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A Verified ODE Solver and the Lorenz Attractor.

Fabian Immler1

  • 1Institut für Informatik, Technische Universität München, Munich, Germany.

Journal of Automated Reasoning
|August 3, 2018
PubMed
Summary
This summary is machine-generated.

This study formally verifies computations proving chaos in the Lorenz attractor using a rigorous numerical algorithm. The verified algorithm ensures the reliability of mathematical proofs for complex dynamical systems.

Keywords:
Isabelle/HOLLorenz attractorOrdinary differential equationPoincaré mapRigorous numerics

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Area of Science:

  • * Computational Mathematics
  • * Dynamical Systems Theory
  • * Formal Verification

Background:

  • * The Lorenz attractor is a foundational model in chaos theory, demonstrating sensitive dependence on initial conditions.
  • * Tucker's original computations provided evidence of chaos but lacked formal verification.
  • * Formal methods are increasingly applied to validate complex scientific computations.

Purpose of the Study:

  • * To formally verify the numerical computations used to prove chaos for the Lorenz attractor.
  • * To establish the reliability and correctness of mathematical proofs in chaos theory.
  • * To bridge the gap between theoretical mathematics and computational evidence.

Main Methods:

  • * Development of a rigorous numerical algorithm with formal verification using Isabelle/HOL.
  • * Formalization of key mathematical concepts including ordinary differential equations and Poincaré maps.
  • * Implementation of approximation schemes (Runge-Kutta, affine arithmetic) and reachability analysis with adaptive control.

Main Results:

  • * Successful formal verification of the numerical computations underpinning the proof of chaos for the Lorenz attractor.
  • * Demonstration of a formalized mathematical framework encompassing differential equations and algorithmic methods.
  • * Systematic refinement of algorithms for execution on original input data.

Conclusions:

  • * The formal verification confirms the validity of Tucker's findings on the Lorenz attractor's chaotic nature.
  • * This work highlights the power of formal methods in certifying complex scientific results.
  • * The developed methodology can be applied to verify other dynamical systems and computational proofs.