## Self-similarity and Fine Structures in the Linear-logistic Map

**B. L. Tan****T. T. Chia***Department of Physics,**National University of Singapore,**Kent Ridge, Singapore 119260*

#### Abstract

From the graphs of the Lyapunov exponent of the linear-logistic map, (which is linear in the region on the left of its maximum but logistic-like on its right), against its parameter *r*, the stable cycles within the chaotic region can be classified into six different patterns. In each of these patterns, at least one increasing arithmetic progression is present and decreasing arithmetic progressions may also be present as well; all the terms of these progressions are intermingled with chaos. One of these patterns is self-similar, while in another the period-adding phenomenon occurs. All the patterns are interrelated through their dependence on the same variables.