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## EE代写|连续线性系统代写Continous Time Linear System代考|A tool for detecting chaos: Lyapunov exponents

Consider again a differential equation of the form
$$\dot{\boldsymbol{x}}=f(\boldsymbol{x}), \quad \boldsymbol{x} \in \mathbb{R}^n .$$
where $f$ is a $C^r$ map with $r \geq 1$. Let $\boldsymbol{x}\left(t, \boldsymbol{x}_0\right.$ ) be the solution of (18) with initial condition $\boldsymbol{x}\left(0, \boldsymbol{x}_0\right)=\boldsymbol{x}_0$. To describe the geometry associated with the attraction and/or repulsion of orbits of (18) relative to $\boldsymbol{x}\left(t, \boldsymbol{x}_0\right)$. One considers the orbit structure of the linearization of (18) about $\boldsymbol{x}\left(t, \boldsymbol{x}_0\right)$, which is given by
$$\dot{\boldsymbol{y}}=D f\left(\boldsymbol{x}\left(t, \boldsymbol{x}_0\right)\right) \boldsymbol{y}$$
Let $X\left(t, \boldsymbol{x}\left(t, \boldsymbol{x}_0\right)\right)$ denote the fundamental solution matrix of (19) and let $\boldsymbol{e} \neq 0$ be a vector in $\mathbb{R}^n$. We define the coefficient of expansion in the direction $e$ along the trajectory through $x_0$ to be
$$\lambda_t\left(\boldsymbol{x}_0, \boldsymbol{e}\right)=\frac{\left|X\left(t, \boldsymbol{x}\left(t, \boldsymbol{x}_0\right)\right) \boldsymbol{e}\right|}{|\boldsymbol{e}|} .$$
Note that the coefficient $\lambda_t\left(\boldsymbol{x}_0, \boldsymbol{e}\right)$ depends on $t$, on the orbit of (18) through $\boldsymbol{x}_0$ and on $e$.

The Lyapunov exponent in the direction $e$ along the trajectory through $\boldsymbol{x}0$ is defined as $$\chi\left(\boldsymbol{x}_0, \boldsymbol{e}\right)=\varlimsup{t \rightarrow+\infty} \frac{1}{t} \lambda_t\left(\boldsymbol{x}0, \boldsymbol{e}\right) .$$ For the zero vector it is common to define $\chi\left(\boldsymbol{x}_0, \mathbf{0}\right)=-\infty$. Proposition 4.1.1 (Properties of Lyapunov exponents). The following properties hold: (i) For any vectors $\boldsymbol{e}_1, \boldsymbol{e}_2 \in \mathbb{R}^n, \chi\left(\boldsymbol{x}_0, \boldsymbol{e}_1+\boldsymbol{e}_2\right) \leq \max \left{\chi\left(\boldsymbol{x}_0, \boldsymbol{e}_1\right), \chi\left(\boldsymbol{x}_0, \boldsymbol{e}_1\right)\right}$. (ii) For any vector $\boldsymbol{e} \in \mathbb{R}^n$ and constant $c \in \mathbb{R}, \chi\left(\boldsymbol{x}_0, c \boldsymbol{e}\right)=\chi\left(\boldsymbol{x}_0, \boldsymbol{e}\right)$. (iii) The set of numbers $\left{\chi\left(\boldsymbol{x}_0, \boldsymbol{e}\right)\right}{\boldsymbol{e} \in \mathbb{R}^n \backslash{0}}$ takes at most $n$ values. It is called Lyapunov spectrum.

In practical applications the Lyapunov exponents of a trajectory are typically computed numerically. There has been much rigorous work in recent years involving the development of algorithms to accurately compute Lyapunov exponents

## EE代写|连续线性系统代写Continous Time Linear System代考|Chaotic behaviour and strange attractors

Let $\phi^t(\boldsymbol{x})$ denote the flow of (18) and assume $\Lambda \subset \mathbb{R}^n$ is a compact and invariant set $\phi^t(\boldsymbol{x})$, i.e. $\phi^t(\Lambda) \subseteq \Lambda$ for all $t \in \mathbb{R}$.

