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Dynamics of a parametrically excited simple pendulum.

Gabriela I Depetri1, Felipe A C Pereira1, Boris Marin2

  • 1Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05315-970 São Paulo, Brazil.

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

This study investigates parametric pendulum dynamics, revealing hidden period-3 resonances under vertical excitation. Tilted excitation breaks attractor degeneracy, altering bifurcation loci and observed dynamics.

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Area of Science:

  • Nonlinear Dynamics
  • Mechanical Vibrations
  • Chaos Theory

Background:

  • Parametric pendulums exhibit complex behaviors, including resonances and bifurcations.
  • Melnikov's method is a powerful tool for analyzing chaotic dynamics but has limitations.

Purpose of the Study:

  • Investigate the dynamics of a parametric simple pendulum under arbitrary excitation angles.
  • Explore the existence and characteristics of odd resonances, particularly period-3 resonances.
  • Analyze the effect of excitation tilt on attractor degeneracy and bifurcation loci.

Main Methods:

  • Experimental investigation using simulations.
  • Analytical calculations applying Melnikov's method.
  • Construction of bifurcation diagrams.

Main Results:

  • Period-3 resonances were experimentally confirmed for vertical excitation, despite analytical limitations.
  • Tilted excitation breaks attractor degeneracy, creating distinct saddle-node bifurcation loci.
  • Bifurcation diagrams revealed self-excited and hidden oscillations for specific excitation angles.

Conclusions:

  • The study demonstrates the existence of previously unpredicted resonances in parametric pendulums.
  • Excitation geometry significantly influences the system's dynamical behavior and attractor structure.
  • Findings contribute to understanding complex nonlinear phenomena in mechanical systems.