# 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|Periodic operators

We will now suppose that $r(x), p(x)$, and $q(x)$ are $\ell$-periodic functions. Denote by
$$\Pi\left(z, x, x_0\right)=\left(\begin{array}{cc} c\left(z, x, x_0\right) & s\left(z, x, x_0\right) \ p(x) c^{\prime}\left(z, x, x_0\right) & p(x) s^{\prime}\left(z, x, x_0\right) \end{array}\right)$$
the principal matrix solution of $(5.40)$. Here $c\left(z, x, x_0\right)$ and $s\left(z, x, x_0\right)$ is a fundamental system of solutions of $(5.39)$ corresponding to the initial conditions $c\left(z, x_0, x_0\right)=p\left(x_0\right) s^{\prime}\left(z, x_0, x_0\right)=1, s\left(z, x_0, x_0\right)=p\left(x_0\right) c^{\prime}\left(z, x_0, x_0\right)=$ 0 . Since the base point will not play an important role we will just set it equal to $x_0=0$ and write $c(z, x)=c(z, x, 0), s(z, x)=s(z, x, 0)$.

In Section $3.5$ we have introduced the monodromy matrix $M(z)=$ $\Pi\left(z, x_0+\ell, x_0\right)$ and its eigenvalues, the Floquet multipliers,
$$\rho_{\pm}(z)=\Delta(z) \pm \sqrt{\Delta(z)^2-1}, \quad \rho_{+}(z) \rho_{-}(z)=1 .$$
We will choose the branch of the square root such that $\left|\rho_{+}(z)\right| \leq 1$. Here the Floquet discriminant is given by
$$\Delta(z)=\frac{\operatorname{tr}(M(z))}{2}=\frac{c(z, \ell)+p(\ell) s^{\prime}(z, \ell)}{2},$$
Moreover, we have found two solutions
$$u_{\pm}(z, x)=c(z, x)+m_{\pm}(z) s(z, x),$$
the Floquet solutions, satisfying
$$\left(\begin{array}{c} u_{\pm}(z, \ell) \ p(\ell) u_{\pm}^{\prime}(z, \ell) \end{array}\right)=\rho_{\pm}(z)\left(\begin{array}{c} u_{\pm}(z, 0) \ p(0) u_{\pm}^{\prime}(z, 0) \end{array}\right)=\rho_{\pm}(z)\left(\begin{array}{c} 1 \ m_{\pm}(z) \end{array}\right) .$$
Here
$$m_{\pm}(z)=\frac{\rho_{\pm}(z)-c(z, \ell)}{s(z, \ell)}=\frac{\dot{s}(z, \ell)}{\rho_{\pm}(z)-\dot{c}(z, \ell)}$$
are the Weyl $m$-functions. Note that at a point $z$ with $s(z, \ell)=0$, the functions $m_{\pm}(z)$ and hence $u_{\pm}(z, x)$ is not well defined. This is due to our normalization $u_{\pm}(z, 0)=1$ which is not possible if the first component of the eigenvector of the monodromy matrix vanishes.

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

You can think of a dynamical system as the time evolution of some physical system, such as the motion of a few planets under the influence of their respective gravitational forces. Usually you want to know the fate of the system for long times, for instance, will the planets eventually collide or will the system persist for all times?

For some systems (e.g., just two planets) these questions are relatively simple to answer since it turns out that the motion of the system is regular and converges, for example, to an equilibrium.

However, many interesting systems are not that regular! In fact, it turns out that for many systems even very close initial conditions might get spread far apart in short times. For example, you probably have heard about the motion of a butterfly which can produce a perturbance of the atmosphere resulting in a thunderstorm a few weeks later.

We begin with the definition: A dynamical system is a semigroup $G$ acting on a space $M$. That is, there is a map
such that
$$T_g \circ T_h=T_{g \circ h} .$$
If $G$ is a group, we will speak of an invertible dynamical system.
We are mainly interested in discrete dynamical systems where
$$G=\mathbb{N}_0 \quad \text { or } \quad G=\mathbb{Z}$$ and in continuous dynamical systems where
$$G=\mathbb{R}^{+} \quad \text { or } \quad G=\mathbb{R} .$$
Of course this definition is quite abstract and so let us look at some examples first.

Example. The prototypical example of a discrete dynamical system is an iterated map. Let $f$ map an interval $I$ into itself and consider
$$T_n=f^n=f \circ f^{n-1}=\underbrace{f \circ \cdots \circ f}_{n \text { times }}, \quad G=\mathbb{N}_0$$

# 常微分方程代考

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

$$\Pi\left(z, x, x_0\right)=\left(c\left(z, x, x_0\right) \quad s\left(z, x, x_0\right) p(x) c^{\prime}\left(z, x, x_0\right) \quad p(x) s^{\prime}\left(z, x, x_0\right)\right)$$

$$\rho_{\pm}(z)=\Delta(z) \pm \sqrt{\Delta(z)^2-1}, \quad \rho_{+}(z) \rho_{-}(z)=1 .$$

$$\Delta(z)=\frac{\operatorname{tr}(M(z))}{2}=\frac{c(z, \ell)+p(\ell) s^{\prime}(z, \ell)}{2},$$

$$u_{\pm}(z, x)=c(z, x)+m_{\pm}(z) s(z, x),$$
Floquet 解决方案，满足
$$\left(u_{\pm}(z, \ell) p(\ell) u_{\pm}^{\prime}(z, \ell)\right)=\rho_{\pm}(z)\left(u_{\pm}(z, 0) p(0) u_{\pm}^{\prime}(z, 0)\right)=\rho_{\pm}(z)\left(1 m_{\pm}(z)\right) .$$

$$m_{\pm}(z)=\frac{\rho_{\pm}(z)-c(z, \ell)}{s(z, \ell)}=\frac{\dot{s}(z, \ell)}{\rho_{\pm}(z)-\dot{c}(z, \ell)}$$

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

$$T_g \circ T_h=T_{g \circ h} .$$

$$G=\mathbb{N}0 \quad \text { or } \quad G=\mathbb{Z}$$ 在连续动力系统中 $$G=\mathbb{R}^{+} \quad \text { or } \quad G=\mathbb{R} .$$ 当然这个定义是相当抽象的，所以让我们先看一些例子。 例子。离散动力系统的典型示例是迭代映射。让 $f$ 映射一个区间 $I$ 进入自身并考虑 $$T_n=f^n=f \circ f^{n-1}=\underbrace{f \circ \cdots \circ f}{n \text { times }}, \quad G=\mathbb{N}_0$$

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