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Variance Continuity for Lorenz Flows.

Wael Bahsoun1, Ian Melbourne2, Marks Ruziboev3

  • 1Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU UK.

Annales Henri Poincare
|July 7, 2020
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The classical Lorenz flow and similar systems obey a Central Limit Theorem (CLT). Researchers proved that the variance within this CLT changes continuously, offering new insights into dynamical systems.

Keywords:
37C1037E05Primary 37A05

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Statistical Mechanics
  • Ergodic Theory

Background:

  • The classical Lorenz flow is a fundamental model in chaos theory, exhibiting sensitive dependence on initial conditions.
  • Central Limit Theorems (CLTs) describe the convergence of probability distributions in statistical mechanics and dynamical systems.
  • Understanding the statistical properties of chaotic systems is crucial for their long-term prediction and analysis.

Purpose of the Study:

  • To investigate the applicability of the Central Limit Theorem (CLT) to the classical Lorenz flow and nearby dynamical systems.
  • To rigorously prove that the variance term within the CLT exhibits continuous variation for these flows.
  • To contribute to the mathematical understanding of statistical properties in chaotic dynamical systems.

Main Methods:

  • Utilizing advanced mathematical analysis and topology to study the Lorenz flow.
  • Applying rigorous proof techniques to establish the properties of the Central Limit Theorem's variance.
  • Employing the epsilon-topology to define the neighborhood of the classical Lorenz flow.

Main Results:

  • Demonstrated that the classical Lorenz flow satisfies a Central Limit Theorem (CLT).
  • Proved the continuous variation of the variance in the CLT for the Lorenz flow and flows in its neighborhood.
  • Established a key mathematical property concerning the statistical behavior of these chaotic systems.

Conclusions:

  • The study confirms the existence of a CLT for the classical Lorenz flow and closely related systems.
  • The continuous variation of the CLT variance provides a refined understanding of the statistical stability of these chaotic flows.
  • This work advances the theoretical framework for analyzing chaotic dynamics using probabilistic methods.