CDS - Definition of ODE • Definition of a Solution • Definition of Flow • Dynamics • Definition of Phase Plane or Phase Space

Remember:

Definition of ODE (Ordinary Differential Equation):$\dot{x}=f(x):\begin{array}{l}x\in {\mathbb{R}}^n,\\ f\in U\to {\mathbb{R}}^n\end{array}$Where:$\dot{x}=\frac{dx}{dt}$And: $U\subset {\mathbb{R}}^n$ is a differentiable setnot-sure-about-this

Definition of solution: $x(x_0,t):I\to {\mathbb{R}}^n$Where:

• $x_0$ are the “initial condtitons”, $x_0\in {\mathbb{R}}^n$ and are fixed, choosen at priori.
• $t$ is time, $t\in I$, and $I\subset \mathbb{R}$, and in this case is a variable.

We are not interested/we don’t care about solving for $x(t)$ analytically, we will not find the formula, at most we will solve it numerically, that means, solving using numerical data, giving $x(t)$ and $t$ actual values. #IMPORTANT

Definition of ‘flow’:${\varphi }_t:U\to {\mathbb{R}}^n$Where:

• $U$ is the space where $f$ is defined.
• ${\varphi }_t(x)$ is just a fancy way of writing $\varphi (x,t)$.
• Consider the flow as an “image” of the function $\dot{x}=f(x)$ for some $t\in [t_0,t_f]$ and initial state $x_0$, where: $x_0=x(t_0)$.

Here we can see how the flow ${\varphi }_t$ is represented:
In first picture we use a single initial condition. In the second we use many initial conditions, to see how the system evolves, each inital condition corresponds to a curve of the entire flow. The arrows on the curves represent the time.

A single ODE like we have seen before with $x\in {\mathbb{R}}^n$, can also be seen as a system of equation, like so:$\{\begin{array}{l}{\dot{x}}_1=f_1(x_1,\dots ,x_n)\\ {\dot{x}}_2=f_2(x_1,\dots ,x_n)\\ ⋮\\ {\dot{x}}_n=f_n(x_1,\dots ,x_n)\end{array}$Where: $x_1,\dots ,x_n$ are all scalars (meaning $\in {\mathbb{R}}^1$). ${\mathbb{R}}^n$ is called the phase space, (also called phase plane if ${\mathbb{R}}^2$).

Here’s the example in which $x\in {\mathbb{R}}^2$:

• In the first graph we represent the dynamics $x_1(t)$ and $x_2(t)$
  • $x_1(t)$ is called $x(t)$
  • $x_2(t)$ is called $y(t)$
• In the second graph we represnt the flow, however it misses the arrows that indicate the flowing of time.
• If we were to take $x\in {\mathbb{R}}^3$ we would need a 3D graph to represent flow.
• ==The first graph represents the dynamics (and is always represented as a 2D graph), the second phase plane or more in general it is called the phase space==.

If we take many different initial conditions:
When we talk about ${\varphi }_t$ convenitionally we the motion of a familiy of particles, so many particles not just one.

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• We are not interested/we don’t care about solving for $x(t)$ analytically, we will not find the formula, at most we will solve it numerically, that means, solving using numerical data, giving $x(t)$ and $t$ actual values.

• Consider the flow as an “image” of the function $\dot{x}=f(x)$ for some $t=\tau$ and initial state $x_0$, where: $x_0=x(t={\tau }_0)$.

• In first picture we use a single initial condition, in the second we use many initial conditions, to see how the system evolves.

• $x$, so $x_0$ as well, are vectors, while $t$ the time is a scalar, so $t\in \mathbb{R}$.

• Here’s the example in which $x\in {\mathbb{R}}^2$
• In the first graph we represent $x_1(t)$ and $x_2(t)$
  • $x_1(t)$ is called $x(t)$
  • $x_2(t)$ is called $y(t)$
• In the second graph we represnt the flow, however it misses the arrows that indicate the flowing of time.
• If we were to take $x\in {\mathbb{R}}^3$ we would need a 3D graph to represent flow.
• The first graph is called/represents dynamics, the second phase space.

• If we take many different initial conditions