It is well known that non-periodic behavior is one of the most puzzling cha
racteristics of chaotic oscillators. So far chaotic dynamical systems have
been investigated in Euclidean spaces. Tn this paper, the concept of non-au
tonomous dynamical systems and that of Hausdorff phase spaces are proposed.
The behavior of chaotic impact oscillators is investigated in Hausdorff ph
ase spaces. It is discovered that, although the non-autonomous dynamical sy
stems described by chaotic impact oscillators are non-periodic in Euclidean
phase spaces, they are periodic in Hausdorff phase spaces. This shows that
Euclidean spaces in which we stayed for hundreds of years may no longer be
suitable for the investigation into chaotic phenomena. In addition, the pe
riodicity of chaotic dynamical systems in Hausdorff metric spaces induces a
new class of strange invariant sets in Euclidean spaces. Such strange inva
riant sets may be an ideal symbol of chaotic dynamical systems. (C) 2000 Ac
ademic Press.