Unstable attractors: existence and stability indices

We show that unstable attractors do not exist for smooth invertible dynamics. In systems lacking these properties, we draw simple conclusions about their stability indices and look at examples highlighting extreme cases of stability and attractiveness - characterized in terms of stability indices. In particular, we investigate the possibilities for great discrepancies between the local and non-local indices sigma(loc)(x) and sigma(x), also depending on properties of the system. We show that while sigma(loc)(x) = -infinity holds for all unstable attractors, it is not straightforward to uniquely identify them using stability indices.