26 lines
1.2 KiB
Markdown
26 lines
1.2 KiB
Markdown
We're talking all about stability
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>[!note] Autonomous vs. Nonautonomous Systems
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>**Autonomous**: $\dot x = X(x)$
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>**Non-Autonomous:** $\dot x = X(x,t)$
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We talk about stability usually meaning that things settle to an equilibrium point. But this isn't the only way to look at things...
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# Poincare Stability (Path Stability)
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For autonomous systems. Basically, adhere to a path for disturbances.
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## Types of Paths
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### Standard Path
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$x^*$ is a phase path or equilibrium point whose stability is in question.
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This is a solution of $\dot x = X$
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### 'Half-path' or 'Half-orbit' or 'Semi-orbit'
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1. Start on $a^*$ and travel on half-path $\mathcal{H}^*$
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2. $x^*(t_0) = a^*$
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$x^*$ is **Poincare stable** if all sufficiently small disturbances of the initial value $a^*$ lead to half-paths that remain a small distance from $\mathcal{H}^*$.
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![[Pasted image 20241028151006.png]]![[Pasted image 20241028151117.png]]
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## How do we define distances?
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$$\text{dist}[x, c] = \min_{y \in C}|x-y|$$
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Where c is a curve.
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Where in the plane we're using the minimum of the 2 norm.
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Stable half-paths can be generally stable, approaching an equilibrium, or periodic.
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Unstable half-paths exceed the bound $\epsilon$ somewhere. |