更改

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

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

让我们成为一个具有固定点的连续可微函数。考虑通过迭代函数得到的动力系统:

另{{Math|''f'': '''R''' → '''R'''}}是一个连续可微函数且存在一个不动点{{Math|''a''}}，{{Math|1=''f''(''a'') = ''a''}}。考虑一个通过迭代函数得到的动力系统:

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 :

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 at 的导数绝对值严格小于1时，不动点是稳定的; 当 at 的导数绝对值严格大于1时，不稳定。这是因为在点附近，函数有一个带斜率的线性近似:

当 {{Math|''a''}} 点的导数{{Math|''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:

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.

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.

这意味着导数测量连续迭代接近不动点或偏离不动点的速率。如果导数 at 恰好是1或-1，那么需要更多的信息来决定稳定性。

这意味着导数度量连续迭代接近或远离不动点{{Math|''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  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.

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.

对于具有不动点的连续可微映射，存在一个类似的判据，它的雅可比矩阵表示为，。如果所有的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绝对值严格大于1，则是不稳定的。与1一样，最大绝对值为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。

对于具有一个不动点{{Math|''a''}}的连续可微映射{{Math|''f'': '''R'''^''n'' → '''R'''^''n''}}，存在一个类似的判据，由{{Math|''a''}}的雅可比矩阵{{Math|''J''_''a''(''f'')}}表示。如果{{Math|''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{{Math|''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 ===

=== Linear autonomous systems ===