# 稳定性理论

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 stability theory研究微分方程 differential equation解的稳定性和动力系统 dynamical system在初始条件的小扰动下的轨迹的稳定性。例如，热传导方程 heat equation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 maximum principle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 orbit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 Lyapunov stable。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 eigenvalue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 Lyapunov function。在实践中，很多稳定性判据 stability criterion中的任何一个都是适用的。

Poincaré maps as stable or unstable according to their features. Stability generally increases to the left of the diagram.]]

根据它们的特征，庞加莱映射是稳定的还是不稳定的。稳定性通常增加到图的左边。]

## 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 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 equilibrium points或不动点，以及周期轨道 periodic orbit。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 stable的；在后一种情况下，轨道是渐近稳定 asymptotically stable 的，给定的轨道称为吸引子。

An equilibrium solution [math]\displaystyle{ f_e }[/math] to an autonomous system of first order ordinary differential equations is called:

An equilibrium solution [math]\displaystyle{ 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]\displaystyle{ f_e }[/math]：

- stable if for every (small) [math]\displaystyle{ \epsilon \gt 0 }[/math], there exists a [math]\displaystyle{ \delta \gt 0 }[/math] such that every solution [math]\displaystyle{ f(t) }[/math] having initial conditions within distance [math]\displaystyle{ \delta }[/math] i.e. [math]\displaystyle{ \| f(t_0) - f_e \| \lt \delta }[/math] of the equilibrium remains within distance [math]\displaystyle{ \epsilon }[/math] i.e. [math]\displaystyle{ \| f(t) - f_e \| \lt \epsilon }[/math] for all [math]\displaystyle{ t \ge t_0 }[/math].

- asymptotically stable if it is stable and, in addition, there exists [math]\displaystyle{ \delta_0 \gt 0 }[/math] such that whenever [math]\displaystyle{ \| f(t_0) - f_e \| \lt \delta_0 }[/math] then [math]\displaystyle{ f(t) \rightarrow f_e }[/math]as [math]\displaystyle{ t \rightarrow \infty }[/math].

- 如果对于任意（小的）[math]\displaystyle{ \epsilon \gt 0 }[/math]，存在[math]\displaystyle{ \delta \gt 0 }[/math]，使得只要初始条件与平衡点的距离在[math]\displaystyle{ \delta }[/math]范围内，例如[math]\displaystyle{ \| f(t_0) - f_e \| \lt \delta }[/math]，就有，对任何[math]\displaystyle{ t \ge t_0 }[/math]满足解 [math]\displaystyle{ f(t) }[/math] 与平衡点的距离在 [math]\displaystyle{ \epsilon }[/math] 范围内，例如[math]\displaystyle{ \| f(t) - f_e \| \lt \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]\displaystyle{ \delta_0 \gt 0 }[/math]，使得对于任何[math]\displaystyle{ \| f(t_0) - f_e \| \lt \delta_0 }[/math]，当[math]\displaystyle{ t \rightarrow \infty }[/math]时都有[math]\displaystyle{ 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.

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。

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 phase space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 Hartman–Grobman theorem)。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 Lyapunov stability和指数稳定性 exponential stability。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

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 映射

Let *f*: **R** → **R** be a continuously differentiable function with a fixed point *a*, *f*(*a*) = *a*. Consider the dynamical system obtained by iterating the function *f*:

Let be a continuously differentiable function with a fixed point , . Consider the dynamical system obtained by iterating the function :

另*f*: **R** → **R**是一个连续可微函数且存在一个不动点*a*，*f*(*a*) = *a*。考虑一个通过迭代函数得到的动力系统:

- [math]\displaystyle{ x_{n+1}=f(x_n), \quad n=0,1,2,\ldots. }[/math]

```
[math]\displaystyle{ x_{n+1}=f(x_n), \quad n=0,1,2,\ldots. }[/math]
```

数学 x { n + 1} f (xn) ， n n 0,1,2， ldots. / math

The fixed point *a* is stable if the absolute value of the derivative of *f* at *a* is strictly less than 1, and unstable if it is strictly greater than 1. This is because near the point *a*, the function *f* has a linear approximation with slope *f'*(*a*):

The fixed point is stable if the absolute value of the derivative of at is strictly less than 1, and unstable if it is strictly greater than 1. This is because near the point , the function has a linear approximation with slope :

