We're talking all about stability

>[!note] Autonomous vs. Nonautonomous Systems
>**Autonomous**: $\dot x = X(x)$
>**Non-Autonomous:** $\dot x = X(x,t)$

We talk about stability usually meaning that things settle to an equilibrium point. But this isn't the only way to look at things...

# Poincare Stability (Path Stability)
For autonomous systems. Basically, adhere to a path for disturbances.
## Types of Paths
### Standard Path
$x^*$ is a phase path or equilibrium point whose stability is in question.
This is a solution of $\dot x = X$
### 'Half-path' or 'Half-orbit' or 'Semi-orbit'
1. Start on $a^*$ and travel on half-path $\mathcal{H}^*$ 
2. $x^*(t_0) = a^*$
$x^*$ is **Poincare stable** if all sufficiently small disturbances of the initial value $a^*$ lead to half-paths that remain a small distance from $\mathcal{H}^*$.
![[Pasted image 20241028151006.png]]![[Pasted image 20241028151117.png]]
## How do we define distances?
$$\text{dist}[x, c] = \min_{y \in C}|x-y|$$
Where c is a curve. 
Where in the plane we're using the minimum of the 2 norm.

## Summary
Stable half-paths can be generally stable, approaching an equilibrium, or periodic.
Unstable half-paths exceed the bound $\epsilon$ somewhere.

Poincare cannot handle the time dependency of systems. As a result, we can't really use Poincare to handle real systems. That leads us to.....

# Lyapunov Stability
Basically extend the 2D distance formula we talked about last time to include n dimensions. (May need to analyze complex solutions as well).

Let's define Lyapunov Stability:
>[!note] Lyapunov Stability Definition
>Let $x^*$ be a real or complex solution of $x = X(x,t)$. Then, 
>1. $x^*$ is lyapunov stable iff for each value of $\epsilon>0$ however small there is a corresponding value of $\delta>0$ such that 
>![[Pasted image 20241028152704.png]]
>2. If the system is autonomous, then we can disregard the idea of $t_0$ in 1. 
>3. Otherwise, we call the system unstable in the sense of Lyapunov.

This stability definition defines that for an autonomous system, Lyapunov stability is sufficient for Poincare stability.

**Uniform Stability:** A solution that is stable and $\delta$ does not change with time. For autonomous systems this is the same

**Asymptotic Stability**: