 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'''.
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。
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:
对于一个一阶常微分方程自治系统的平衡解<math>f_e</math>：
*stable if for every (small) <math>\epsilon > 0</math>, there exists a <math>\delta > 0 </math> such that every solution <math>f(t) </math> having initial conditions within distance <math> \delta </math> i.e. <math> \| f(t_0) - f_e \| < \delta</math> of the equilibrium remains within distance <math> \epsilon </math> i.e. <math>\| f(t) - f_e \| < \epsilon</math> for all <math> t \ge t_0 </math>.
*stable if for every (small) <math>\epsilon > 0</math>, there exists a <math>\delta > 0 </math> such that every solution <math>f(t) </math> having initial conditions within distance <math> \delta </math> i.e. <math> \| f(t_0) - f_e \| < \delta</math> of the equilibrium remains within distance <math> \epsilon </math> i.e. <math>\| f(t) - f_e \| < \epsilon</math> for all <math> t \ge t_0 </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.
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.
稳定性意味着在微小的扰动下轨迹不会发生太大的变化。相反的情况，即附近的轨道与给定的轨道互相排斥，这也是有趣的。一般来说，在某些方向扰动初始状态使得轨迹渐近地接近给定轨迹，而在其他方向扰动则使得轨迹远离给定轨迹。也可能存在某些方向扰动轨道行为比较复杂(既不会收敛也不会完全逃逸)，从而稳定性理论不能给出关于这样的动力学的充分信息。
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方法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，存在一个 n n 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负实数的复数，那么这个点就是一个稳定的吸引不动点，并且附近的点以指数速率收敛到它，cf 李雅普诺夫稳定性和指数稳定。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，其特征值的实部分分别为负和正。对于更复杂的轨道的扰动，人们已经知道类似的陈述。
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方法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，存在一个 n n 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负实数的复数，那么这个点就是一个稳定的吸引不动点，并且附近的点以指数速率收敛到它，cf 李雅普诺夫稳定性和指数稳定。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，其特征值的实部分分别为负和正。对于更复杂的轨道的扰动，人们已经知道类似的陈述。
== Stability of fixed points ==
== Stability of fixed points ==