当 *a* 点的导数*f*的绝对值严格小于1时，不动点是稳定的; 当其严格大于1时是不稳定。这是因为在该点附近，函数线性近似的斜率为:

- [math]\displaystyle{ f(x) \approx f(a)+f'(a)(x-a). }[/math]

```
[math]\displaystyle{ f(x) \approx f(a)+f'(a)(x-a). }[/math]
```

数学 f (x)大约 f (a) + f’(a)(x-a)。数学

Thus

Thus

因此

- [math]\displaystyle{ x_{n+1}-x_{n} = f(x_n)-x_n \simeq f(a) + f'(a)(x_n-a)-x_n = a + f'(a)(x_n-a)-x_n = (f'(a)-1)(x_n-a) \to \frac{x_{n+1}-x_{n}}{x_n-a}=f'(a)-1 }[/math]

```
[math]\displaystyle{ x_{n+1}-x_{n} = f(x_n)-x_n \simeq f(a) + f'(a)(x_n-a)-x_n = a + f'(a)(x_n-a)-x_n = (f'(a)-1)(x_n-a) \to \frac{x_{n+1}-x_{n}}{x_n-a}=f'(a)-1 }[/math]
```

数学 x { n + 1}-x { n } f (x n)-x n simeq f (a) + f’(a)(x n-a)-x n a + f’(a)(x n-a)-x n (f’(a)-1)(x n-a) to frac { x { n + 1}-x { n-a } f’(a)-1 / math

which means that the derivative measures the rate at which the successive iterates approach the fixed point *a* or diverge from it. If the derivative at *a* is exactly 1 or −1, then more information is needed in order to decide stability.

which means that the derivative measures the rate at which the successive iterates approach the fixed point or diverge from it. If the derivative at is exactly 1 or −1, then more information is needed in order to decide stability.

这意味着导数度量连续迭代接近或远离不动点*a*的速率。如果不动点处的导数恰好是1或-1，那么需要更多的信息来决定稳定性。

There is an analogous criterion for a continuously differentiable map *f*: **R**^{n} → **R**^{n} with a fixed point *a*, expressed in terms of its Jacobian matrix at *a*, *J*_{a}(*f*). If all eigenvalues of *J* are real or complex numbers with absolute value strictly less than 1 then *a* is a stable fixed point; if at least one of them has absolute value strictly greater than 1 then *a* is unstable. Just as for *n*=1, the case of the largest absolute value being 1 needs to be investigated further — the Jacobian matrix test is inconclusive. The same criterion holds more generally for diffeomorphisms of a smooth manifold.

There is an analogous criterion for a continuously differentiable map with a fixed point , expressed in terms of its Jacobian matrix at , . If all eigenvalues of are real or complex numbers with absolute value strictly less than 1 then is a stable fixed point; if at least one of them has absolute value strictly greater than 1 then is unstable. Just as for =1, the case of the largest absolute value being 1 needs to be investigated further — the Jacobian matrix test is inconclusive. The same criterion holds more generally for diffeomorphisms of a smooth manifold.

对于具有一个不动点*a*的连续可微映射*f*: **R**^{n} → **R**^{n}，存在一个类似的判据，由*a*的雅可比矩阵*J*_{a}(*f*)表示。如果*J*的所有特征值都是绝对值严格小于1的实数或复数，则是稳定不动点; 如果其中至少有一个特征值的绝对值严格大于1，则它是不稳定的。就像对于*n*=1，最大本征值绝对值为1的情况也需要进一步研究ーー雅可比矩阵检验是不确定的。同样的准则对光滑流形的微分同胚也有更广泛的适用性。

### Linear autonomous systems 线性自治系统

The stability of fixed points of a system of constant coefficient linear differential equations of first order can be analyzed using the eigenvalues of the corresponding matrix.

The stability of fixed points of a system of constant coefficient linear differential equations of first order can be analyzed using the eigenvalues of the corresponding matrix.

