An experimental study of bifurcation, chaos, and dimensionality in a system forced through a bifurcation parameter

J. P. Cusumano, M. T. Sharkady

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

An experimental study of a system that is parametrically excited through a bifurcation parameter is presented. The system consits of a lightly-damped, flexible beam which is buckled and unbuckled magnetically: it is parametrically excited by driving an electromagnet with a low-frequency sine wave. For voltage amplitudes in excess of the static bifurcation value, the beam slowly switches between the one-and two-well configurations. Experimental static and dynamic bifurcation results are presented. Static bifurcatons for the system are shown to involve a butterfly catastrophe. The dynamic bifurcation diagram, obtained with an automated data acquisition system, shows several period-doubling sequences, jump phenomena, and a chaotic region. Poincaré sections of a chaotic steady-state are obtained for various values of the driving phase, and the correlation dimension of the chaotic attractor is estimated over a large scaling region. Singular system analysis is used to demonstrate the effect of delay time on the noise level in delay-reconstructions, and to provide an independent check on the dimension estimate by directly estimating the number of independent coordinates from time series data. The correlation dimension is also estimated using the delay-reconstructed data and shown to be in good agrement with the value obtained from the Poincaré sections. The bifurcation and dimension results are used together with physical sonsiderations to derive the general form of a single-degree-of-freedom model for the experimental system.

Original languageEnglish (US)
Pages (from-to)467-489
Number of pages23
JournalNonlinear Dynamics
Volume8
Issue number4
DOIs
StatePublished - Dec 1 1995

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Bifurcation and Chaos
Chaos theory
Dimensionality
Experimental Study
Bifurcation
Electromagnets
Correlation Dimension
Time series
Data acquisition
Time delay
Systems analysis
Switches
Electric potential
Flexible Beam
Catastrophe
Period Doubling
Singular Systems
Delay Time
Chaotic Attractor
Bifurcation Diagram

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering

Cite this

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abstract = "An experimental study of a system that is parametrically excited through a bifurcation parameter is presented. The system consits of a lightly-damped, flexible beam which is buckled and unbuckled magnetically: it is parametrically excited by driving an electromagnet with a low-frequency sine wave. For voltage amplitudes in excess of the static bifurcation value, the beam slowly switches between the one-and two-well configurations. Experimental static and dynamic bifurcation results are presented. Static bifurcatons for the system are shown to involve a butterfly catastrophe. The dynamic bifurcation diagram, obtained with an automated data acquisition system, shows several period-doubling sequences, jump phenomena, and a chaotic region. Poincar{\'e} sections of a chaotic steady-state are obtained for various values of the driving phase, and the correlation dimension of the chaotic attractor is estimated over a large scaling region. Singular system analysis is used to demonstrate the effect of delay time on the noise level in delay-reconstructions, and to provide an independent check on the dimension estimate by directly estimating the number of independent coordinates from time series data. The correlation dimension is also estimated using the delay-reconstructed data and shown to be in good agrement with the value obtained from the Poincar{\'e} sections. The bifurcation and dimension results are used together with physical sonsiderations to derive the general form of a single-degree-of-freedom model for the experimental system.",
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An experimental study of bifurcation, chaos, and dimensionality in a system forced through a bifurcation parameter. / Cusumano, J. P.; Sharkady, M. T.

In: Nonlinear Dynamics, Vol. 8, No. 4, 01.12.1995, p. 467-489.

Research output: Contribution to journalArticle

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