We say that the flow $\phi^t(\boldsymbol{x})$ has sensitive dependence on initial conditions if for any point $\boldsymbol{x} \in \Lambda$, there is at least one point arbitrarily close to $x$ that diverges from x. More precisely:

Definition 4.2.1 (Sensitive dependence on initial conditions). The flow $\phi^t(\boldsymbol{x})$ is said to have sensitive dependence on initial conditions on $\Lambda$ if there exists $\epsilon>0$ such that, for any $\boldsymbol{x} \in \Lambda$ and any neighborhood $U$ of $\boldsymbol{x}$, there exists $\boldsymbol{y} \in U$ and $t>0$ such that $\left|\phi^t(\boldsymbol{x})-\phi^t(\boldsymbol{y})\right|>\epsilon$

Taken just by itself, sensitive dependence on initial conditions is a fairly common property in many dynamical systems. For a set to be chaotic, a couple of other properties need to be added:

Definition 4.2.2 (Chaotic invariant set). An invariant set $\Lambda$ is said to be chaotic if
(i) $\phi^t(\boldsymbol{x})$ has sensitive dependence on initial conditions on $\Lambda$.
(ii) $\phi^t(\boldsymbol{x})$ is topologically transitive on $\Lambda$.
(iii) The periodic orbits of $\phi^t(\boldsymbol{x})$ are dense in $\Lambda$.

## EE代写|连续线性系统代写Continous Time Linear System代考|A tool for detecting chaos: Lyapunov exponents

$$\dot{\boldsymbol{x}}=f(\boldsymbol{x}), \quad \boldsymbol{x} \in \mathbb{R}^n .$$

$$\dot{\boldsymbol{y}}=D f\left(\boldsymbol{x}\left(t, \boldsymbol{x}_0\right)\right) \boldsymbol{y}$$

$$\lambda_t\left(\boldsymbol{x}_0, \boldsymbol{e}\right)=\frac{\left|X\left(t, \boldsymbol{x}\left(t, \boldsymbol{x}_0\right)\right) \boldsymbol{e}\right|}{|\boldsymbol{e}|}$$

$$\chi\left(\boldsymbol{x}_0, \boldsymbol{e}\right)=\varlimsup \overline{\lim } t \rightarrow+\infty \frac{1}{t} \lambda_t(\boldsymbol{x} 0, \boldsymbol{e}) .$$

〈left 缺少或无法识别的分隔符 集}left 缺少或无法识别的分隔等 最多需要 $n$ 值。称为李雅普诺夫谱。

## EE代写|连续线性系统代写Continous Time Linear System代茰|Chaotic behaviour and strange attractors

(i) $\phi^t(\boldsymbol{x})$ 对初始条件具有敏感的依赖性 $\Lambda$.
(二) $\phi^t(\boldsymbol{x})$ 在上拓扑传递 $\Lambda$.
(iii) 的周期轨道 $\phi^t(\boldsymbol{x})$ 密集在 $\Lambda$.

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## 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。

Posted on Categories:EE代写, 电子代写, 连续线性系统

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## EE代写|连续线性系统代写Continous Time Linear System代考|Periodic orbits and Poincar´e maps

We consider again the differential equation (3) and let $\gamma$ denote a periodic orbit of the flow of (3) with period $T$ and $\boldsymbol{p} \in \gamma$. Then, for some $k$, the $k^{\text {th }}$ coordinate of the map $f$ must be non-zero at $\boldsymbol{p}$, i.e. $f_k(\boldsymbol{p}) \neq 0$. Take the hyperplane given by
$$\Sigma=\left{\boldsymbol{x}: x_k=p_k\right} .$$
The hyperplane $\Sigma$ is called a cross section at $\boldsymbol{p}$. For some $\boldsymbol{x} \in \Sigma$ near $\boldsymbol{p}$, the flow $\phi^t(\boldsymbol{x})$ returns to $\Sigma$ in time $\tau(\boldsymbol{x})$ close to $T$. We call $\tau(\boldsymbol{x})$ the first return time.

