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Arnol'd cat map lattices.

Minos Axenides1, Emmanuel Floratos1,2, Stam Nicolis3

  • 1Institute for Nuclear and Particle Physics, NCSR "Demokritos", Aghia Paraskevi 15310, Greece.

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|July 19, 2023
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Summary
This summary is machine-generated.

We developed lattice field theories using the Arnol'd cat map, revealing chaotic dynamics in coupled systems. This research explores spatiotemporal chaos and its dependence on interaction parameters.

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Area of Science:

  • Mathematical Physics
  • Dynamical Systems Theory
  • Chaos Theory

Background:

  • Arnol'd cat map is a fundamental model in chaos theory.
  • Understanding coupled map lattices is crucial for complex system dynamics.
  • Generalizing single-map properties to lattice field theories presents theoretical challenges.

Purpose of the Study:

  • To construct Arnol'd cat map lattice field theories in both phase and configuration spaces.
  • To generalize the symplectic group constraint to linearly coupled maps.
  • To analyze the spatiotemporal chaotic properties of these novel systems.

Main Methods:

  • Construction of lattice field theories using the Arnol'd cat map.
  • Imposing symplectic group constraints on evolution operators for coupled maps.
  • Exploiting the cat map-Fibonacci sequence correspondence.
  • Analyzing equations of motion in configuration space, including inverted harmonic oscillators.
  • Utilizing benchmarks for deterministic chaos, such as unstable periodic orbits.

Main Results:

  • Successful construction of Arnol'd cat map lattice field theories.
  • Demonstration of chaotic properties arising from coupled inverted harmonic oscillators.
  • Identification of spatiotemporal chaos through dense unstable periodic orbits.
  • Observation of ergodicity and mixing for long periods.
  • Strong dependence of the period spectrum on interaction strength and range.

Conclusions:

  • The constructed lattice field theories exhibit rich chaotic dynamics.
  • The interplay between potential runaway and phase space compactification drives chaos.
  • The findings provide insights into spatiotemporal chaos in complex dynamical systems.