Unusual period doublings in the linear-logistic map

Abstract

In the route to chaos, the linear-logistic map g(x), consisting of a left linearly increasing portion joined to a right logistic portion at its maximum, has an unusual behavior. At every other period-doubling bifurcation point, the ''slope'' of gm(x) at any of the m stable fixed points is 0 instead of -1 for the usual period-doubling route. This map possesses rather unusual features in the Lyapunov exponent versus parameter graphs, the gm(x) graphs, the values of the scale reduction factor, and that of the superstable parameters. These properties suggest that either every other period doubling is different from the Feigenbaum type, or that all the period doublings are usual but with changes occurring extremely rapidly at every other one. (c) 1995 The American Physical Society © 1995 The American Physical Society.