微积分网课代修|常微分方程代写Ordinary Differential Equation代考|MATH275 Geometric Interpretation of Autonomous Systems

微积分网课代修|常微分方程代写Ordinary Differential Equation代考|MATH275 Geometric Interpretation of Autonomous Systems

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微积分网课代修|常微分方程代写Ordinary Differential Equation代考|MATH275 Geometric Interpretation of Autonomous Systems

微积分网课代修|常微分方程代写Ordinary Differential Equation代考|Geometric Interpretation of Autonomous Systems

In this section we will describe a very important geometric interpretation of the autonomous differential equation
$$
\dot{x}=f(x), \quad x \in \mathbb{R}^{n} .
$$
The function given by $x \mapsto(x, f(x))$ defines a vector field on $\mathbb{R}^{n}$ associated with the differential equation (1.7). Here the first component of the function specifies the base point and the second component specifies the vector at this base point. A solution $t \mapsto \phi(t)$ of (1.7) has the property that its tangent vector at each time $t$ is given by
$$
(\phi(t), \dot{\phi}(t))=(\phi(t), f(\phi(t))) .
$$
In other words, if $\xi \in \mathbb{R}^{n}$ is on the orbit of this solution, then the tangent line to the orbit at $\xi$ is generated by the vector $(\xi, f(\xi))$, as depicted in Figure $1.1$.

We have just mentioned two essential facts: (i) There is a one-to-one correspondence between vector fields and autonomous differential equations. (ii) Every tangent vector to a solution curve is given by a vector in the vector field. These facts suggest that the geometry of the associated vector field is closely related to the geometry of the solutions of the differential equation when the solutions are viewed as curves in a Euclidean space. This geometric interpretation of the solutions of autonomous differential equations provides a deep insight into the general nature of the solutions of differential equations, and at the same time suggests the “geometric method” for studying differential equations: qualitative features expressed geometrically are paramount; analytic formulas for solutions are of secondary importance. Finally, let us note that the vector field associated with a differential equation is given explicitly. Thus, one of the main goals of the geometric method is to derive qualitative properties of solutions directly from the vector field without “solving” the differential equation.

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

An important property of the set of solutions of the autonomous differential equation (1.7),
$$
\dot{x}=f(x), \quad x \in \mathbb{R}^{n},
$$
is the fact that these solutions form a one-parameter group that defines a phase flow. More precisely, let us define the function $\phi: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ as follows: For $x \in \mathbb{R}^{n}$, let $t \mapsto \phi(t, x)$ denote the solution of the autonomous differential equation (1.7) such that $\phi(0, x)=x$.

We know that solutions of a differential equation may not exist for all $t \in \mathbb{R}$. However, for simplicity, let us assume that every solution does exist for all time. If this is the case, then each solution is called complete, and the fact that $\phi$ defines a one-parameter group is expressed concisely as follows:
$$
\phi(t+s, x)=\phi(t, \phi(s, x)) .
$$
In view of this equation, if the solution starting at time zero at the point $x$ is continued until time $s$, when it reaches the point $\phi(s, x)$, and if a new solution at this point with initial time zero is continued until time $t$, then this new solution will reach the same point that would have been reached if the original solution, which started at time zero at the point $x$, is continued until time $t+s$.

微积分网课代修|常微分方程代写Ordinary Differential Equation代考|MATH275 Geometric Interpretation of Autonomous Systems

微积分网课代修|常微分方程代写Ordinary Differential Equation代考|Geometric Interpretation of Autonomous Systems


在本节中,我们将描述自治微分方程的一个非常重要的几何解释
$$
\dot{x}=f(x), \quad x \in \mathbb{R}^{n} .
$$
给出的函数 $x \mapsto(x, f(x))$ 定义一个向量场 $\mathbb{R}^{n}$ 与微分方程 (1.7) 相关。这里函数的第 一个分量指定基点,第二个分量指定该基点处的向量。一个解法 $t \mapsto \phi(t)(1.7)$ 的有 性质,它的切向量在每个时间 $t$ 是 (准) 给的
$$
(\phi(t), \dot{\phi}(t))=(\phi(t), f(\phi(t))) .
$$
换句话说,如果 $\xi \in \mathbb{R}^{n}$ 在这个解的轨道上,那么轨道的切线在 $\xi$ 由向量生成 $(\xi, f(\xi))$ ,如图所示1.1.
我们刚刚提到了两个基本事实: (i) 向量场和自治微分方程之间存在一一对应的关系。
(ii) 解曲线的每个切向量由向量场中的向量给出。这些事实表明,当解被视为欧几里得 空间中的曲线时,相关矢量场的几何与微分方程解的几何密切相关。这种对自治微分方 程解的几何解释提供了对微分方程解的一般性质的深刻见解,同时提出了研究微分方程 的“几何方法”: 几何表达的定性特征是最重要的;解的解析公式是次要的。最后,让我 们注意,与微分方程相关的矢量场是明确给出的。因此,几何方法的主要目标之一是直 接从矢量场导出解的定性属性,而不“求解”微分方程。

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

自治微分方程 (1.7) 的解集的一个重要性质,
$$
\dot{x}=f(x), \quad x \in \mathbb{R}^{n},
$$
事实上,这些解决方案形成了一个定义相流的单参数组。更准确地说,让我们定义函数 $\phi: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ 如下: 对于 $x \in \mathbb{R}^{n}$ ,让 $t \mapsto \phi(t, x)$ 表示自治微分方程 (1.7) 的 解,使得 $\phi(0, x)=x$.
我们知道微分方程的解可能并非对所有人都存在 $t \in \mathbb{R}$. 然而,为简单起见,让我们假 设每个解决方案都确实存在。如果是这种情况,那么每个解决方案都称为完整的,而事 实是 $\phi$ 定义一个单参数组简明如下:
$$
\phi(t+s, x)=\phi(t, \phi(s, x)) .
$$
鉴于这个方程,如果解在时间 0 处开始 $x$ 一直持续到时间 $s$ ,当它到达点 $\phi(s, x)$ ,并且 如果此时初始时间为零的新解决方案持续到时间 $t$ ,那么这个新的解决方案将达到与原 始解决方案相同的点,原始解决方案在该点的时间为零开始 $x$ ,直持续到时间 $t+s$.

微积分网课代修|常微分方程代写Ordinary Differential Equation代考|MATH275 Geometric Interpretation of Autonomous Systems
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