Dynamical systems: Is there always an attractor?

Question

I am just starting out with dynamical systems, and I am wondering if you can have a dynamical system with out any attractors (if so, can you give an example).

Explanations or references backing up answers are appreciated.

Answer 1

There are several systems that don't have any attractors. E.g. all unstable systems are without attractors, as illustrated by $$x_1' = −2 x_1 − 6 x_2 + 8, \; x_2' = 8 x_1 + 4x_2 − 12,$$ as shown in on page 19 of this note (provided by Moo in a comment)