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300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-10-28 Stability.md
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@ -7,11 +7,12 @@ We're talking all about stability
We talk about stability usually meaning that things settle to an equilibrium point. But this isn't the only way to look at things... 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) # Poincare Stability (Path Stability)
For autonomous systems. Basically, adhere to a path for distrubances. For autonomous systems. Basically, adhere to a path for disturbances.
## Types of Paths ## Types of Paths
### Standard Path ### Standard Path
$x^*$ is a phase path or equilibrium point whose stability is in question. $x^*$ is a phase path or equilibrium point whose stability is in question.
This is a solution of $\dot x = X$ This is a solution of $\dot x = X$
### 'Half-path' or 'Half-orbit' or 'Semi-orbit' ### 'Half-path' or 'Half-orbit' or 'Semi-orbit'
1. Start on $a^*$ and travel on half-path $\mathcal{H}^*$ 1. Start on $a^*$ and travel on half-path $\mathcal{H}^*$
2. $ 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 $\mathcal{H}^*$.