Definition 2.8.1. Let $V \subset \Sigma$ be an open set in $\Sigma$ on which $\tau(\boldsymbol{x})$ is a differentiable function. The Poincaré map, $P: V \rightarrow \Sigma$, is defined by
$$P(\boldsymbol{x})=\phi^{\tau(\boldsymbol{x})}(\boldsymbol{x}) .$$
Thus, the Poincaré map reduces the analysis of a continuous time dynamical system to the analysis of a discrete time dynamical system. This is very useful for the analysis of the behaviour of periodic orbits of flows since such orbits are fixed points of the Poincaré map. We list below some properties of the Poincaré map of a flow near a periodic orbit.

## EE代写|连续线性系统代写Continous Time Linear System代考|Theorem

Theorem 2.8.2. Let $\phi^t$ be the $C^r$ flow $(r \geq 1)$ of $\dot{\boldsymbol{x}}=f(\boldsymbol{x})$.
i) If $\boldsymbol{p}$ is on a periodic orbit of period $T$ and $\Sigma$ is transversal at $\boldsymbol{p}$, then the first return time $\tau(\boldsymbol{x})$ is defined in a neighbourhood $V$ of $\boldsymbol{p}$ and $\tau: V \rightarrow \mathbb{R}$ is $C^r$.
ii) The Poincaré map (8) is $C^r$.
iii) If $\gamma$ is a periodic orbit of period $T$ and $\boldsymbol{p} \in \gamma$, then $D \phi_{\boldsymbol{p}}^T$ has 1 as an eigenvalue with eigenvector $f(\boldsymbol{p})$.
iv) If $\gamma$ is a periodic orbit of period $T$ and $\boldsymbol{p}, \boldsymbol{q} \in \gamma$, then the derivatives $D \phi_{\boldsymbol{p}}^T$ and $D \phi_{\boldsymbol{q}}^T$ are linearly conjugate and so have the same eigenvalues.

We will now use the Poincaré map to study the stability of periodic orbits of the dynamical system defined by (3). Let $\gamma$ be a periodic orbit of period $T$ for the flow of (3) with $\boldsymbol{p} \in \gamma$ and let $1, \lambda_1, \ldots, \lambda_{m-1}$ be the eigenvalues of $D \phi_{\boldsymbol{p}}^T$. The $m-1$ eigenvalues $\lambda_1, \ldots, \lambda_{m-1}$ are called the characteristic multipliers of the periodic orbit $\gamma$. We say that

• $\gamma$ is hyperbolic if $\left|\lambda_j\right| \neq 1$ for all $j \in{1, \ldots, m-1}$.
• $\gamma$ is elliptic if $\left|\lambda_j\right|=1$ for all $j \in{1, \ldots, m-1}$.
Moreover, in the case where $\gamma$ is a hyperbolic periodic orbit, we say that
• $\gamma$ is a periodic sink if $\left|\lambda_j\right|<1$ for all $j \in{1, \ldots, m-1}$.
• $\gamma$ is a periodic source if $\left|\lambda_j\right|>1$ for all $j \in{1, \ldots, m-1}$.
• $\gamma$ is a saddle periodic orbit if $\gamma$ is neither a periodic sink nor a periodic source.
The next result establishes the relation between the characteristic multipliers and the Poincaré map near a periodic orbit.

## EE代写|连续线性系统代写Continous Time Linear System代考|Periodic orbits and Poincar

${ }^{-}$e maps $\boldsymbol{p} ， 1 E f_k(\boldsymbol{p}) \neq 0$. 取由绐出的超平面
\left 缺少或无法识别的分隔符

