Mappings and Lyapunov/Asymptotic stability

[fionamb831]

Hi! This is a very general question. I am trying to get my head around the concept of mappings in relation to non-linear dynamical systems. I have that
The autonomous vector field $$\dot{x} = f(x)$$,
$$x \in \bf{R}^n$$
has a fixed point
$f(\bar{x}) = 0 $

and

DEFINITION (LIAPUNOV STABILITY) $\bar{x}(t)$ is said to be stable (or
Liapunov stable) if, given $\epsilon  > 0$, there exists a $\delta = \delta(\epsilon) > 0$ such that,
for any other solution, $y(t)$, of the equation above satisfying $|x(t_o) - y(t_o)| < \delta$, then
$|\bar{x}(t) - y(t)| < \epsilon$  for $t > t_o, t_o \in  \bf{R}$

There is also a definition for asymptotic stability. At the start of the book Introduction to Applied linear dynamical systems and Chaos, by Stephen Wiggins, he talks for only a small part about maps, but then asks the reader to go about finding the definitions for liapunov and asymptotic stability for maps yourself. He also says that it is very similar to the vector field methods that he has outlined.

So I suppose what I want to know here is how I would go about describing these concepts using a mapping. I know the mapping I should be using is a bi-infinite sequence of points, I have never worked with maps up to this point so this is completely new to me. If anyone knows what the definitions are or how to go about finding them it would be a great help. It's right at the start of the book so I feel I need to get my head around this before I go further.

 

[Aaron Bex]

To answer your question, you can think of a mapping in terms of a discrete version of a vector field. That is, rather than having a continuous flow of a vector field, you have a discrete set of points that are mapped from one iteration to the next. In this case, Liapunov stability for a map would mean that given some epsilon (a tolerance for error), there exists a delta (the distance between two points) such that if two points are within this delta, then their distance will remain less than epsilon for all future iterations. Asymptotic stability for a map would mean that the distance between two points will tend to zero as the number of iterations increases.

Hope this helps!