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Characterization of a spring pendulum phase-space trajectories.

Karla P Acosta-Zamora1, José Núñez González2, Ahtziri González3

  • 1Instituto de Energías Renovables, Universidad Nacional Autónoma de México, Temixco, Morelos 62580, Mexico.

Chaos (Woodbury, N.Y.)
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Summary
This summary is machine-generated.

This study reveals that spring pendulum orbits relate to torus and cable knots. A new parameter, Ω, characterizes these dynamics, showing predictable distributions and linking orbits with similar behavior.

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Area of Science:

  • Dynamical systems
  • Chaos theory
  • Geometric mechanics

Background:

  • Phase-space trajectories of spring pendulums exhibit complex behaviors.
  • Understanding these dynamics is crucial for analyzing nonlinear systems.
  • Poincaré maps are a standard tool for visualizing dynamical system properties.

Purpose of the Study:

  • To investigate the geometrical properties of spring pendulum orbits.
  • To establish a connection between orbit geometry and knot theory.
  • To introduce and validate a new parameter for characterizing orbital dynamics.

Main Methods:

  • Utilized Poincaré maps to analyze phase-space trajectories.
  • Examined toroidal and poloidal turns of orbits.
  • Developed algorithms to calculate a rational parameter Ω.
  • Related orbital structures to torus knots and cable knots.

Main Results:

  • Regular orbits correspond to torus knots, forming segments in Poincaré maps.
  • A rational parameter Ω, analogous to frequency, characterizes orbits via Farey sequences.
  • Orbits with identical Ω values exhibit similar dynamical behaviors.
  • Chains of islands in Poincaré maps are linked to cable knots, some non-trivial.

Conclusions:

  • The rational parameter Ω effectively describes spring pendulum dynamics and their geometric properties.
  • Orbital dynamics are intrinsically linked to topological structures (knots).
  • Ω parameter distributions in (z,Ω) space are predictable, offering insights into system behavior.