Li and Yorke (1975) proved that any one-dimensional system which exhibits a regular cycle of period three will also display regular cycles of every other length as well as completely chaotic cycles.
Period Three Theorem
See also
Chaos, Map Cycle, Sharkovsky's TheoremExplore with Wolfram|Alpha
References
Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, p. 79, 2003.Li, T. Y. and Yorke, J. A. "Period Three Implies Chaos." Amer. Math. Monthly 82, 985-992, 1975.Referenced on Wolfram|Alpha
Period Three TheoremCite this as:
Weisstein, Eric W. "Period Three Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PeriodThreeTheorem.html