Riddled Basins and Unstable Dimension Variability in Chaotic Systems With and Without Symmetry

Celso Grebogi, R. L. Viana

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

Riddling occurs in dissipative dynamical systems with more than one attractor, when the basin of one attractor is punctured with holes belonging to the basins of the other attractors. The basin of a chaotic attractor is riddled if (i) it has a positive Lebesgue measure; (ii) in the vicinity of every point belonging to the basin of the attractor, there is a positive Lebesgue measure set of points that asymptote to another attractor. We investigate the presence of riddled basins in a two-dimensional noninvertible map with a symmetry-breaking term. In the symmetric case the onset of riddling is characterized by an unstable-unstable pair bifurcation, which also leads to unstable dimension variability in the invariant chaotic set. The nonsymmetric case exhibits a chaotic attractor, but a riddled basin occurs only at the bifurcation point, since after that the attractor becomes a chaotic saddle. We analyze the presence of unstable dimension variability in the symmetric case by computing the finite-time transverse Lyapunov exponents. We point out some consequences of those facts to the synchronization properties of coupled chaotic systems.

Original languageEnglish
Pages (from-to)2689-2698
Number of pages9
JournalInternational Journal of Bifurcation and Chaos
Volume11
Publication statusPublished - 2001

Keywords

  • SYNCHRONIZATION
  • BIFURCATION
  • TRANSIENTS

Cite this

Riddled Basins and Unstable Dimension Variability in Chaotic Systems With and Without Symmetry. / Grebogi, Celso; Viana, R. L.

In: International Journal of Bifurcation and Chaos, Vol. 11, 2001, p. 2689-2698.

Research output: Contribution to journalArticle

@article{d7aed16630214760a895a96e3a619a82,
title = "Riddled Basins and Unstable Dimension Variability in Chaotic Systems With and Without Symmetry",
abstract = "Riddling occurs in dissipative dynamical systems with more than one attractor, when the basin of one attractor is punctured with holes belonging to the basins of the other attractors. The basin of a chaotic attractor is riddled if (i) it has a positive Lebesgue measure; (ii) in the vicinity of every point belonging to the basin of the attractor, there is a positive Lebesgue measure set of points that asymptote to another attractor. We investigate the presence of riddled basins in a two-dimensional noninvertible map with a symmetry-breaking term. In the symmetric case the onset of riddling is characterized by an unstable-unstable pair bifurcation, which also leads to unstable dimension variability in the invariant chaotic set. The nonsymmetric case exhibits a chaotic attractor, but a riddled basin occurs only at the bifurcation point, since after that the attractor becomes a chaotic saddle. We analyze the presence of unstable dimension variability in the symmetric case by computing the finite-time transverse Lyapunov exponents. We point out some consequences of those facts to the synchronization properties of coupled chaotic systems.",
keywords = "SYNCHRONIZATION, BIFURCATION, TRANSIENTS",
author = "Celso Grebogi and Viana, {R. L.}",
year = "2001",
language = "English",
volume = "11",
pages = "2689--2698",
journal = "International Journal of Bifurcation and Chaos",
issn = "0218-1274",
publisher = "WORLD SCIENTIFIC PUBL CO PTE LTD",

}

TY - JOUR

T1 - Riddled Basins and Unstable Dimension Variability in Chaotic Systems With and Without Symmetry

AU - Grebogi, Celso

AU - Viana, R. L.

PY - 2001

Y1 - 2001

N2 - Riddling occurs in dissipative dynamical systems with more than one attractor, when the basin of one attractor is punctured with holes belonging to the basins of the other attractors. The basin of a chaotic attractor is riddled if (i) it has a positive Lebesgue measure; (ii) in the vicinity of every point belonging to the basin of the attractor, there is a positive Lebesgue measure set of points that asymptote to another attractor. We investigate the presence of riddled basins in a two-dimensional noninvertible map with a symmetry-breaking term. In the symmetric case the onset of riddling is characterized by an unstable-unstable pair bifurcation, which also leads to unstable dimension variability in the invariant chaotic set. The nonsymmetric case exhibits a chaotic attractor, but a riddled basin occurs only at the bifurcation point, since after that the attractor becomes a chaotic saddle. We analyze the presence of unstable dimension variability in the symmetric case by computing the finite-time transverse Lyapunov exponents. We point out some consequences of those facts to the synchronization properties of coupled chaotic systems.

AB - Riddling occurs in dissipative dynamical systems with more than one attractor, when the basin of one attractor is punctured with holes belonging to the basins of the other attractors. The basin of a chaotic attractor is riddled if (i) it has a positive Lebesgue measure; (ii) in the vicinity of every point belonging to the basin of the attractor, there is a positive Lebesgue measure set of points that asymptote to another attractor. We investigate the presence of riddled basins in a two-dimensional noninvertible map with a symmetry-breaking term. In the symmetric case the onset of riddling is characterized by an unstable-unstable pair bifurcation, which also leads to unstable dimension variability in the invariant chaotic set. The nonsymmetric case exhibits a chaotic attractor, but a riddled basin occurs only at the bifurcation point, since after that the attractor becomes a chaotic saddle. We analyze the presence of unstable dimension variability in the symmetric case by computing the finite-time transverse Lyapunov exponents. We point out some consequences of those facts to the synchronization properties of coupled chaotic systems.

KW - SYNCHRONIZATION

KW - BIFURCATION

KW - TRANSIENTS

M3 - Article

VL - 11

SP - 2689

EP - 2698

JO - International Journal of Bifurcation and Chaos

JF - International Journal of Bifurcation and Chaos

SN - 0218-1274

ER -