Chaos in the cubic mapping

AbstractMotivated by a problem in genetics involving one locus with two alleles, R. M. May gave the example of the family of cubic maps x→ax3 + (1−a)x of the interval [-1,1] and established the existence of a bifurcating sequence of cycles of period 2n over an interval of parameter values. This paper extends the analysis of May beyond that region. We find a critical value a∗ beyond which the map has a snapback repeller and hence chaotic behavior; this value further marks the onset of the first cycle of odd order >1. When a=4 the map is onto the interval and we find an associated invariant measure