Properties of a logistic map with a sectional discontinuity

Abstract

We have studied numerically the properties of the logistic map with a single sectional discontinuity at x=xd. We give the main features of these maps over a wide range of xd, including accumulation points, inverse cascades, bifurcation diagrams, basins of attraction, and a new superimposition rule. We find that the main characteristics of the logistic map with a discontinuity at the origin [T. T. Chia and B. L. Tan, Phys. Rev. A 45, 8441 (1992)], such as the occurrence of inverse cascades, and the validity of rule I, rule II (which are rules for determining whether higher-level cascades exist), and the summation rule, are still retained in these new discontinuous maps, implying that these properties and rules are universal in discontinuous maps. However, there are important differences as well, such as the number of inverse cascades and the types of routes to chaos which may include a period-doubling route, with period doublings occurring at the same values of aPD for different values of xd, and an "alternating" route. Further, we find that in the chaotic regions of these maps, either the modified summation rule holds or there exists the following period-doubling sequence: 2->4→8→16→32→..., which may also exist in the periodic regions, depending on the values of xd. © 1993 The American Physical Society.