  In this article we prove that for a diffeomorphism on a compact Riemannian
manifold, if there is a nontrival homoclinic class that is not uniformly
hyperbolic or the diffeomorphism is a $C^{1+\alpha}$ and there is a hyperbolic
ergodic measure whose support is not uniformly hyperbolic, then we find a type
of strongly distributional chaos which is stronger than usual distributional
chaos and Li-Yorke chaos in the set of irregular orbits that are not uniformly
hyperbolic. Meanwhile, we prove that various fractal sets are strongly
distributional chaotic, such as irregular sets, level sets, several recurrent
level sets of points with different recurrent frequency, and some intersections
of these fractal sets. In the process of proof, we give an abstract general
mechanism to study strongly distributional chaos provided that the system has a
sequence of nondecreasing invariant compact subsets such that every subsystem
has exponential specification property, or has exponential shadowing property
and transitivity. The advantage of this abstract framework is that it is not
only applicable in systems with specification property including transitive
Anosov diffeomorphisms, mixing expanding maps, mixing subshifts of finite type
and mixing sofic subshifts but also applicable in systems without specification
property including $\beta$-shifts, Katok map, generic systems in the space of
robustly transitive diffeomorphisms and generic volume-preserving
diffeomorphisms.
