  Let (X,T) be a topologically transitive dynamical system. We show that if
there is a subsystem (Y,T) of (X,T) such that (X\times Y, T\times T) is
transitive, then (X,T) is strongly chaotic in the sense of Li and Yorke. We
then show that many of the known sufficient conditions in the literature, as
well as a few new results, are corollaries of this statement. In fact, the kind
of chaotic behavior we deduce in these results is a much stronger variant of
Li-Yorke chaos which we call uniform chaos. For minimal systems we show, among
other results, that uniform chaos is preserved by extensions and that a minimal
system which is not uniformly chaotic is PI.
