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# EE代写|连续线性系统代写Continous Time Linear System代考|EE235 Differential equations and Dynamical Systems

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## EE代写|连续线性系统代写Continous Time Linear System代考|Differential equations and Dynamical Systems

A dynamical system is a triple $\left(M, \phi^t, K\right)$ where $M$ is called the phase space and is usually a smooth manifold or a subset of $\mathbb{R}^n, \phi^t: M \times K \rightarrow M$, called the evolution, is a smooth action of $K$ in $M$ and $K$ is either a subset of $\mathbb{R}$ in the case of a continuous time dynamical system or a subset of $\mathbb{Z}$ in the case of a discrete time dynamical system. The smooth action $\phi^t(\boldsymbol{x})$ describes the evolution with time $t \in K$ of a point $\boldsymbol{x}$ in the phase space $M$.

In this notes we study dynamical systems in continuous time, determined by ordinary differential equations. We will introduce some of the basic concepts of the theory, with a special emphasis on the study of examples to illustrate such concepts. All the concepts, statements and its proofs in this notes can be found in classical references such as $[5,8,9,10]$.

Before proceeding to the rigorous treatment of the subject, we will try to provide some motivation with some very simple examples. First of all, we may say that a differential equation is an equality involving a function, and its derivatives. One of the most simple examples of differential equations is provided by
$$\dot{x}=a x, \quad x \in \mathbb{R},$$
where $a$ is a fixed parameter. This equation can be used to model many different situations. For instance, in the case where $a$ is a positive real number, this equation may be used to model the growth of a population with unlimited resources or the effect of continuously compounding interest. When $a$ is a negative number, it can model radioactive decay. Due to its very simple form, an explicit expression for the solutions can be obtained:
$$x(t)=x_0 \mathrm{e}^{a t}, \quad t \in \mathbb{R}$$
where $x_0$ is the value of $x(t)$ when $t=0$. If $a>0$, the solution tends to infinity as $t$ goes to infinity, while the solution tends to zero if $a<0$. If $a=0$ all solutions are constant.

## EE代写|连续线性系统代写Continous Time Linear System代考|Existence and uniqueness of solutions

We will study equations of the form
$$\dot{\boldsymbol{x}}=f(\boldsymbol{x}, t ; \mu),$$
with $x \in U \subseteq \mathbb{R}^n, t \in \mathbb{R}$, and $\mu \in V \subseteq \mathbb{R}^p$ where $U$ and $V$ are open sets in $\mathbb{R}^n$ and $\mathbb{R}^p$, respectively. The overdot in (1) means $\frac{\mathrm{d}}{\mathrm{d} t}$, and $\mu$ is seen as a parameter. The independent variable $t$ is often referred to as time. We refer to (1) as a ordinary differential equation.

A solution of (1) is a map, $\boldsymbol{x}$, from some interval $I \subseteq \mathbb{R}$ into $\mathbb{R}^n$, which we represent as
\begin{aligned} \boldsymbol{x}: & I \rightarrow \mathbb{R}^n \ & t \mapsto \boldsymbol{x}(t) \end{aligned}
such that $\boldsymbol{x}(t)$ satisfies (1), i.e.
$$\dot{\boldsymbol{x}}(t)=f(\boldsymbol{x}(t), t ; \mu) .$$
The map $\boldsymbol{x}(t)$ has the geometrical interpretation of a curve in $\mathbb{R}^n$, and (1) gives the tangent vector at each point of the curve.

## EE代写|连续线性系统代写CONTINOUS TIME LINEAR SYSTEM 代考|DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS

$$\dot{x}=a x, \quad x \in \mathbb{R},$$

$$x(t)=x_0 \mathrm{e}^{a t}, \quad t \in \mathbb{R}$$

## EE代写|连续线性系统代写CONTINOUS TIME LINEAR SYSTEM 代考|EXISTENCE AND UNIQUENESS OF SOLUTIONS

$$\dot{x}=f(x, t ; \mu)$$

$$\boldsymbol{x}: I \rightarrow \mathbb{R}^n \quad t \mapsto \boldsymbol{x}(t)$$

$$\dot{\boldsymbol{x}}(t)=f(\boldsymbol{x}(t), t ; \mu) .$$

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