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Multidimensional period doubling structures.

Jeong Yup Lee1, Dvir Flom2, Shelomo I Ben-Abraham2

  • 1Department of Mathematical Education, Catholic Kwandong University, Gangneung, Republic of Korea.

Acta Crystallographica. Section A, Foundations and Advances
|April 30, 2016
PubMed
Summary
This summary is machine-generated.

This study generalizes period doubling sequences to any dimension using substitution and recursion rules. The research demonstrates these higher-dimensional structures exhibit pure point diffractive properties and reveals their inherent symmetries.

Keywords:
aperiodic structuresperiod doubling structurespure point diffraction

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Fractal Geometry
  • Mathematical Physics

Background:

  • Period doubling bifurcations are a fundamental route to chaos in dynamical systems.
  • Existing theories primarily focus on one-dimensional systems.
  • Generalizing these concepts to higher dimensions is crucial for understanding complex phenomena.

Purpose of the Study:

  • To develop a mathematical framework for generalizing period doubling sequences to arbitrary dimensions.
  • To investigate the diffraction properties of these generalized structures.
  • To identify and analyze the symmetries inherent in high-dimensional period doubling structures.

Main Methods:

  • Extension of established substitution and recursion rules.
  • Formal mathematical development of the generalized formalism.
  • Analysis of diffraction patterns and symmetry properties.

Main Results:

  • Successful generalization of the period doubling sequence to arbitrary dimensions.
  • Proof that these higher-dimensional period doubling structures are pure point diffractive.
  • Identification of specific symmetries within these structures.

Conclusions:

  • The formalism provides a robust method for studying high-dimensional dynamical systems.
  • The pure point diffractive nature has implications for understanding wave phenomena and material science.
  • The identified symmetries offer insights into the underlying order of complex systems.