### Abstract

The nonhyperbolic structure of synchronization dynamics is investigated in this work. We argue analytically and confirm numerically that the chaotic dynamics on the synchronization manifold exhibits an unstable dimension variability, which is an extreme form of nonhyperbolicity. We analyze the dynamics in the synchronization manifold and in its transversal direction, where a tonguelike structure is formed, through a system of two coupled chaotic maps. The unstable dimension variability is revealed in the statistical distribution of the finite-time transversal Lyapunov exponent, having both negative and positive values. We also point out that unstable dimension variability is a cause of severe modeling difficulty.

Original language | English |
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Pages (from-to) | 462-468 |

Number of pages | 7 |

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

Volume | 62 |

Issue number | 1 |

Publication status | Published - Jul 2000 |

### Keywords

- LYAPUNOV EXPONENTS
- DYNAMICAL-SYSTEMS
- TRAJECTORIES
- BIFURCATION
- SETS

### Cite this

**Unstable dimension variability and synchronization of chaotic systems.** / Viana, R L ; Grebogi, C .

Research output: Contribution to journal › Article

*Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*, vol. 62, no. 1, pp. 462-468.

}

TY - JOUR

T1 - Unstable dimension variability and synchronization of chaotic systems

AU - Viana, R L

AU - Grebogi, C

PY - 2000/7

Y1 - 2000/7

N2 - The nonhyperbolic structure of synchronization dynamics is investigated in this work. We argue analytically and confirm numerically that the chaotic dynamics on the synchronization manifold exhibits an unstable dimension variability, which is an extreme form of nonhyperbolicity. We analyze the dynamics in the synchronization manifold and in its transversal direction, where a tonguelike structure is formed, through a system of two coupled chaotic maps. The unstable dimension variability is revealed in the statistical distribution of the finite-time transversal Lyapunov exponent, having both negative and positive values. We also point out that unstable dimension variability is a cause of severe modeling difficulty.

AB - The nonhyperbolic structure of synchronization dynamics is investigated in this work. We argue analytically and confirm numerically that the chaotic dynamics on the synchronization manifold exhibits an unstable dimension variability, which is an extreme form of nonhyperbolicity. We analyze the dynamics in the synchronization manifold and in its transversal direction, where a tonguelike structure is formed, through a system of two coupled chaotic maps. The unstable dimension variability is revealed in the statistical distribution of the finite-time transversal Lyapunov exponent, having both negative and positive values. We also point out that unstable dimension variability is a cause of severe modeling difficulty.

KW - LYAPUNOV EXPONENTS

KW - DYNAMICAL-SYSTEMS

KW - TRAJECTORIES

KW - BIFURCATION

KW - SETS

M3 - Article

VL - 62

SP - 462

EP - 468

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 - 1

ER -