Posted on Categories:Linear algebra, 数学代写, 线性代数

# 数学代写|线性代数代写Linear algebra代考|MA405 DISCRETE DYNAMICAL SYSTEMS

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|线性代数代写Linear algebra代考|DISCRETE DYNAMICAL SYSTEMS

Eigenvalues and eigenvectors provide the key to understanding the long-term behavior, or evolution, of a dynamical system described by a difference equation $\mathbf{x}_{k+1}=A \mathbf{x}_k$. Such an equation was used to model population movement in Section 1.10, various Markov chains in Section 4.9, and the spotted owl population in the introductory example for this chapter. The vectors $\mathbf{x}_k$ give information about the system as time (denoted by $k$ ) passes. In the spotted owl example, for instance, $\mathbf{x}_k$ listed the numbers of owls in three age classes at time $k$.

The applications in this section focus on ecological problems because they are easier to state and explain than, say, problems in physics or engineering. However, dynamical systems arise in many scientific fields. For instance, standard undergraduate courses in control systems discuss several aspects of dynamical systems. The modern statespace design method in such courses relies heavily on matrix algebra. ${ }^1$ The steady-state response of a control system is the engineering equivalent of what we call here the “long-term behavior” of the dynamical system $\mathbf{x}_{k+1}=A \mathbf{x}_k$.

Until Example 6, we assume that $A$ is diagonalizable, with $n$ linearly independent eigenvectors, $\mathbf{v}_1, \ldots, \mathbf{v}_n$, and corresponding eigenvalues, $\lambda_1, \ldots, \lambda_n$. For convenience, assume the eigenvectors are arranged so that $\left|\lambda_1\right| \geq\left|\lambda_2\right| \geq \cdots \geq\left|\lambda_n\right|$. Since $\left{\mathbf{v}_1, \ldots, \mathbf{v}_n\right}$ is a basis for $\mathbb{R}^n$, any initial vector $\mathbf{x}_0$ can be written uniquely as
$$\mathbf{x}_0=c_1 \mathbf{v}_1+\cdots+c_n \mathbf{v}_n$$
This eigenvector decomposition of $\mathbf{x}_0$ determines what happens to the sequence $\left{\mathbf{x}_k\right}$. The next calculation generalizes the simple case examined in Example 5 of Section 5.2. Since the $\mathbf{v}_i$ are eigenvectors,
\begin{aligned} \mathbf{x}_1=A \mathbf{x}_0 &=c_1 A \mathbf{v}_1+\cdots+c_n A \mathbf{v}_n \ &=c_1 \lambda_1 \mathbf{v}_1+\cdots+c_n \lambda_n \mathbf{v}_n \end{aligned}
In general,
$$\mathbf{x}_k=c_1\left(\lambda_1\right)^k \mathbf{v}_1+\cdots+c_n\left(\lambda_n\right)^k \mathbf{v}_n \quad(k=0,1,2, \ldots)$$
The examples that follow illustrate what can happen in (2) as $k \rightarrow \infty$.

## 数学代写|线性代数代写Linear algebra代考|A Predator–Prey System

Deep in the redwood forests of California, dusky-footed wood rats provide up to $80 \%$ of the diet for the spotted owl, the main predator of the wood rat. Example 1 uses a linear dynamical system to model the physical system of the owls and the rats. (Admittedly, the model is unrealistic in several respects, but it can provide a starting point for the study of more complicated nonlinear models used by environmental scientists.)

EXAMPLE 1 Denote the owl and wood rat populations at time $k$ by $\mathbf{x}k=\left[\begin{array}{c}O_k \ R_k\end{array}\right]$, where $k$ is the time in months, $O_k$ is the number of owls in the region studied, and $R_k$ is the number of rats (measured in thousands). Suppose \begin{aligned} &O{k+1}=(.5) O_k+(.4) R_k \ &R_{k+1}=-p \cdot O_k+(1.1) R_k \end{aligned}
where $p$ is a positive parameter to be specified. The (.5) $O_k$ in the first equation says that with no wood rats for food, only half of the owls will survive each month, while the (1.1) $R_k$ in the second equation says that with no owls as predators, the rat population will grow by $10 \%$ per month. If rats are plentiful, the $(.4) R_k$ will tend to make the owl population rise, while the negative term $-p \cdot O_k$ measures the deaths of rats due to predation by owls. (In fact, $1000 p$ is the average number of rats eaten by one owl in one month.) Determine the evolution of this system when the predation parameter $p$ is $.104$

SOLUTION When $p=.104$, the eigenvalues of the coefficient matrix $A$ for the equations in (3) turn out to be $\lambda_1=1.02$ and $\lambda_2=.58$. Corresponding eigenvectors are
$$\mathbf{v}_1=\left[\begin{array}{l} 10 \ 13 \end{array}\right], \quad \mathbf{v}_2=\left[\begin{array}{l} 5 \ 1 \end{array}\right]$$
An initial $\mathbf{x}_0$ can be written as $\mathbf{x}_0=c_1 \mathbf{v}_1+c_2 \mathbf{v}_2$. Then, for $k \geq 0$,
\begin{aligned} \mathbf{x}_k &=c_1(1.02)^k \mathbf{v}_1+c_2(.58)^k \mathbf{v}_2 \ &=c_1(1.02)^k\left[\begin{array}{l} 10 \ 13 \end{array}\right]+c_2(.58)^k\left[\begin{array}{l} 5 \ 1 \end{array}\right] \end{aligned}

## 数学代写线性代数代写Linear algebra代考|A Predator-Prey System

$$\mathbf{v}_1=\left[\begin{array}{ll} 10 & 13 \end{array}\right], \quad \mathbf{v}_2=\left[\begin{array}{ll} 5 & 1 \end{array}\right]$$

$$\mathbf{x}_k=c_1(1.02)^k \mathbf{v}_1+c_2(.58)^k \mathbf{v}_2 \quad=c_1(1.02)^k[1013]+c_2(.58)^k[51]$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。