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# EE代写|连续线性系统代写Continous Time Linear System代考|ECE321 A tool for detecting chaos: Lyapunov exponents

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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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