1.1 Basic Concepts and Geometric Approach

Phase Space

The set of all $\mathbf{x}=(x_1,x_2,\dots ,x_n)\in \mathrm{\Omega }\subseteq {\mathbb{R}}^n$ possible states of a system, $\mathrm{\Omega }$, is the phase space

Example: $x(t)$ is the amount of population, the phase space $\mathrm{\Omega }=[0,+\mathrm{\infty }]$

These processes are usually deterministic, finite dimensional, and differentiable

Vector Field

Vector Fields is an assignment of a vector $\mathbf{F}(t,x)=(f_1(t,x),\dots ,f_n(t,x))\in {\mathbb{R}}^n$ for each $t\in I\subseteq \mathbb{R}$ and $x\in \mathrm{\Omega }\subseteq {\mathbb{R}}^n$

The vector $x$ is a state and the vector fields generate the dynamic

First-Order ODE

Flow

Let $x_0\in \mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be the initial state and $I\subseteq \mathbb{R}$ be an open interval containing $0$.
Assume that $x(t),t\in I$ solves the dynamical system (1.2) with initial condition $x(0)=x_0$.
We call the map:

the flow of the system, denoted as ${\varphi }^t(x_0)$.
Where ${\varphi }^t(x_0)=x(t)$ but it's dependant on the initial state.

Semigroup Property

For all $s_1,s_2\in \mathbb{R}$ and for all $\mathbf{x}\in U$, the following hold:

1. ${\varphi }^0(\mathbf{x})=\mathbf{x};$

2. ${\varphi }^{s_2}({\varphi }^{s_1}(\mathbf{x}))={\varphi }^{s_1}({\varphi }^{s_2}(\mathbf{x}))={\varphi }^{s_1+s_2}(\mathbf{x}),$
   as long as the flow is well-defined for $s_1,s_2$.

Autonomous System

Consider the dynamical system $\dot{\mathbf{x}}(t)=F(t,\mathbf{x}(t))$.
If $F$ does not depend on $t$ explicitly, then the system is called autonomous, otherwise the system is non-autonomous.

More precisely, an autonomous system is of the following form:

Second Order

If we have $\ddot{x}(t)=\mathbf{F}(t,x(t),\dot{x}(t))$ then we can add an addition variable $y=\dot{x}$