Dynamics of a parametrically excited simple pendulum

Gabriela I. Depetri, Felipe A C Pereira, Boris Marin, Murilo S Baptista, J C Sartorelli

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Abstract

The dynamics of a parametric simple pendulum submitted to an arbitrary angle of excitation ϕ was investigated experimentally by simulations and analytically. Analytical calculations for the loci of saddle-node bifurcations corresponding to the creation of resonant orbits were performed by applying Melnikov's method. However, this powerful perturbative method cannot be used to predict the existence of odd resonances for a vertical excitation within first order corrections. Yet, we showed that period-3 resonances indeed exist in such a configuration. Two degenerate attractors of different phases, associated with the same loci of saddle-node bifurcations in parameter space, are reported. For tilted excitation, the degeneracy is broken due to an extra torque, which was confirmed by the calculation of two distinct loci of saddle-node bifurcations for each attractor. This behavior persists up to ϕ≈7π/180, and for inclinations larger than this, only one attractor is observed. Bifurcation diagrams were constructed experimentally for ϕ=π/8 to demonstrate the existence of self-excited resonances (periods smaller than three) and hidden oscillations (for periods greater than three).
Original languageEnglish
Article number033103
JournalChaos
Volume28
Issue number3
DOIs
Publication statusPublished - Mar 2018

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Saddle-node Bifurcation
saddles
pendulums
Pendulum
loci
Pendulums
Locus
Attractor
Bifurcation (mathematics)
Excitation
excitation
Melnikov Method
Inclination
Bifurcation Diagram
Degeneracy
inclination
Torque
torque
Parameter Space
Orbits

Keywords

  • parametric pendulum
  • parametric resonances
  • subharmonic resonances
  • Melnikov method
  • numerical continuation

Cite this

Depetri, G. I., Pereira, F. A. C., Marin, B., Baptista, M. S., & Sartorelli, J. C. (2018). Dynamics of a parametrically excited simple pendulum. Chaos, 28(3), [033103]. https://doi.org/10.1063/1.5016819

Dynamics of a parametrically excited simple pendulum. / Depetri, Gabriela I.; Pereira, Felipe A C; Marin, Boris; Baptista, Murilo S; Sartorelli, J C.

In: Chaos, Vol. 28, No. 3, 033103, 03.2018.

Research output: Contribution to journalArticle

Depetri, GI, Pereira, FAC, Marin, B, Baptista, MS & Sartorelli, JC 2018, 'Dynamics of a parametrically excited simple pendulum', Chaos, vol. 28, no. 3, 033103. https://doi.org/10.1063/1.5016819
Depetri GI, Pereira FAC, Marin B, Baptista MS, Sartorelli JC. Dynamics of a parametrically excited simple pendulum. Chaos. 2018 Mar;28(3). 033103. https://doi.org/10.1063/1.5016819
Depetri, Gabriela I. ; Pereira, Felipe A C ; Marin, Boris ; Baptista, Murilo S ; Sartorelli, J C. / Dynamics of a parametrically excited simple pendulum. In: Chaos. 2018 ; Vol. 28, No. 3.
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abstract = "The dynamics of a parametric simple pendulum submitted to an arbitrary angle of excitation ϕ was investigated experimentally by simulations and analytically. Analytical calculations for the loci of saddle-node bifurcations corresponding to the creation of resonant orbits were performed by applying Melnikov's method. However, this powerful perturbative method cannot be used to predict the existence of odd resonances for a vertical excitation within first order corrections. Yet, we showed that period-3 resonances indeed exist in such a configuration. Two degenerate attractors of different phases, associated with the same loci of saddle-node bifurcations in parameter space, are reported. For tilted excitation, the degeneracy is broken due to an extra torque, which was confirmed by the calculation of two distinct loci of saddle-node bifurcations for each attractor. This behavior persists up to ϕ≈7π/180, and for inclinations larger than this, only one attractor is observed. Bifurcation diagrams were constructed experimentally for ϕ=π/8 to demonstrate the existence of self-excited resonances (periods smaller than three) and hidden oscillations (for periods greater than three).",
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N2 - The dynamics of a parametric simple pendulum submitted to an arbitrary angle of excitation ϕ was investigated experimentally by simulations and analytically. Analytical calculations for the loci of saddle-node bifurcations corresponding to the creation of resonant orbits were performed by applying Melnikov's method. However, this powerful perturbative method cannot be used to predict the existence of odd resonances for a vertical excitation within first order corrections. Yet, we showed that period-3 resonances indeed exist in such a configuration. Two degenerate attractors of different phases, associated with the same loci of saddle-node bifurcations in parameter space, are reported. For tilted excitation, the degeneracy is broken due to an extra torque, which was confirmed by the calculation of two distinct loci of saddle-node bifurcations for each attractor. This behavior persists up to ϕ≈7π/180, and for inclinations larger than this, only one attractor is observed. Bifurcation diagrams were constructed experimentally for ϕ=π/8 to demonstrate the existence of self-excited resonances (periods smaller than three) and hidden oscillations (for periods greater than three).

AB - The dynamics of a parametric simple pendulum submitted to an arbitrary angle of excitation ϕ was investigated experimentally by simulations and analytically. Analytical calculations for the loci of saddle-node bifurcations corresponding to the creation of resonant orbits were performed by applying Melnikov's method. However, this powerful perturbative method cannot be used to predict the existence of odd resonances for a vertical excitation within first order corrections. Yet, we showed that period-3 resonances indeed exist in such a configuration. Two degenerate attractors of different phases, associated with the same loci of saddle-node bifurcations in parameter space, are reported. For tilted excitation, the degeneracy is broken due to an extra torque, which was confirmed by the calculation of two distinct loci of saddle-node bifurcations for each attractor. This behavior persists up to ϕ≈7π/180, and for inclinations larger than this, only one attractor is observed. Bifurcation diagrams were constructed experimentally for ϕ=π/8 to demonstrate the existence of self-excited resonances (periods smaller than three) and hidden oscillations (for periods greater than three).

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