# 数学代写|常微分方程代写ordinary differential equation代考|MATH211

## 数学代写|常微分方程代写ordinary differential equation代考|Floquet Theory

In this section, we will begin the study of linear systems of the form
$$\dot{x}=A(t) x, \quad x \in \mathbb{R}^n$$
where $t \rightarrow A(t)$ is a $T$-periodic continuous matrix-valued function. The main theorem in this section, Floquet’s theorem, gives a canonical form for each fundamental matrix solution. This result will be used to show that there is a periodic time-dependent change of coordinates that transforms system (2.26) into a homogeneous linear system with constant coefficients. Floquet’s theorem is a corollary of the following result about the range of the exponential map.

Theorem 2.47. If $C$ is a nonsingular $n \times n$ matrix, then there is an $n \times n$ matrix $B$, possibly complex, such that $e^B=C$. If $C$ is a nonsingular real $n \times n$ matrix, then there is a real $n \times n$ matrix $B$ such that $e^B=C^2$.
Proof. If $S$ is a nonsingular $n \times n$ matrix such that $S^{-1} C S=J$ is in Jordan canonical form, and if $e^K=J$, then $S e^K S^{-1}=C$. As a result, $e^{S K S^{-1}}=C$ and $B=S K S^{-1}$ is the desired matrix. Thus, it suffices to consider the nonsingular matrix $C$ or $C^2$ to be a Jordan block.

For the first statement of the theorem, assume that $C=\lambda I+N$ where $N$ is nilpotent; that is, $N^m=0$ for some integer $m$ with $0 \leq m<n$. Because $C$ is nonsingular, $\lambda \neq 0$ and we can write $C=\lambda(I+(1 / \lambda) N)$. A computation using the series representation of the function $t \mapsto \ln (1+t)$ at $t=0$ shows that, formally (that is, without regard to the convergence of the series), if $B=(\ln \lambda) I+M$ where
$$M=\sum_{j=1}^{m-1} \frac{(-1)^{j+1}}{j \lambda^j} N^j,$$
then $e^B=C$. But because $N$ is nilpotent, the series are finite. Thus, the formal series identity is an identity. This proves the first statement of the theorem.

## 数学代写|常微分方程代写ordinary differential equation代考|Hill’s Equation

A famous example where Floquet theory applies to give good stability results is Hill’s equation,
$$\ddot{u}+a(t) u=0, \quad a(t+T)=a(t) .$$
This equation was introduced by George W. Hill in his study of the motions of the moon. Roughly speaking, the motion of the moon can be viewed as a harmonic oscillator in a periodic gravitational field. However, this model equation arises in many areas of applied mathematics where the stability of periodic motions is an issue. A prime example, mentioned in the previous section, is the stability analysis of small oscillations of a pendulum whose length varies with time.
If we define
$$x:=\left(\begin{array}{c} u \ \dot{u} \end{array}\right),$$
then Hill’s equation is equivalent to the first order system $\dot{x}=A(t) x$ where
$$A(t)=\left(\begin{array}{cc} 0 & 1 \ -a(t) & 0 \end{array}\right)$$

# 常微分方程代考

## 数学代写|常微分方程代写ordinary differential equation代考|Floquet Theory

$$\dot{x}=A(t) x, \quad x \in \mathbb{R}^n$$

$$M=\sum_{j=1}^{m-1} \frac{(-1)^{j+1}}{j \lambda^j} N^j$$

## 数学代写|常微分方程代写ordinary differential equation代考|Hill’s Equation

$$\ddot{u}+a(t) u=0, \quad a(t+T)=a(t) .$$

$$x:=(u \dot{u}),$$

$$A(t)=\left(\begin{array}{lll} 0 & 1-a(t) & 0 \end{array}\right)$$

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