# 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|MATH2410 Periodic Solutions

• 单变量微积分
• 多变量微积分
• 傅里叶级数
• 黎曼积分
• ODE
• 微分学

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|Periodic Solutions

We have seen that the stability of a rest point can often be determined by linearization or by an application of Lyapunov’s direct method. In both cases, the stability can be determined by analysis in an arbitrary open set (no matter how “small”) containing the rest point. For this reason, we say that the stability of a rest point is a local problem. However, it is not possible to determine the stability of a periodic solution without considering the ordinary differential equation in a neighborhood of the entire periodic orbit. In other words, global methods must be employed. This fact makes the analysis of periodic solutions much more difficult (and more interesting) than the analysis of rest points. In this section we will introduce some of the basic ideas that are used to study the existence and stability of periodic solutions.

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|The Poincar´e Map

A very powerful concept in the study of periodic orbits is the Poincaré map. It is a corner stone of the “geometric theory” of Henri Poincaré [143], the father of our subject. To define the Poincaré map, also called the return map, let $\phi_{t}$ denote the flow of the differential equation $\dot{x}=f(x)$, and suppose that $S \subseteq \mathbb{R}^{n}$ is an $(n-1)$-dimensional submanifold. If $p \in S$ and $(p, f(p)) \notin T_{p} S$, then we say that the vector $(p, f(p))$ is transverse to $S$ at $p$. If $(p, f(p))$ is transverse to $S$ at each $p \in S$, we say that $S$ is a section for $\phi_{t}$. If $p$ is in $S$, then the curve $t \mapsto \phi_{t}(p)$ “passes through” $S$ as $t$ passes through $t=0$. Perhaps there is some $T=T(p)>0$ such that $\phi_{T}(p) \in S$. In this case, we say that the point p returns to $S$ at time $T$. If there is an open subset $\Sigma \subseteq S$ such that each point of $\Sigma$ returns to $S$, then $\Sigma$ is called a Poincaré section. In this case, let us define $P: \Sigma \rightarrow S$ as follows: $P(p):=\phi_{T(p)}(p)$ where $T(p)>0$ is the time of the first return to $S$. The map $P$ is called the Poincaré map, or the return map on $\Sigma$ and $T: \Sigma \rightarrow \mathbb{R}$ is called the return time map (see Figure 1.22). Using the fact that the solution of a differential equation is smoothly dependent on its initial value and the implicit function theorem, it can be proved that both $P$ and $T$ are smooth functions on $\Sigma$ (see Exercise 1.96).

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代 考|The Poincar’ e Map

$(p, f(p)) \notin T_{p} S$ ，那么我们说向量 $(p, f(p))$ 是横向的 $S$ 在 $p$. 如果 $(p, f(p))$ 是横向的
$S$ 在每一个 $p \in S$ ，我们说 $S$ 是一个部分 $\phi_{t}$. 如果 $p$ 在 $S$ ，那么曲线 $t \mapsto \phi_{t}(p)$ “经过” $S$

$T: \Sigma \rightarrow \mathbb{R}$ 称为返回时间图 (见图 1.22) 。利用微分方程的解平滑依赖于其初值和隐 函数定理这一事实，可以证明 $P$ 和 $T$ 是光滑的函数 $\Sigma$ (见刃题 1.96) 。