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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，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 ==