46 lines
2.2 KiB
Markdown
46 lines
2.2 KiB
Markdown
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**: |