The anharmonic route to chaos: kneading theory

Abstract

The kneading theory for maps which model the anharmonic route to chaos is developed. The authors show that the transition to chaos in a range of problems is via a sequence of period-doubling and homoclinic bifurcations, and that this route to chaos is robust in the sense that if a family of differential equations undergoes this transition to chaos, then so do sufficiently close families. The sequence of bifurcations generates orbits of period (pn), n>or=1, related by pn+1=2pn+(-1)nk which exist for maps on the boundary of chaos.