THE PERIODICITY OF CHAOTIC IMPACT OSCILLATORS IN HAUSDORFF PHASE SPACES

It is well known that non-periodic behavior is one of the most puzzling characteristics of chaotic oscillators. So far chaotic dynamical systems have been investigated in Euclidean spaces. In this paper, the concept of non-autonomous dynamical systems and that of Hausdorff phase spaces are proposed. The behavior of chaotic impact oscillators is investigated in Hausdorff phase spaces. It is discovered that, although the non-autonomous dynamical systems 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 periodicity of chaotic dynamical systems in Hausdorff metric spaces induces a new class of strange invariant sets in Euclidean spaces. Such strange invariant sets may be an ideal symbol of chaotic dynamical systems.