一阶常系数线性微分方程组的不动点的稳定性，可以通过分析相应矩阵的特征值得到。

An autonomous system

一个自治系统 autonomous system

- [math]\displaystyle{ x' = Ax, }[/math]

[math]\displaystyle{ x' = Ax, }[/math]

数学 x’ Ax，/ 数学

where *x*(*t*) ∈ **R**^{n} and *A* is an *n*×*n* matrix with real entries, has a constant solution

where and is an matrix with real entries, has a constant solution

其中 *x*(*t*) ∈ **R**^{n} 且 *A* 是一个*n*×*n*的实矩阵，它具有常数解

- [math]\displaystyle{ x(t)=0. }[/math]

```
[math]\displaystyle{ x(t)=0. }[/math]
```

数学 x (t)0. / 数学

(In a different language, the origin 0 ∈ **R**^{n} is an equilibrium point of the corresponding dynamical system.) This solution is asymptotically stable as *t* → ∞ ("in the future") if and only if for all eigenvalues *λ* of *A*, Re(*λ*) < 0. Similarly, it is asymptotically stable as *t* → −∞ ("in the past") if and only if for all eigenvalues *λ* of *A*, Re(*λ*) > 0. If there exists an eigenvalue *λ* of *A* with Re(*λ*) > 0 then the solution is unstable for *t* → ∞.

(In a different language, the origin is an equilibrium point of the corresponding dynamical system.) This solution is asymptotically stable as ("in the future") if and only if for all eigenvalues of , . Similarly, it is asymptotically stable as ("in the past") if and only if for all eigenvalues of , . If there exists an eigenvalue of with then the solution is unstable for .

(在另一种语言中，原点是该动力系统的平衡点。)这个解是随着*t* → ∞（未来）是渐近稳定的当且仅当对于*A*的所有特征值*λ*有Re(*λ*) < 0。类似地，它随着*t* → -∞（过去）是渐近稳定的当且仅当对于*A*的所有特征值*λ*有Re(*λ*) > 0。如果存在一个*A*的特征值*λ*使得Re(*λ*) > 0，则该解在*t* → ∞时是不稳定的。

Application of this result in practice, in order to decide the stability of the origin for a linear system, is facilitated by the Routh–Hurwitz stability criterion. The eigenvalues of a matrix are the roots of its characteristic polynomial. A polynomial in one variable with real coefficients is called a Hurwitz polynomial if the real parts of all roots are strictly negative. The Routh–Hurwitz theorem implies a characterization of Hurwitz polynomials by means of an algorithm that avoids computing the roots.

Application of this result in practice, in order to decide the stability of the origin for a linear system, is facilitated by the Routh–Hurwitz stability criterion. The eigenvalues of a matrix are the roots of its characteristic polynomial. A polynomial in one variable with real coefficients is called a Hurwitz polynomial if the real parts of all roots are strictly negative. The Routh–Hurwitz theorem implies a characterization of Hurwitz polynomials by means of an algorithm that avoids computing the roots.

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 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.

非线性系统不动点的渐近稳定性通常可以用 Hartman-Grobman 定理来建立。

Suppose that *v* is a *C*^{1}-vector field in **R**^{n} which vanishes at a point *p*, *v*(*p*) = 0. Then the corresponding autonomous system

Suppose that is a -vector field in which vanishes at a point , . Then the corresponding autonomous system

假设*v*是**R**^{n}上的一个*C*^{1}-向量场，并且下降至某一点*p*有*v*(*p*) = 0。那么相应的自治系统

- [math]\displaystyle{ x'=v(x) }[/math]

[math]\displaystyle{ x'=v(x) }[/math]

数学 x’ v (x) / 数学

has a constant solution

has a constant solution

有一个常数解

- [math]\displaystyle{ x(t)=p. }[/math]

```
[math]\displaystyle{ x(t)=p. }[/math]
```

数学 x (t) p / 数学

Let *J*_{p}(*v*) be the *n*×*n* Jacobian matrix of the vector field *v* at the point *p*. If all eigenvalues of *J* 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*J*_{p}(*v*)为向量场 *v*在点*p*的*n*×*n*雅可比矩阵 Jacobian matrix。如果*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

A general way to establish Lyapunov stability or asymptotic stability of a dynamical system is by means of Lyapunov functions.

A general way to establish Lyapunov stability or asymptotic stability of a dynamical system is by means of Lyapunov functions.

建立动力系统的李雅普诺夫稳定性或渐近稳定的一般方法是利用李亚普诺夫函数。

## See also 参见

## References

- ↑ Egwald Mathematics - Linear Algebra: Systems of Linear Differential Equations: Linear Stability Analysis Accessed 10 October 2019.

## External links

- Stable Equilibria by Michael Schreiber, The Wolfram Demonstrations Project.

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This page was moved from wikipedia:en:Stability theory. Its edit history can be viewed at 稳定性理论/edithistory