Floquet theory：周期系数线性系统与 monodromy matrix

旧博客原文

原题：Eloquent theory

Consider matrix ODE:

$\dot{\phi}(t)=A(t)\phi(t)$

Where $A(t)$ is a given periodic matrix with period $T$ , i.e. $A(x)=A(x+T), \forall x\in R$ .

Then the solution $\phi(t)$ satisfied identity:

$\phi(t+T)=\phi(t)\phi^{-1}(0)\phi(T)$ .

This could be explained as $\phi^{-1}\phi(T)=\int_{0}^T\phi(t)$ .

Now we consider to solve the equation: $e^{TB}=\phi^{-1}(0)\phi(T)$ . At least formally it could be solved:

$B=\frac{1}{T}log(\frac{\phi(T)}{\phi(0)})$ .

(Unfortunately $log$ is a multi-value function so $B=B_0+2\pi ik I$ , where $I$ is the identity matrix and $B$ is a solution of $e^{TB}=\phi^{-1}(0)\phi(T)$ .) This argument is false.

In fact matrix is not like numbers, the $log$ function is much more complicated. we have,

$log(A)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{A^n}{n} ...(*)$

So to solve $e^{TB}=\frac{1}{T}(\frac{\phi(T)}{\phi(I)})$ , it is equivalent to :

$B=\frac{1}{T}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}(\frac{\phi(T)}{\phi(I)})^n}{n}$

But this type of identity only meaningful when $||\frac{\phi(T)}{\phi(I)}||<1$ , so is it true that for $||\frac{\phi(T)}{\phi(I)}||<1$ the equation is solved by $(*)$ , and for $||\frac{\phi(T)}{\phi(I)}||\geq 1$ it do not have solution?

The naive inspirit is wrong, the situation is similar to the $\mathbb Q_p$ case while $log_p$ could extend to $D(p^{\frac{-1}{p-1}-})$ and the identity

$exp_p(log_p(1+x))=1+x$

always holds for $x\in D(p^{\frac{-1}{p-1}-})$ . The key observation is $log[(1+Y)(1+Y)]=log(1+X)+log(1+Y)$ always holds when $||X||,||Y||<1$ , this will lead to a reasonable value of

$log[(1+X)(1+Y)]=log(1+X+Y+XY)$

even when $||X+Y+XY||\geq 1$ and this process could be continue to the whole matrix space and the identity enjoy the accosted principle so $log(X)$ is well-defined for all $X\in M_{2\times 2}$ .

Now it is time to consider the rotation number, which is defined by $\lim_{n\to \infty}\frac{f^{n}(x)-x}{n}$ for $f:R\to R$ is a continuous increasing function.

And I do not know how to associated a dynamic system for the matrix $B$ given here, but in any case it seems iff it is given by a hemoermorphifm then the rotation number is zero due to the following reason:

Consider $\mathbb S^1$ as the quotient $\mathbb R/\mathbb Z$ . Your homeomorphism $f$ lifts to a homeomorphism

$\phi : \mathbb R \to \mathbb R$ such that $\phi(x+1)=\phi(x)+1$ .

Form the map $h:=\frac{1}{q} \sum _{n=1} ^q (\phi^{\circ n}-pn)$ , where $\phi ^{\circ n}$ is the composition $n$ times of $\phi$ with itself. By construction $h\circ \phi = h+\frac{p}{q}$ and $h(x+1)=1+h(x)$ , so that $h$ factors as a homeomorphism of the circle conjugating $f$ to the rotation.
By the way this approach wors in $\mathbb R^n$ too.

Maslov index of a holomorphic disk

A natural way to understand the rotation number here is according the way of maslov index, we have the following formula:

$f(A)=\int_{\Gamma}\frac{1}{2\pi i}\frac{f(\lambda)}{\lambda I-A}f(\lambda)d\lambda$

$TB=log(\frac{\phi(T)}{\phi(0)})=log(\int_0^T \phi'(\lambda)d\lambda)=\int_0^T log(\phi'(\lambda))d\lambda=\int_0^T log(A+F(t))d\lambda$

Proof sketch:

1. $B=\frac{1}{T}log(e^{\int_0^T A+f(t)dt})= \frac{1}{T}(\int_{0}^T A+f(t)dt)$ .

2. The dynamic system is defined by : $W: R^2-\{0\} \to R^2-\{0\}$ , $W( x)=B x$ .

3. this dynamic system $(R^2-\{0\},W)$ is conjugate to the dynamic system $T:S_1\to S_1$ , Not difficult to proof it is a homomorphism on $S_1$ and it is zero entropy by Pesin’s formula

If $T: S_1\to S_1$ could lifting to $\hat T:R \to R$ the rotation number is defined as :

$\lim_{n\to \infty}\frac{\hat T^n(x)}{n}$

This problem is not a good problem due to the philosophy, i.e. use rotation number to describe the information of a hamiltonian flow is not satisfied, in fact it is difficult to establish a suitable definition of “rotation number”! But this is the first crucial thing to establish a theorem!

Hamiltonian flow

In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field, defined for any energy function or Hamiltonian. A Hamiltonian vector field is a geometric manifestation of Hamilton’s equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics.[1]

Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions f and g on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the Poisson bracket of f and g.

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

Floquet theory 研究周期系数线性微分方程

$$\dot x=A(t)x,\qquad A(t+T)=A(t).$$

它告诉我们，周期系统的长期行为由一个周期部分和一个指数部分共同决定。

Floquet theory：周期系数线性系统与 monodromy matrix — Floquet theory 把周期系数线性系统分解成周期部分和指数部分，monodromy matrix 控制稳定性。

1. 基本解矩阵

令 $X(t)$ 是基本解矩阵，$X(0)=I$。周期性给出

$$X(t+T)=X(t)X(T).$$

矩阵 $X(T)$ 称为 monodromy matrix。它记录系统经过一个周期后的净变化。

2. Floquet 分解

Floquet theorem 说，在复数域上可以写成

$$X(t)=P(t)e^{tB},$$

其中 $P(t+T)=P(t)$，$B$ 是常矩阵。也就是说，周期系统可以拆成周期振荡和指数增长/衰减。

3. 矩阵对数的细节

形式上想令

$$B=\frac1T\log X(T).$$

但矩阵对数是多值的，而且实矩阵上未必能选到实对数。正确表述通常在复数域成立；若要实形式，需要加入额外周期或 Jordan 分解的讨论。

4. 稳定性

monodromy matrix 的特征值称为 Floquet multipliers。若所有 multiplier 的模都小于 $1$，零解渐近稳定；若有模大于 $1$ 的 multiplier，则出现不稳定方向。

5. 与 rotation number 的关系

二维或辛系统中，monodromy 的作用可能诱导圆周或射影线上的动力系统，此时 rotation number 可以描述方向的平均旋转。这把 Floquet theory 和一维动力系统联系起来。

Floquet theory：周期系数线性系统与 monodromy matrix

旧博客原文

补充说明

1. 基本解矩阵

2. Floquet 分解

3. 矩阵对数的细节

4. 稳定性

5. 与 rotation number 的关系

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