Computational Complexity of Iterated Maps on the Interval

TL;DR

This paper presents an algorithm for accurately computing orbits of interval maps in dynamical systems, analyzing its complexity in relation to Lyapunov exponents, and proving its optimality.

Contribution

It introduces a general, correct, and optimal algorithm for computing orbits of interval maps with complexity analysis linked to Lyapunov exponents.

Findings

Algorithm is proven correct and optimal.

Complexity measure relates to Lyapunov exponent.

Provides a general approach for arbitrary-precision orbit computation.

Abstract

The correct computation of orbits of discrete dynamical systems on the interval is considered. Therefore, an arbitrary-precision floating-point approach based on automatic error analysis is chosen and a general algorithm is presented. The correctness of the algorithm is shown and the computational complexity is analyzed. There are two main results. First, the computational complexity measure considered here is related to the Lyapunov exponent of the dynamical system under consideration. Second, the presented algorithm is optimal with regard to that complexity measure.