The limit case response of archetypal oscillator for smooth and discontinuous dynamics

Qingjie Cao, Marian Wiercigroch, Ekaterina E. Pavlovskaia, Celso Grebogi, J. Michael T. Thompson

Research output: Contribution to journalArticle

68 Citations (Scopus)

Abstract

In this paper, the limit case of the SD (smooth and discontinuous) oscillator is studied. This system exhibits standard dynamics governed by the hyperbolic structure associated with the stationary state of the double-well. The substantial deviation from the standard dynamics is the non-smoothness of the velocity in crossing from one well to another, caused by the loss of local hyperbolicity due to the discontinuity. Without dissipation, the KAM structure on the Poincaré section is constructed with generic KAM curves and a series of fixed points associated with surrounded islands of quasi-periodic orbits and the chaotic connection orbits. It is found that, for a fixed set of parameters, a special chaotic orbit exits there which fills a finite region and connects a series of islands dominated by different chains of fixed points. As one adds weak dissipation, the periodic solutions in this finite region remain unchanged while the quasi-periodic solutions (isolated islands) are converted to the corresponding periodic solutions. The relevant dynamics for the system with weak dissipation under external excitation is shown having period doubling bifurcation leading to chaos, and multi-stable solutions.
Original languageEnglish
Pages (from-to)462-473
Number of pages12
JournalInternational Journal of Non-Linear Mechanics
Volume43
Issue number6
Early online date20 Jan 2008
DOIs
Publication statusPublished - Jul 2008

Keywords

  • SD oscillator
  • discontinuity
  • KAM structure
  • chaotic sea
  • impact oscillator
  • bifurcations
  • systems
  • chaos
  • attractors
  • vibrations
  • maps
  • form

Cite this

The limit case response of archetypal oscillator for smooth and discontinuous dynamics. / Cao, Qingjie; Wiercigroch, Marian; Pavlovskaia, Ekaterina E.; Grebogi, Celso; Thompson, J. Michael T.

In: International Journal of Non-Linear Mechanics, Vol. 43, No. 6, 07.2008, p. 462-473.

Research output: Contribution to journalArticle

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title = "The limit case response of archetypal oscillator for smooth and discontinuous dynamics",
abstract = "In this paper, the limit case of the SD (smooth and discontinuous) oscillator is studied. This system exhibits standard dynamics governed by the hyperbolic structure associated with the stationary state of the double-well. The substantial deviation from the standard dynamics is the non-smoothness of the velocity in crossing from one well to another, caused by the loss of local hyperbolicity due to the discontinuity. Without dissipation, the KAM structure on the Poincar{\'e} section is constructed with generic KAM curves and a series of fixed points associated with surrounded islands of quasi-periodic orbits and the chaotic connection orbits. It is found that, for a fixed set of parameters, a special chaotic orbit exits there which fills a finite region and connects a series of islands dominated by different chains of fixed points. As one adds weak dissipation, the periodic solutions in this finite region remain unchanged while the quasi-periodic solutions (isolated islands) are converted to the corresponding periodic solutions. The relevant dynamics for the system with weak dissipation under external excitation is shown having period doubling bifurcation leading to chaos, and multi-stable solutions.",
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author = "Qingjie Cao and Marian Wiercigroch and Pavlovskaia, {Ekaterina E.} and Celso Grebogi and Thompson, {J. Michael T.}",
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AU - Cao, Qingjie

AU - Wiercigroch, Marian

AU - Pavlovskaia, Ekaterina E.

AU - Grebogi, Celso

AU - Thompson, J. Michael T.

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N2 - In this paper, the limit case of the SD (smooth and discontinuous) oscillator is studied. This system exhibits standard dynamics governed by the hyperbolic structure associated with the stationary state of the double-well. The substantial deviation from the standard dynamics is the non-smoothness of the velocity in crossing from one well to another, caused by the loss of local hyperbolicity due to the discontinuity. Without dissipation, the KAM structure on the Poincaré section is constructed with generic KAM curves and a series of fixed points associated with surrounded islands of quasi-periodic orbits and the chaotic connection orbits. It is found that, for a fixed set of parameters, a special chaotic orbit exits there which fills a finite region and connects a series of islands dominated by different chains of fixed points. As one adds weak dissipation, the periodic solutions in this finite region remain unchanged while the quasi-periodic solutions (isolated islands) are converted to the corresponding periodic solutions. The relevant dynamics for the system with weak dissipation under external excitation is shown having period doubling bifurcation leading to chaos, and multi-stable solutions.

AB - In this paper, the limit case of the SD (smooth and discontinuous) oscillator is studied. This system exhibits standard dynamics governed by the hyperbolic structure associated with the stationary state of the double-well. The substantial deviation from the standard dynamics is the non-smoothness of the velocity in crossing from one well to another, caused by the loss of local hyperbolicity due to the discontinuity. Without dissipation, the KAM structure on the Poincaré section is constructed with generic KAM curves and a series of fixed points associated with surrounded islands of quasi-periodic orbits and the chaotic connection orbits. It is found that, for a fixed set of parameters, a special chaotic orbit exits there which fills a finite region and connects a series of islands dominated by different chains of fixed points. As one adds weak dissipation, the periodic solutions in this finite region remain unchanged while the quasi-periodic solutions (isolated islands) are converted to the corresponding periodic solutions. The relevant dynamics for the system with weak dissipation under external excitation is shown having period doubling bifurcation leading to chaos, and multi-stable solutions.

KW - SD oscillator

KW - discontinuity

KW - KAM structure

KW - chaotic sea

KW - impact oscillator

KW - bifurcations

KW - systems

KW - chaos

KW - attractors

KW - vibrations

KW - maps

KW - form

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JO - International Journal of Non-Linear Mechanics

JF - International Journal of Non-Linear Mechanics

SN - 0020-7462

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