Li and Yorke (1975) proved that any 1-D system which exhibits a regular Cycle of period three will also display regular Cycles of every other length as well as completely Chaotic Cycles.
See also Chaos, Cycle (Map)
Li, T. Y. and Yorke, J. A. ``Period Three Implies Chaos.'' Amer. Math. Monthly 82, 985-992, 1975.