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分类： Floquet theory

  • 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。