Archetypal oscillator for smooth and discontinuous dynamics

Qingjie Cao, Marian Wiercigroch, Ekaterina Evgenievna Pavlovskaia, Celso Grebogi, John Michael Tutill Thompson

Research output: Contribution to journalArticle

112 Citations (Scopus)

Abstract

We propose an archetypal system to investigate transitions from smooth to discontinuous dynamics. In the smooth regime, the system bears significant similarities to the Duffing oscillator, exhibiting the standard dynamics governed by the hyperbolic structure associated with the stationary state of the double well. At the discontinuous limit, however, there is a substantial departure in the dynamics from the standard one. In particular, the velocity flow suffers a jump in crossing from one well to another, caused by the loss of local hyperbolicity due to the collapse of the stable and unstable manifolds of the stationary state. In the presence of damping and external excitation, the system has coexisting attractors and also a chaotic saddle which becomes a chaotic attractor when a smoothness parameter drops to zero. This attractor can bifurcate to a high-period periodic attractor or a chaotic sea with islands of quasiperiodic attractors depending on the strength of damping.

Original languageEnglish
Article number046218
Number of pages5
JournalPhysical Review. E, Statistical, Nonlinear and Soft Matter Physics
Volume74
Issue number4
DOIs
Publication statusPublished - 30 Oct 2006

Keywords

  • impact oscillator
  • systems
  • reconstruction
  • bifurcations
  • behavior

Cite this

Archetypal oscillator for smooth and discontinuous dynamics. / Cao, Qingjie; Wiercigroch, Marian; Pavlovskaia, Ekaterina Evgenievna; Grebogi, Celso; Thompson, John Michael Tutill.

In: Physical Review. E, Statistical, Nonlinear and Soft Matter Physics, Vol. 74, No. 4, 046218, 30.10.2006.

Research output: Contribution to journalArticle

@article{e74929e8a8ca45c2a3e5d6a0f0d0bd7b,
title = "Archetypal oscillator for smooth and discontinuous dynamics",
abstract = "We propose an archetypal system to investigate transitions from smooth to discontinuous dynamics. In the smooth regime, the system bears significant similarities to the Duffing oscillator, exhibiting the standard dynamics governed by the hyperbolic structure associated with the stationary state of the double well. At the discontinuous limit, however, there is a substantial departure in the dynamics from the standard one. In particular, the velocity flow suffers a jump in crossing from one well to another, caused by the loss of local hyperbolicity due to the collapse of the stable and unstable manifolds of the stationary state. In the presence of damping and external excitation, the system has coexisting attractors and also a chaotic saddle which becomes a chaotic attractor when a smoothness parameter drops to zero. This attractor can bifurcate to a high-period periodic attractor or a chaotic sea with islands of quasiperiodic attractors depending on the strength of damping.",
keywords = "impact oscillator, systems, reconstruction, bifurcations, behavior",
author = "Qingjie Cao and Marian Wiercigroch and Pavlovskaia, {Ekaterina Evgenievna} and Celso Grebogi and Thompson, {John Michael Tutill}",
year = "2006",
month = "10",
day = "30",
doi = "10.1103/PhysRevE.74.046218",
language = "English",
volume = "74",
journal = "Physical Review. E, Statistical, Nonlinear and Soft Matter Physics",
issn = "1539-3755",
publisher = "AMER PHYSICAL SOC",
number = "4",

}

TY - JOUR

T1 - Archetypal oscillator for smooth and discontinuous dynamics

AU - Cao, Qingjie

AU - Wiercigroch, Marian

AU - Pavlovskaia, Ekaterina Evgenievna

AU - Grebogi, Celso

AU - Thompson, John Michael Tutill

PY - 2006/10/30

Y1 - 2006/10/30

N2 - We propose an archetypal system to investigate transitions from smooth to discontinuous dynamics. In the smooth regime, the system bears significant similarities to the Duffing oscillator, exhibiting the standard dynamics governed by the hyperbolic structure associated with the stationary state of the double well. At the discontinuous limit, however, there is a substantial departure in the dynamics from the standard one. In particular, the velocity flow suffers a jump in crossing from one well to another, caused by the loss of local hyperbolicity due to the collapse of the stable and unstable manifolds of the stationary state. In the presence of damping and external excitation, the system has coexisting attractors and also a chaotic saddle which becomes a chaotic attractor when a smoothness parameter drops to zero. This attractor can bifurcate to a high-period periodic attractor or a chaotic sea with islands of quasiperiodic attractors depending on the strength of damping.

AB - We propose an archetypal system to investigate transitions from smooth to discontinuous dynamics. In the smooth regime, the system bears significant similarities to the Duffing oscillator, exhibiting the standard dynamics governed by the hyperbolic structure associated with the stationary state of the double well. At the discontinuous limit, however, there is a substantial departure in the dynamics from the standard one. In particular, the velocity flow suffers a jump in crossing from one well to another, caused by the loss of local hyperbolicity due to the collapse of the stable and unstable manifolds of the stationary state. In the presence of damping and external excitation, the system has coexisting attractors and also a chaotic saddle which becomes a chaotic attractor when a smoothness parameter drops to zero. This attractor can bifurcate to a high-period periodic attractor or a chaotic sea with islands of quasiperiodic attractors depending on the strength of damping.

KW - impact oscillator

KW - systems

KW - reconstruction

KW - bifurcations

KW - behavior

U2 - 10.1103/PhysRevE.74.046218

DO - 10.1103/PhysRevE.74.046218

M3 - Article

VL - 74

JO - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

JF - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

SN - 1539-3755

IS - 4

M1 - 046218

ER -