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Arithmetical method to detect integrability in maps.

J A G Roberts1, F Vivaldi

  • 1School of Mathematics, The University of New South Wales, Sydney NSW 2052, Australia.

Physical Review Letters
|February 7, 2003
PubMed
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We developed a new method to detect integrals of motion in symplectic rational maps using finite fields. Orbit structure analysis reveals distinct statistics for integrable versus non-integrable maps.

Area of Science:

  • Mathematics
  • Dynamical Systems
  • Number Theory

Background:

  • Symplectic rational maps are fundamental in dynamical systems.
  • Detecting integrability is crucial for understanding map behavior.
  • Existing methods for integrability detection can be limited.

Purpose of the Study:

  • To develop a novel method for detecting integrals of motion in symplectic rational maps.
  • To analyze the orbit structure of these maps over finite fields.
  • To differentiate between integrable and non-integrable maps based on orbit statistics.

Main Methods:

  • Representing symplectic rational maps over finite fields.
  • Examining the orbit structure of the maps.
  • Statistical analysis of orbit properties.

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Main Results:

  • A new method for detecting integrals of motion was successfully developed.
  • Distinct orbit statistics were observed for integrable and non-integrable maps.
  • The method provides a clear way to distinguish between integrable and non-integrable symplectic rational maps.

Conclusions:

  • The developed method is effective for detecting integrability in symplectic rational maps.
  • Finite field representation offers a powerful tool for analyzing dynamical systems.
  • Orbit structure analysis provides key insights into the integrability of mathematical maps.