You dived in deep to one of the questions we asked in our exploration of the Feigenbaum Plot: how many cycles of each type are there? It turns out that for every cycle born out of chaos, there is an “evil twin” repelling cycle born as well. As a result, there are actually two 3-cycles. Moreover, not all cycles are necessarily bound by real numbers, instead moving to the realm of complex numbers. This results in many more 4-, 5-, 6-cycles and beyond.
Furthermore, these “evil twin” repelling cycles serve a purpose. As the members of the attracting cycle bifurcate, their patterns branch out further and further. When this pattern of branching reaches one of the values of the repelling cycle – indeed it will reach all members of the repelling cycle simultaneously – the branching stops and the system dissolves into chaos once more.