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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。
== Stability of fixed points ==
== 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.
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.
最简单的一种轨道是一个不动点，或者叫做平衡点。如果一个力学系统处于稳定的平衡状态，那么一个小的推力会导致局部运动，例如，象钟摆那样的小的振动。在有阻尼的系统中，稳定的平衡态是渐近稳定的。另一方面，对于一个不稳定的平衡，例如一个球停留在山顶上，某些小的推力就会导致一个大幅度的运动，这个运动可能会也可能不会收敛到原始状态。
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.
关于线性系统的稳定性有许多有用的检验方法。非线性系统的稳定性常常可以从其线性化的系统的稳定性中推断出来。
=== Maps ===
=== 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''}}:
设为该点向量场的雅可比矩阵。如果所有的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。
设为该点向量场的雅可比矩阵。如果所有的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 ==
== Lyapunov function for general dynamical systems ==