Period 3 Bifurcation for the Logistic Mapping

Abstract

In the context of continuous mappings of the interval, one of the most striking features may be Sharkovsky's theorem [6] which, among other thing, shows that the existence of a period 3 point implies the existence of periodic points of every period (see also [2, 5]). Therefore, for a one-parameter family of interval mappings, the determination of period 3 bifurcation points may be interesting. In recent years, the logistic mapping f(x) = 1 - x2 has been entensively studied ([1, 4]). By using computer simulation for this family f(x), as the parameter is increased from 0, we can observe the Feigenbaum "cascades" [3]. That is, stable periodic points of double periods accumulate in a geometric and universal way. As the parameter is approximately equal to 1.7498 ([1, p.129]), there seems to be a period 3 bifurcation. In this note, we show that this family f(x) does have a period 3 bifurcation exactly at = 7/4.