Stability of Dynamical Systems: Linearization, Eigenvalue Tests, and Lyapunov Methods

All flashcards

What is a perturbation?

A small change to the initial state or initial data of a system, used to test how its later behavior changes.

What is an orbit?

The trajectory traced by a solution as the system evolves through its state space.

What does Lyapunov stability mean for an equilibrium?

For every desired closeness $\epsilon>0$, there is a sufficiently small initial-distance threshold $\delta>0$ that keeps the solution within $\epsilon$ of the equilibrium for all future time.

What extra property defines asymptotic stability?

In addition to remaining close under small perturbations, nearby solutions converge to the equilibrium as time tends to infinity.

Give one distance measure used to compare functions.

An $L^p$ norm or the supremum norm can measure distances between functions.

Quiz questions

1. For a continuous-time linear system $\dot{x}=Ax$, what does a strictly negative real part of an eigenvalue indicate about its associated direction?Answer: Perturbations in that direction decay in forward time.
2. For a two-dimensional linear system $\dot{X}=AX$, what does $\det A<0$ imply about the origin?Answer: The origin is a saddle.
3. Near what kind of equilibrium does the Hartman–Grobman theorem relate nonlinear dynamics to the linearization?Answer: A hyperbolic equilibrium, with no linearization eigenvalue having zero real part.
4. For the linear flow $\dot{X}=AX$, what does $\operatorname{tr}A=0$ imply about the flow map?Answer: It preserves area.
5. For a differentiable discrete map near a fixed point, what does an eigenvalue of the Jacobian with magnitude greater than one indicate?Answer: The fixed point is unstable.