$$P(\boldsymbol{x})=\phi^{\tau(\boldsymbol{x})}(\boldsymbol{x}) .$$

## EE代写|连续线性系统代与Continous Time Linear System代考|Theorem

$\rightarrow$ 如果 $\boldsymbol{p}$ 在周期的周期轨道上 $T$ 和 横向于 $\boldsymbol{p}$, 那 $\angle$ 第一次返回时间 $\tau(\boldsymbol{x})$ 在邻域中定义 $V$ 的 $\boldsymbol{p}$ 和 $\tau: V \rightarrow \mathbb{R}$ 是 $C^r$.
ii) 庞加莱㘨射 (8) 是 $C^r$.
iii) 如果 $\gamma$ 是周期的周期轨道 $T$ 和 $\boldsymbol{p} \in \gamma$ ，然后 $D \phi_{\boldsymbol{p}}^T$ 具有 1 作为具有特征向量的特征值 $f(\boldsymbol{p})$.
iv) 如果 $\gamma$ 是周期的周期轨道 $T$ 和 $\boldsymbol{p}, \boldsymbol{q} \in \gamma$ ，那/㝵数 $D \phi_{\boldsymbol{p}}^T$ 和 $D \phi_{\boldsymbol{q}}^T$ 是线性共轪的，因此具有相同的特征值。

• $\gamma$ 是双曲线的如果 $\left|\lambda_j\right| \neq 1$ 对所有人 $j \in 1, \ldots, m-1$.
• $\gamma$ 是椭圆的如果 $\left|\lambda_j\right|=1$ 对所有人 $j \in 1, \ldots, m-1$.
此外，在这种情况下 $\gamma$ 是一个双曲周期轨道，我们说
• $\gamma$ 是一个周期性汇如果 $\left|\lambda_j\right|<1$ 对所有人 $j \in 1, \ldots, m-1$.
• $\gamma$ 是周期性来源, 如果 $\left|\lambda_j\right|>1$ 对所有人 $j \in 1, \ldots, m-1$.
• $\gamma$ 是一个鞍周期肍九道如果 $\gamma$ 既不是周期:汇也不是周期源。
下一个结果建立了特征乘子与周期轨道附近的 Poincaré 映射之间的关系。

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## 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。

Posted on Categories:EE代写, 电子代写, 连续线性系统

## avatest™帮您通过考试

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## EE代写|连续线性系统代写Continous Time Linear System代考|Invariant sets

Invariant sets can be seen as the most basic building blocks for the understanding of the behaviour of a given dynamical system. These sets have the following property: trajectories starting in the invariant set, remain in the invariant set, for all of their future, and all of their past.

Definition 2.2.1. Let $S \subset \mathbb{R}^n$ be a set. Then $S$ is said to be invariant under the dynamics of (3) if for any $\boldsymbol{x}_0 \in S$ we have $\phi^t\left(\boldsymbol{x}_0\right) \in S$ for all $t \in \mathbb{R}$. If we restrict ourselves to positive times $t$ then we refer to $S$ as a positively invariant set and, for negative time, as a negatively invariant set.

We will now discuss some special invariant sets: orbits, equilibria, periodic orbits, and limit sets. We will introduce other invariant sets later in this notes.

Let $\boldsymbol{x}0 \in \mathbb{R}^n$ be a point in the phase space of (3). The orbit through $\boldsymbol{x}{\mathbf{o}}$, which we denote by $\mathcal{O}\left(\boldsymbol{x}0\right)$, is the set of points in phase space that lie on a trajectory of (3) passing through $\boldsymbol{x}_0$ : $$\mathcal{O}\left(\boldsymbol{x}_0\right)=\left{\phi^t\left(\boldsymbol{x}{\mathbf{o}}\right): t \in \mathbb{R}\right}$$
The positive semiorbit through $\boldsymbol{x}0$ is the set $$\mathcal{O}^{+}\left(x_0\right)=\left{\phi^t\left(x{\mathbf{o}}\right): t \geq 0\right} .$$
and the negative semiorbit through $\boldsymbol{x}0$ is the set $$\mathcal{O}^{-}\left(\boldsymbol{x}_0\right)=\left{\phi^t\left(\boldsymbol{x}{\mathbf{o}}\right): t \leq 0\right} .$$
Note that for any $t \in \mathbb{R}$ we have that $\mathcal{O}\left(\phi^t\left(\boldsymbol{x}_0\right)\right)=\mathcal{O}\left(\boldsymbol{x}_0\right)$.
A point $\boldsymbol{p} \in \mathbb{R}^n$ is an equilibrium for the flow of (3) if $\phi^t(\boldsymbol{p})=\boldsymbol{p}$ for all $t \in \mathbb{R}$. Since the flows we consider here are solutions of differential equations, we obtain that an equilibrium $\boldsymbol{p}$ for the flow $\phi^t$ of the differential equation (3) must satisfy $f(\boldsymbol{p})=\mathbf{0}$. Equilibria are solutions that do not change in time, thus providing the most simple example of invariant sets.

A point $\boldsymbol{p} \in M$ is a periodic point of period $T$ for the flow of (3) if there exists some positive number $T \in \mathbb{R}$ such that $\phi^T(\boldsymbol{p})=\boldsymbol{p}$ and $\phi^t(\boldsymbol{p}) \neq \boldsymbol{p}$ for every $0<t<T$. The orbit $\mathcal{O}(\boldsymbol{p})$ of a periodic point is called a periodic orbit.

## EE代写|连续线性系统代写Continous Time Linear System代考|Stability

We will now discuss the notions of Lyapunov stability and asymptotic stability. Such notions can be intuitively stated as follows: if $\boldsymbol{p}$ is a Lyapunov stable point then for every point $\boldsymbol{q}$ close enough to $\boldsymbol{p}$ its orbit stays close to the orbit of $\boldsymbol{p}$; if $\boldsymbol{p}$ is asymptotically stable if it is Lyapunov stable and for every point $\boldsymbol{q}$ close enough to $\boldsymbol{p}$ the forward orbit of $\boldsymbol{q}$ will converge to the forward orbit of $\boldsymbol{p}$.

Definition 2.3.1 (Lyapunov stability). The orbit of a point $\boldsymbol{p} \in \mathbb{R}^n$ is Lyapunov stable by the flow $\phi^t$ of (3) if for any $\epsilon>0$ there is $\delta>0$ such that if $|\boldsymbol{q}-\boldsymbol{p}|<\delta$, then $$\left|\phi^t(\boldsymbol{q})-\phi^t(\boldsymbol{p})\right|<\epsilon$$ for all $t \geq 0$. The orbit of a point $\boldsymbol{p} \in \mathbb{R}^n$ which is not stable is said to be unstable. Definition 2.3.2 (Asymptotic stability). The orbit of a point $\boldsymbol{p} \in \mathbb{R}^n$ is asymptotically stable by the flow $\phi^t$ of (3) if it is Lyapunov stable and there exists a neighbourhood $V$ of $\boldsymbol{p}$ such that for every $\boldsymbol{q} \in V$, $$\left|\phi^t(\boldsymbol{q})-\phi^t(\boldsymbol{p})\right| \rightarrow 0,$$ as $t$ tends to infinity. Example 2.3.3. Consider again the logistic model for population growth: $$\dot{x}=a x\left(1-\frac{x}{K}\right), \quad x \in \mathbb{R}, a, K>0 .$$
From the analysis of its phase portrait, we obtain that $x=0$ is an asymptotically stable equilibrium while $x=K$ is an unstable equilibrium.

For an example of a Lyapunov stable equilibrium which is not asymptotically stable, consider the two dimensional system
$$\left{\begin{array}{l} \dot{u}=v \ \dot{v}=-u, \quad(u, v) \in \mathbb{R} \times \mathbb{R} . \end{array}\right.$$

## EE代写|连续线性系统代写Continous Time Linear System代考|Invariant sets

、left 缺少或无法识别的分隔符

《left 缺少或无法识别的分隔符

、left 缺少或无法识别的分隔符

## EE代写|连续线性系统代写Continous Time Linear System代考|Stability

\$\$
$\backslash$ left {
$$\dot{u}=v \dot{v}=-u, \quad(u, v) \in \mathbb{R} \times \mathbb{R} .$$
、正确的。
$\$ \

EE代写|连续线性系统代写Continous Time Linear System代考 请认准UprivateTA™. UprivateTA™为您的留学生涯保驾护航。

## 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。