谱子流形
W
(
E
)
{\displaystyle {\mathcal {W}}(E)}
E 产生（emanate）出来的示意图。既约坐标中的轨迹
p
(
t
)
{\displaystyle p(t)}
W
(
p
)
{\displaystyle W(p)}
[ 1] 动力系统 中，谱子流形 （SSM）是唯一的最光滑不变流形 ，是线性动力系统的谱子空间添加非线性因素后的非线性推广。[ 2] 非线性降维 中的应用。
考虑非线性常微分方程
d
x
d
t
=
A
x
+
f
0
(
x
)
,
x
∈
R
n
,
{\displaystyle {\frac {dx}{dt}}=Ax+f_{0}(x),\quad x\in \mathbb {R} ^{n},}
常矩阵
A
∈
R
n
×
n
{\displaystyle \ A\in \mathbb {R} ^{n\times n}}
f
0
=
O
(
|
x
|
2
)
{\displaystyle f_{0}={\mathcal {O}}(|x|^{2})}
假设对A 的所有特征值
λ
j
,
j
=
1
,
…
,
n
{\displaystyle \lambda _{j},\ j=1,\ldots ,n}
Re
λ
j
<
0
{\displaystyle {\text{Re}}\lambda _{j}<0}
m 个特征向量
v
i
E
{\displaystyle v_{i}^{E}}
E
=
span
{
v
1
E
,
…
v
m
E
}
{\displaystyle E={\text{span}}\,\{v_{1}^{E},\ldots v_{m}^{E}\}}
E 是线性化系统的不变子空间
d
x
d
t
=
A
x
,
x
∈
R
n
.
{\displaystyle {\frac {dx}{dt}}=Ax,\quad x\in \mathbb {R} ^{n}.}
在线性系统中加入非线性项
f
0
{\displaystyle f_{0}}
E 通常会扰动成无穷多个不变流形。当中，唯一最平滑的流形就是谱子流形。
对
Re
λ
j
>
0
{\displaystyle {\text{Re}}\lambda _{j}>0}
只要E 的谱中的特征值
λ
i
E
{\displaystyle \lambda _{i}^{E}}
E 相切的谱子流形。[ 3] A 的某个特征值相等的
λ
i
E
{\displaystyle \lambda _{i}^{E}}
E ，并将分析推广到与广义谱子空间有关的高维SSM。
关于谱子流形的理论可推广到非线性非自治系统 ，形式为
d
x
d
t
=
A
x
+
f
0
(
x
)
+
ϵ
f
1
(
x
,
Ω
t
)
,
Ω
∈
T
k
,
0
≤
ϵ
≪
1
,
{\displaystyle {\frac {dx}{dt}}=Ax+f_{0}(x)+\epsilon f_{1}(x,\Omega t),\quad \Omega \in \mathbb {T} ^{k},\ 0\leq \epsilon \ll 1,}
其中
f
1
:
R
n
×
T
k
→
R
n
{\displaystyle f_{1}:\mathbb {R} ^{n}\times \mathbb {T} ^{k}\to \mathbb {R} ^{n}}
准周期 强迫项。[ 4]
谱子流形对动力系统中严格的非线性降维非常有用。高维相空间还原为低维流形后，便可精确描述系统的主要渐进行为，从而大大简化系统结构。[ 5] [ 6]
此外，还可利用机器学习算法，直接从动力系统的轨迹数据中提取这些流形。[ 7]
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