# Robust chaos with prescribed natural invariant measure and Lyapunov exponent

## TL;DR

This paper develops methods to construct one-dimensional chaotic maps with guaranteed robustness, specific invariant measures, and fixed or varying Lyapunov exponents, enhancing control over chaotic system properties.

## Contribution

It introduces new techniques for designing robust chaos in one-dimensional maps with prescribed invariant measures and Lyapunov exponents, including parameter-dependent and parameter-independent approaches.

## Key findings

- Constructed maps with robust chaos and fixed invariant measure.
- Developed methods for maps with varying Lyapunov exponents.
- Provided exact invariant measure calculations in specific cases.

## Abstract

We extend in several ways a recently proposed method to construct one-dimensional chaotic maps with exactly known natural invariant measure [Sogo 1999, 2009]. First, we assume that the given invariant measure depends on a continuous parameter and show how to construct maps with robust chaos --i.e., chaos that is not destroyed by arbitrarily small changes of the parameter-- and prescribed invariant measure and constant Lyapunov exponent. Then, by relaxing one condition in the approach of Refs. \cite{Sogo1,Sogo2}, we describe a method to construct robust chaos with prescribed constant invariant measure and varying Lyapunov exponent. Another extension of a condition in Refs. [Sogo 1999, 2009] provides a new method to get robust chaos with known varying Lyapunov exponent. In this third approach the invariant measure can be computed exactly in many particular cases. Finally we discuss how to use diffeomorphisms to construct maps with robust chaos, any number of parameters and prescribed invariant measure and Lyapunov exponent.