Chaotic attractor of the normal form map for grazing bifurcations of impact oscillators

Pengcheng Miao, Denghui Li* (Corresponding Author), Yuan Yue, Jianhua Xie, Celso Grebogi

*Corresponding author for this work

Research output: Contribution to journalArticle

Abstract

Grazing bifurcations can cause impact oscillators to exhibit chaotic motions. Such dynamical behaviour can be described by a normal form map (called the Nordmark map). A main feature of the Nordmark map is that it has a square-root term. The purpose of this paper is to study the structure of the chaotic attractor of the Nordmark map from the topological point of view. First, the trapping region of the asymptotic dynamics of the map is constructed. It is then proven that, for some set of parameter values having positive Lebesgue measure, the ω-limit set of each point of the trapping region is contained in a invariant set which is just the closure of the unstable manifold of the hyperbolic fixed point of the map. Besides, the dynamics on the invariant set is topologically mixing. Accordingly the invariant set is a chaotic attractor of the map.

Original languageEnglish
Pages (from-to)164-170
Number of pages7
JournalPhysica D: Nonlinear Phenomena
Volume398
Early online date29 Mar 2019
DOIs
Publication statusPublished - Nov 2019

Fingerprint

Grazing Bifurcation
Chaotic Attractor
grazing
Normal Form
oscillators
Invariant Set
Trapping
trapping
Unstable Manifold
Chaotic Motion
Limit Set
Lebesgue Measure
Square root
Dynamical Behavior
Set of points
closures
Closure
Fixed point
causes

Keywords

  • Chaotic attractor
  • Impact oscillator
  • Nordmark map
  • Trapping region
  • DYNAMICS
  • SYSTEMS
  • STRANGE ATTRACTORS
  • BORDER-COLLISION BIFURCATIONS

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Applied Mathematics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Chaotic attractor of the normal form map for grazing bifurcations of impact oscillators. / Miao, Pengcheng; Li, Denghui (Corresponding Author); Yue, Yuan; Xie, Jianhua; Grebogi, Celso.

In: Physica D: Nonlinear Phenomena, Vol. 398, 11.2019, p. 164-170.

Research output: Contribution to journalArticle

Miao, Pengcheng ; Li, Denghui ; Yue, Yuan ; Xie, Jianhua ; Grebogi, Celso. / Chaotic attractor of the normal form map for grazing bifurcations of impact oscillators. In: Physica D: Nonlinear Phenomena. 2019 ; Vol. 398. pp. 164-170.
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AU - Li, Denghui

AU - Yue, Yuan

AU - Xie, Jianhua

AU - Grebogi, Celso

N1 - This work is supported by the National Natural Science Foundation of China (11572263, 11672249 and 11732014).

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