Phase Portrait

To understand the behaviour of a system we can either get an explicit solution or we use the phase portrait.

Given a 1D dynamical system:

We can analyze its phase flow by:

1. Draw a graph of the function
2. Find the fixed points $x^{\ast }$ such that $f(x^{\ast })=0$
3. For a non-fixed point figure out is sign. If it's positive it will go right, otherwise it will go left

No Oscillation Phenomenon

In a 1D system we cant have periodic orbit since by IVT if a function increases then returns, it must cross through a fixed point stopping essentially freezing the system.

First Variation Equation of a Scalar System

Let $\dot{x}=f(x)$, $x\in \mathrm{\Omega }\subseteq \mathbb{R}$ be a 1-dimensional dynamical system, where $f(x)$ is differentiable and $f^{\prime }(x)$ is continuous. Suppose that $x_0$ is a fixed point. Then the system

is called the First Variation Equation of the original system.
Since the solution ${\varphi }^t(x_0+v_0)$ will in general not be a fixed point. We are checking our new point $x(0)=x_0+v_0$

So we’re looking at how the perturbation $v(t)$ evolves over time.
Define:
$v(t)={\varphi }^t(x_0+v_0)-x_0$

Consider the fixed point $x_0$ for the diffeq $\dot{x}=f(x)$ where $f$ is differentiable and $C^1$

$f^{\prime }(x_0)<0⟹$ attracting fixed point
$f^{\prime }(x_0)>0⟹$ repelling fixed point
$f^{\prime }(x_0)=0⟹$ the derivative does not determine the stability type