2.2 KiB
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'
- Start on
a^*and travel on half-path\mathcal{H}^* x^*(t_0) = a^*x^*is Poincare stable if all sufficiently small disturbances of the initial valuea^*lead to half-paths that remain a small distance from\mathcal{H}^*. !
!
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 ofx = X(x,t). Then,
x^*is lyapunov stable iff for each value of\epsilon>0however small there is a corresponding value of\delta>0such that !- If the system is autonomous, then we can disregard the idea of
t_0in 1.- 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: