### Abstract

Dynamical systems possessing symmetries have invariant manifolds. According to the transversal stability properties of this invariant manifold, nearby trajectories may spend long stretches of time in its vicinity before being repelled from it as a chaotic burst, after which the trajectories return to their original laminar behavior. The onset of chaotic bursting is determined by the loss of transversal stability of low-period periodic orbits embedded in the invariant manifold, in such a way that the shadowability of chaotic orbits is broken due to unstable dimension variability, characterized by finite-time Lyapunov exponents fluctuating about zero. We use a two-dimensional map with an invariant subspace to estimate shadowing distances and times from the statistical properties of the bursts in the transversal direction. A stochastic model (biased random walk with reflecting barrier) is used to relate the shadowability properties to the distribution of the finite-time Lyapunov exponents.

Original language | English |
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Article number | 046213 |

Number of pages | 9 |

Journal | Physical Review. E, Statistical, Nonlinear and Soft Matter Physics |

Volume | 66 |

Issue number | 4 |

DOIs | |

Publication status | Published - Oct 2002 |

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### Keywords

- on-off intermittency
- periodic-orbits
- Lyapunov exponents
- systems
- attractors
- synchronization
- oscillators
- bifurcation
- transients
- crises

### Cite this

*Physical Review. E, Statistical, Nonlinear and Soft Matter Physics*,

*66*(4), [046213]. https://doi.org/10.1103/PhysRevE.66.046213

**Chaotic bursting at the onset of unstable dimension variability.** / Viana, R L ; Pinto, S E D ; Grebogi, C .

Research output: Contribution to journal › Article

*Physical Review. E, Statistical, Nonlinear and Soft Matter Physics*, vol. 66, no. 4, 046213. https://doi.org/10.1103/PhysRevE.66.046213

}

TY - JOUR

T1 - Chaotic bursting at the onset of unstable dimension variability

AU - Viana, R L

AU - Pinto, S E D

AU - Grebogi, C

PY - 2002/10

Y1 - 2002/10

N2 - Dynamical systems possessing symmetries have invariant manifolds. According to the transversal stability properties of this invariant manifold, nearby trajectories may spend long stretches of time in its vicinity before being repelled from it as a chaotic burst, after which the trajectories return to their original laminar behavior. The onset of chaotic bursting is determined by the loss of transversal stability of low-period periodic orbits embedded in the invariant manifold, in such a way that the shadowability of chaotic orbits is broken due to unstable dimension variability, characterized by finite-time Lyapunov exponents fluctuating about zero. We use a two-dimensional map with an invariant subspace to estimate shadowing distances and times from the statistical properties of the bursts in the transversal direction. A stochastic model (biased random walk with reflecting barrier) is used to relate the shadowability properties to the distribution of the finite-time Lyapunov exponents.

AB - Dynamical systems possessing symmetries have invariant manifolds. According to the transversal stability properties of this invariant manifold, nearby trajectories may spend long stretches of time in its vicinity before being repelled from it as a chaotic burst, after which the trajectories return to their original laminar behavior. The onset of chaotic bursting is determined by the loss of transversal stability of low-period periodic orbits embedded in the invariant manifold, in such a way that the shadowability of chaotic orbits is broken due to unstable dimension variability, characterized by finite-time Lyapunov exponents fluctuating about zero. We use a two-dimensional map with an invariant subspace to estimate shadowing distances and times from the statistical properties of the bursts in the transversal direction. A stochastic model (biased random walk with reflecting barrier) is used to relate the shadowability properties to the distribution of the finite-time Lyapunov exponents.

KW - on-off intermittency

KW - periodic-orbits

KW - Lyapunov exponents

KW - systems

KW - attractors

KW - synchronization

KW - oscillators

KW - bifurcation

KW - transients

KW - crises

U2 - 10.1103/PhysRevE.66.046213

DO - 10.1103/PhysRevE.66.046213

M3 - Article

VL - 66

JO - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

JF - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

SN - 1539-3755

IS - 4

M1 - 046213

ER -