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Rigorous computation of expansion in one-dimensional dynamics

Paweł Pilarczyk, Michał Palczewski, Stefano Luzzatto

TL;DR

This work provides a rigorous, computer-assisted framework to certify a lower bound $\lambda$ on the uniform expansion exponent outside a critical neighborhood for one-dimensional maps, notably the quadratic family. It combines interval arithmetic with a graph-theoretic representation and an adaptive, dynamically defined partition to overcome the inefficiencies of uniform partitions, and it includes a method to prove optimality via the existence of a periodic orbit with an upper bound $\lambda_{\max}$. The approach yields substantially tighter expansion bounds at lower computational cost than prior methods and offers detailed comparisons, complexity analyses, and a publicly available software implementation. By delivering explicit, verifiable bounds, this work advances quantitative understanding of chaotic dynamics and supports computer-assisted parameter-exclusion arguments for stochastic parameters, with potential extensions to broader map classes and higher dimensions.

Abstract

We introduce an effective algorithmic method for the computation of a lower bound for uniform expansion in one-dimensional dynamics. The approach employs interval arithmetic and thus provides a rigorous numerical result (computer-assisted proof). The method uses efficient graph algorithms and an iterative approach for optimal performance. A software implementation of the method is made publicly available. This is an example of a quantitative result in the theory of dynamical systems, as opposed to many qualitative results whose assumptions may be difficult to verify and the conclusions may have limited use in practical models that describe natural phenomena. We discuss and illustrate the effectiveness of our method and apply it to the quadratic map family.

Rigorous computation of expansion in one-dimensional dynamics

TL;DR

This work provides a rigorous, computer-assisted framework to certify a lower bound on the uniform expansion exponent outside a critical neighborhood for one-dimensional maps, notably the quadratic family. It combines interval arithmetic with a graph-theoretic representation and an adaptive, dynamically defined partition to overcome the inefficiencies of uniform partitions, and it includes a method to prove optimality via the existence of a periodic orbit with an upper bound . The approach yields substantially tighter expansion bounds at lower computational cost than prior methods and offers detailed comparisons, complexity analyses, and a publicly available software implementation. By delivering explicit, verifiable bounds, this work advances quantitative understanding of chaotic dynamics and supports computer-assisted parameter-exclusion arguments for stochastic parameters, with potential extensions to broader map classes and higher dimensions.

Abstract

We introduce an effective algorithmic method for the computation of a lower bound for uniform expansion in one-dimensional dynamics. The approach employs interval arithmetic and thus provides a rigorous numerical result (computer-assisted proof). The method uses efficient graph algorithms and an iterative approach for optimal performance. A software implementation of the method is made publicly available. This is an example of a quantitative result in the theory of dynamical systems, as opposed to many qualitative results whose assumptions may be difficult to verify and the conclusions may have limited use in practical models that describe natural phenomena. We discuss and illustrate the effectiveness of our method and apply it to the quadratic map family.

Paper Structure

This paper contains 45 sections, 4 theorems, 21 equations, 6 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Uniform expansion outside the critical neighbourhood
  3. Rigorous Explicit Estimates for Uniform Expansion
  4. Background
  5. Our contribution
  6. Formal Assumptions
  7. Assumptions on the map
  8. Assumptions on the numerical methods
  9. Rigorous numerics
  10. Overview of the paper
  11. Weighted Directed Graphs
  12. Main definitions
  13. Graph representation of a map
  14. Minimum cycle mean
  15. Main algorithms
  16. ...and 30 more sections

Key Result

Proposition 1

Let $G$ be a representation of a map $f \colon I \to I$ that has at least one cycle. Let $\lambda = \mu(G)$. Then $\lambda$ is a lower bound on the expansion exponent of $f$ on $I \setminus \Delta$.

Figures (6)

  • Figure 1: The lower bound $\lambda$ on the expansion exponent computed for the quadratic map $f_a$ with the new method and with the previous approach, as discussed in Section \ref{['sec:new_vs_old']}. The bifurcation diagram is shown for reference. (Colour online.)
  • Figure 2: The lower bound $\lambda$ on the expansion exponent computed for the quadratic map $f_a$ outside the critical neighbourhood of various radii $\delta > 0$. The bifurcation diagram is shown for reference. (Colour online.)
  • Figure 3: Comparison of the lower bounds on the expansion exponents obtained using rigorous computation for the quadratic map outside the critical neighbourhood of radius $0.001$ for individual values of parameters and intervals of parameters for the quadratic map and the parameter range $[1.4,2]$. (Colour online.)
  • Figure 4: Boxplots of the differences between estimates obtained for single values of the parameter $a$ and for intervals of parameters. Each box spans from the first quartile to the third quartile of the differences. The whiskers extend to within $1.5$ times the interquartile range from the box.
  • Figure 5: Location of intervals in the phase space $I \setminus \Delta$ that are taken for splitting in the first $100$ iterations of Algorithm \ref{['alg:refine']}.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Definition 1
  • Definition 2: $f$-admissible partition of $I \setminus \Delta$
  • Definition 3: Graph representation of a map
  • Proposition 1: see Day2008
  • proof
  • Proposition 2
  • proof
  • Remark 4
  • Remark 5
  • Proposition 3
  • ...and 3 more