Robust chaos with variable Lyapunov exponent in smooth one-dimensional maps

TL;DR

This paper introduces new methods for creating smooth one-dimensional maps with robust chaos, where the Lyapunov exponent varies with the parameter, challenging previous assumptions about necessary conditions for such chaos.

Contribution

The paper presents novel techniques for generating robust chaotic maps with variable Lyapunov exponents, removing the need for negative Schwarzian derivative conditions.

Findings

Maps exhibit robust chaos over parameter intervals.

Lyapunov exponent varies widely with parameters.

Previous maps are conjugated to the tent map with fixed Lyapunov exponent.

Abstract

We present several new easy ways of generating smooth one-dimensional maps displaying robust chaos, i.e., chaos for whole intervals of the parameter. Unlike what happens with previous methods, the Lyapunov exponent of the maps constructed here varies widely with the parameter. We show that the condition of negative Schwarzian derivative, which was used in previous works, is not a necessary condition for robust chaos. Finally we show that the maps constructed in previous works have always the Lyapunov exponent $\ln 2$ because they are conjugated to each other and to the tent map by means of smooth homeomorphisms. In the methods presented here, the maps have variable Lyapunov coefficients because they are conjugated through non-smooth homeomorphisms similar to Minkowski's question mark function.