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Rigorous computation of expansion in one-dimensional dynamics.

Paweł Pilarczyk1, Michał Palczewski2, Stefano Luzzatto3

  • 1Faculty of Applied Physics and Mathematics & Digital Technologies Centre, Gdańsk University of Technology, ul. Gabriela Narutowicza 11/12, 80-233 Gdańsk, Poland.

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Summary
This summary is machine-generated.

We developed a rigorous numerical method using interval arithmetic to calculate lower bounds for uniform expansion in one-dimensional dynamics. This quantitative approach offers verifiable results for practical models, unlike many qualitative theories.

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Area of Science:

  • Dynamical Systems Theory
  • Numerical Analysis
  • Computer-Assisted Proofs

Background:

  • Qualitative results in dynamical systems often lack verifiable assumptions and practical applicability.
  • Quantitative methods are needed for rigorous analysis of natural phenomena modeled by dynamical systems.

Purpose of the Study:

  • To introduce an effective algorithmic method for computing a rigorous lower bound for uniform expansion in 1D dynamics.
  • To provide a quantitative, computer-assisted proof applicable to real-world models.

Main Methods:

  • Utilizes interval arithmetic for rigorous numerical computation.
  • Employs efficient graph algorithms and an iterative approach for optimal performance.
  • Develops a publicly available software implementation.

Main Results:

  • Successfully computed a lower bound for uniform expansion in one-dimensional dynamics.
  • Demonstrated the effectiveness of the method through application to the quadratic map family.
  • Provided a verifiable quantitative result in dynamical systems theory.

Conclusions:

  • The developed method offers a rigorous and practical approach to analyzing uniform expansion in dynamical systems.
  • This quantitative method enhances the reliability and applicability of theoretical findings in modeling natural phenomena.