### Abstract

The success of deterministic modeling of a physical system relies on whether the solution of the model would approximate the dynamics of the actual system. When the system is chaotic, situations can arise where periodic orbits embedded in the chaotic set have distinct number of unstable directions, a dynamical property known as unstable dimension variability. As a consequence, no model of the system produces reasonably long trajectories that are realized by nature. We argue and present physical examples indicating that, in such a case, though the model is deterministic and low-dimensional, statistical quantities can still be reliably computed. We also argue that unstable dimension variability may be common in high-dimensional chaotic systems such as those arising from discretization of nonlinear partial differential equations.

Original language | English |
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Title of host publication | STATISTICAL PHYSICS |

Editors | M Tokuyama, HE Stanley |

Place of Publication | MELVILLE |

Publisher | AMER INST PHYSICS |

Pages | 531-542 |

Number of pages | 12 |

ISBN (Print) | 1-56396-940-8 |

Publication status | Published - 2000 |

Event | 3rd Tohwa University International Conference on Statistical Physics - FUKUOKA Duration: 9 Nov 1999 → 12 Nov 1999 |

### Conference

Conference | 3rd Tohwa University International Conference on Statistical Physics |
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City | FUKUOKA |

Period | 9/11/99 → 12/11/99 |

### Keywords

- UNSTABLE DIMENSION VARIABILITY
- TRANSVERSE INSTABILITY
- DYNAMIC-SYSTEMS
- TRAJECTORIES
- OSCILLATORS
- ATTRACTORS
- ORBITS

### Cite this

*STATISTICAL PHYSICS*(pp. 531-542). MELVILLE: AMER INST PHYSICS.

**Necessity of statistical modeling of deterministic chaotic systems.** / Lai, Y C ; Grebogi, C ; Lai, Ying-Cheng.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*STATISTICAL PHYSICS.*AMER INST PHYSICS, MELVILLE, pp. 531-542, 3rd Tohwa University International Conference on Statistical Physics, FUKUOKA, 9/11/99.

}

TY - GEN

T1 - Necessity of statistical modeling of deterministic chaotic systems

AU - Lai, Y C

AU - Grebogi, C

AU - Lai, Ying-Cheng

PY - 2000

Y1 - 2000

N2 - The success of deterministic modeling of a physical system relies on whether the solution of the model would approximate the dynamics of the actual system. When the system is chaotic, situations can arise where periodic orbits embedded in the chaotic set have distinct number of unstable directions, a dynamical property known as unstable dimension variability. As a consequence, no model of the system produces reasonably long trajectories that are realized by nature. We argue and present physical examples indicating that, in such a case, though the model is deterministic and low-dimensional, statistical quantities can still be reliably computed. We also argue that unstable dimension variability may be common in high-dimensional chaotic systems such as those arising from discretization of nonlinear partial differential equations.

AB - The success of deterministic modeling of a physical system relies on whether the solution of the model would approximate the dynamics of the actual system. When the system is chaotic, situations can arise where periodic orbits embedded in the chaotic set have distinct number of unstable directions, a dynamical property known as unstable dimension variability. As a consequence, no model of the system produces reasonably long trajectories that are realized by nature. We argue and present physical examples indicating that, in such a case, though the model is deterministic and low-dimensional, statistical quantities can still be reliably computed. We also argue that unstable dimension variability may be common in high-dimensional chaotic systems such as those arising from discretization of nonlinear partial differential equations.

KW - UNSTABLE DIMENSION VARIABILITY

KW - TRANSVERSE INSTABILITY

KW - DYNAMIC-SYSTEMS

KW - TRAJECTORIES

KW - OSCILLATORS

KW - ATTRACTORS

KW - ORBITS

M3 - Conference contribution

SN - 1-56396-940-8

SP - 531

EP - 542

BT - STATISTICAL PHYSICS

A2 - Tokuyama, M

A2 - Stanley, HE

PB - AMER INST PHYSICS

CY - MELVILLE

ER -