### 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 language | English |
---|---|

Pages (from-to) | 2689-2698 |

Number of pages | 9 |

Journal | International Journal of Bifurcation and Chaos |

Volume | 11 |

Publication status | Published - 2001 |

### Keywords

- SYNCHRONIZATION
- BIFURCATION
- TRANSIENTS

### Cite this

*International Journal of Bifurcation and Chaos*,

*11*, 2689-2698.

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

Research output: Contribution to journal › Article

*International Journal of Bifurcation and Chaos*, vol. 11, pp. 2689-2698.

}

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 -