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→Non-linear autonomous systems
为了判定线性系统原点的稳定性，劳斯-赫尔维茨稳定性判据推动了这一结果在实践中的应用。矩阵的特征值是其特征多项式的根。如果所有根的实部都是严格负的，那么一个具有实系数的单变量多项式称为赫尔维茨多项式。劳斯-赫尔维茨定理通过避免计算根的算法暗示了赫尔维茨多项式的特征。
为了判定线性系统原点的稳定性，劳斯-赫尔维茨稳定性判据推动了这一结果在实践中的应用。矩阵的特征值是其特征多项式的根。如果所有根的实部都是严格负的，那么一个具有实系数的单变量多项式称为赫尔维茨多项式。劳斯-赫尔维茨定理通过避免计算根的算法暗示了赫尔维茨多项式的特征。
=== Non-linear autonomous systems ===
=== Non-linear autonomous systems 非线性自治系统===
Asymptotic stability of fixed points of a non-linear system can often be established using the [[Hartman–Grobman theorem]].
Asymptotic stability of fixed points of a non-linear system can often be established using the [[Hartman–Grobman theorem]].
Suppose that is a -vector field in which vanishes at a point , . Then the corresponding autonomous system
Suppose that is a -vector field in which vanishes at a point , . Then the corresponding autonomous system
假设{{Math|''v''}}是{{Math|'''R'''<sup>''n''</sup>}}上的一个{{Math|''C''<sup>1</sup>}}-向量场，并且下降至某一点{{Math|''p''}}有{{Math|1=''v''(''p'') = 0}}。那么相应的自治系统
has a constant solution
has a constant solution
有一个常数解
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''<sub>''p''</sub>(''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。
== Lyapunov function for general dynamical systems ==
== Lyapunov function for general dynamical systems ==