### Abstract

Most existing works on riddling assume that the underlying dynamical system possesses an invariant subspace that usually results from a symmetry. In realistic applications of chaotic systems, however, there exists no perfect symmetry. The aim of this paper is to examine the consequences of symmetry-breaking on riddling. In particular, we consider smooth deterministic perturbations that destroy the existence of invariant subspace, and identify, as a symmetry-breaking parameter is increased from zero, two distinct bifurcations. In the first case, the chaotic attractor in the invariant subspace is transversely stable so that the basin is riddled. We find that a bifurcation from riddled to fractal basins can occur in the sense that an arbitrarily small amount of symmetry breaking can replace the riddled basin by fractal basins. We call this a catastrophe of riddling. In the second case, where the chaotic attractor in the invariant subspace is transversely unstable so that there is no riddling in the unperturbed system, the presence of a symmetry breaking, no matter how small, can immediately create fractal basins in the vicinity of the original invariant subspace. This is a smooth-fractal basin boundary metamorphosis. We analyze the dynamical mechanisms for both catastrophes of riddling and basin boundary metamorphoses, derive scaling laws to characterize the fractal basins induced by symmetry breaking, and provide numerical confirmations. The main implication of our results is that while riddling is robust against perturbations that preserve the system symmetry, riddled basins of chaotic attractors in the invariant subspace, on which most existing works are focused, are structurally unstable against symmetry-breaking perturbations.

Original language | English |
---|---|

Article number | 056228 |

Number of pages | 16 |

Journal | Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 64 |

Issue number | 5 |

DOIs | |

Publication status | Published - 26 Oct 2001 |

### Keywords

- unstable periodic-orbits
- chaotic dynamical-systems
- coupled oscillator-systems
- natural measure
- dimension variability
- strange attractors
- boundaries
- singularities
- metamorphoses
- intermittency

### Cite this

*Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*,

*64*(5), [056228]. https://doi.org/10.1103/PhysRevE.64.056228

**Catastrophic bifurcation from riddled to fractal basins.** / Lai, Ying-Cheng; Andrade, Victor.

Research output: Contribution to journal › Article

*Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*, vol. 64, no. 5, 056228. https://doi.org/10.1103/PhysRevE.64.056228

}

TY - JOUR

T1 - Catastrophic bifurcation from riddled to fractal basins

AU - Lai, Ying-Cheng

AU - Andrade, Victor

PY - 2001/10/26

Y1 - 2001/10/26

N2 - Most existing works on riddling assume that the underlying dynamical system possesses an invariant subspace that usually results from a symmetry. In realistic applications of chaotic systems, however, there exists no perfect symmetry. The aim of this paper is to examine the consequences of symmetry-breaking on riddling. In particular, we consider smooth deterministic perturbations that destroy the existence of invariant subspace, and identify, as a symmetry-breaking parameter is increased from zero, two distinct bifurcations. In the first case, the chaotic attractor in the invariant subspace is transversely stable so that the basin is riddled. We find that a bifurcation from riddled to fractal basins can occur in the sense that an arbitrarily small amount of symmetry breaking can replace the riddled basin by fractal basins. We call this a catastrophe of riddling. In the second case, where the chaotic attractor in the invariant subspace is transversely unstable so that there is no riddling in the unperturbed system, the presence of a symmetry breaking, no matter how small, can immediately create fractal basins in the vicinity of the original invariant subspace. This is a smooth-fractal basin boundary metamorphosis. We analyze the dynamical mechanisms for both catastrophes of riddling and basin boundary metamorphoses, derive scaling laws to characterize the fractal basins induced by symmetry breaking, and provide numerical confirmations. The main implication of our results is that while riddling is robust against perturbations that preserve the system symmetry, riddled basins of chaotic attractors in the invariant subspace, on which most existing works are focused, are structurally unstable against symmetry-breaking perturbations.

AB - Most existing works on riddling assume that the underlying dynamical system possesses an invariant subspace that usually results from a symmetry. In realistic applications of chaotic systems, however, there exists no perfect symmetry. The aim of this paper is to examine the consequences of symmetry-breaking on riddling. In particular, we consider smooth deterministic perturbations that destroy the existence of invariant subspace, and identify, as a symmetry-breaking parameter is increased from zero, two distinct bifurcations. In the first case, the chaotic attractor in the invariant subspace is transversely stable so that the basin is riddled. We find that a bifurcation from riddled to fractal basins can occur in the sense that an arbitrarily small amount of symmetry breaking can replace the riddled basin by fractal basins. We call this a catastrophe of riddling. In the second case, where the chaotic attractor in the invariant subspace is transversely unstable so that there is no riddling in the unperturbed system, the presence of a symmetry breaking, no matter how small, can immediately create fractal basins in the vicinity of the original invariant subspace. This is a smooth-fractal basin boundary metamorphosis. We analyze the dynamical mechanisms for both catastrophes of riddling and basin boundary metamorphoses, derive scaling laws to characterize the fractal basins induced by symmetry breaking, and provide numerical confirmations. The main implication of our results is that while riddling is robust against perturbations that preserve the system symmetry, riddled basins of chaotic attractors in the invariant subspace, on which most existing works are focused, are structurally unstable against symmetry-breaking perturbations.

KW - unstable periodic-orbits

KW - chaotic dynamical-systems

KW - coupled oscillator-systems

KW - natural measure

KW - dimension variability

KW - strange attractors

KW - boundaries

KW - singularities

KW - metamorphoses

KW - intermittency

U2 - 10.1103/PhysRevE.64.056228

DO - 10.1103/PhysRevE.64.056228

M3 - Article

VL - 64

JO - Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

JF - Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

SN - 1063-651X

IS - 5

M1 - 056228

ER -