Multistability and arithmetically period-adding bifurcations in piecewise smooth dynamical systems

Younghae Do, Ying-Cheng Lai

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

Multistability has been a phenomenon of continuous interest in nonlinear dynamics. Most existing works so far have focused on smooth dynamical systems. Motivated by the fact that nonsmooth dynamical systems can arise commonly in realistic physical and engineering applications such as impact oscillators and switching electronic circuits, we investigate multistability in such systems. In particular, we consider a generic class of piecewise smooth dynamical systems expressed in normal form but representative of nonsmooth systems in realistic situations, and focus on the weakly dissipative regime and the Hamiltonian limit. We find that, as the Hamiltonian limit is approached, periodic attractors can be generated through a series of saddle-node bifurcations. A striking phenomenon is that the periods of the newly created attractors follow an arithmetic sequence. This has no counterpart in smooth dynamical systems. We provide physical analyses, numerical computations, and rigorous mathematical arguments to substantiate the finding.

Original languageEnglish
Article number043107
Number of pages9
JournalChaos
Volume18
Issue number4
DOIs
Publication statusPublished - Dec 2008

Keywords

  • bifurcation
  • Jacobian matrices
  • nonlinear dynamical systems
  • stability
  • border-collision bifurcations
  • grazing bifurcations
  • impact oscillators
  • linear-oscillator
  • attractors
  • mode
  • maps

Cite this

Multistability and arithmetically period-adding bifurcations in piecewise smooth dynamical systems. / Do, Younghae; Lai, Ying-Cheng.

In: Chaos, Vol. 18, No. 4, 043107, 12.2008.

Research output: Contribution to journalArticle

@article{e05743a19b574ba9b1f792e83136f3b9,
title = "Multistability and arithmetically period-adding bifurcations in piecewise smooth dynamical systems",
abstract = "Multistability has been a phenomenon of continuous interest in nonlinear dynamics. Most existing works so far have focused on smooth dynamical systems. Motivated by the fact that nonsmooth dynamical systems can arise commonly in realistic physical and engineering applications such as impact oscillators and switching electronic circuits, we investigate multistability in such systems. In particular, we consider a generic class of piecewise smooth dynamical systems expressed in normal form but representative of nonsmooth systems in realistic situations, and focus on the weakly dissipative regime and the Hamiltonian limit. We find that, as the Hamiltonian limit is approached, periodic attractors can be generated through a series of saddle-node bifurcations. A striking phenomenon is that the periods of the newly created attractors follow an arithmetic sequence. This has no counterpart in smooth dynamical systems. We provide physical analyses, numerical computations, and rigorous mathematical arguments to substantiate the finding.",
keywords = "bifurcation, Jacobian matrices, nonlinear dynamical systems, stability, border-collision bifurcations, grazing bifurcations, impact oscillators, linear-oscillator, attractors, mode, maps",
author = "Younghae Do and Ying-Cheng Lai",
year = "2008",
month = "12",
doi = "10.1063/1.2985853",
language = "English",
volume = "18",
journal = "Chaos",
issn = "1054-1500",
publisher = "American Institute of Physics",
number = "4",

}

TY - JOUR

T1 - Multistability and arithmetically period-adding bifurcations in piecewise smooth dynamical systems

AU - Do, Younghae

AU - Lai, Ying-Cheng

PY - 2008/12

Y1 - 2008/12

N2 - Multistability has been a phenomenon of continuous interest in nonlinear dynamics. Most existing works so far have focused on smooth dynamical systems. Motivated by the fact that nonsmooth dynamical systems can arise commonly in realistic physical and engineering applications such as impact oscillators and switching electronic circuits, we investigate multistability in such systems. In particular, we consider a generic class of piecewise smooth dynamical systems expressed in normal form but representative of nonsmooth systems in realistic situations, and focus on the weakly dissipative regime and the Hamiltonian limit. We find that, as the Hamiltonian limit is approached, periodic attractors can be generated through a series of saddle-node bifurcations. A striking phenomenon is that the periods of the newly created attractors follow an arithmetic sequence. This has no counterpart in smooth dynamical systems. We provide physical analyses, numerical computations, and rigorous mathematical arguments to substantiate the finding.

AB - Multistability has been a phenomenon of continuous interest in nonlinear dynamics. Most existing works so far have focused on smooth dynamical systems. Motivated by the fact that nonsmooth dynamical systems can arise commonly in realistic physical and engineering applications such as impact oscillators and switching electronic circuits, we investigate multistability in such systems. In particular, we consider a generic class of piecewise smooth dynamical systems expressed in normal form but representative of nonsmooth systems in realistic situations, and focus on the weakly dissipative regime and the Hamiltonian limit. We find that, as the Hamiltonian limit is approached, periodic attractors can be generated through a series of saddle-node bifurcations. A striking phenomenon is that the periods of the newly created attractors follow an arithmetic sequence. This has no counterpart in smooth dynamical systems. We provide physical analyses, numerical computations, and rigorous mathematical arguments to substantiate the finding.

KW - bifurcation

KW - Jacobian matrices

KW - nonlinear dynamical systems

KW - stability

KW - border-collision bifurcations

KW - grazing bifurcations

KW - impact oscillators

KW - linear-oscillator

KW - attractors

KW - mode

KW - maps

U2 - 10.1063/1.2985853

DO - 10.1063/1.2985853

M3 - Article

VL - 18

JO - Chaos

JF - Chaos

SN - 1054-1500

IS - 4

M1 - 043107

ER -