Obsidian/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-10-28 Stability.md

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}^*.