Chapter 21

An immensely influential paper in the history of the study of chaos was the work of Ruelle and Takens [1], later extended with Newhouse [2] on the robustness of quasiperiodic dynamics with 3 or more incommensurate frequencies. This addressed the question of whether complexity in dynamics was likely to occur through the accumulation of Hopf-like bifurcations adding additional frequencies (and presumably spatial “modes”), or instead through the onset of low dimensional chaos. The statement of the theorem was quite mathematical, which led initially to a misunderstanding of the strength of the result in the physics community, but nevertheless it remains a powerful result.