### Abstract

Systems of coupled chaotic maps and flows arise in many situations of physical and biological interest. The aim of this paper is to analyze and to present numerical evidence for a common type of nonhyperbolic behavior in these systems: unstable dimension variability. We show that unstable periodic orbits embedded in the dynamical invariant set of such a system can typically have different numbers of unstable directions. The consequence of this may be severe: the system cannot be modeled deterministically in the sense that no trajectory of the model can be realized by the natural chaotic system that the model is supposed to describe and quantify. We argue that unstable dimension variability can arise for small values of the coupling parameter. Severe modeling difficulties, nonetheless, occur only for reasonable coupling when the unstable dimension variability is appreciable. We speculate about the possible physical consequences in this case.

Original language | English (US) |
---|---|

Pages (from-to) | 5445-5454 |

Number of pages | 10 |

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

Volume | 60 |

Issue number | 5 A |

State | Published - Nov 1999 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

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

*60*(5 A), 5445-5454.

**Unstable dimension variability in coupled chaotic systems.** / Lai, Ying-Cheng; Lerner, David; Williams, Kaj; Grebogi, Celso.

Research output: Contribution to journal › Article

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*, vol. 60, no. 5 A, pp. 5445-5454.

}

TY - JOUR

T1 - Unstable dimension variability in coupled chaotic systems

AU - Lai, Ying-Cheng

AU - Lerner, David

AU - Williams, Kaj

AU - Grebogi, Celso

PY - 1999/11

Y1 - 1999/11

N2 - Systems of coupled chaotic maps and flows arise in many situations of physical and biological interest. The aim of this paper is to analyze and to present numerical evidence for a common type of nonhyperbolic behavior in these systems: unstable dimension variability. We show that unstable periodic orbits embedded in the dynamical invariant set of such a system can typically have different numbers of unstable directions. The consequence of this may be severe: the system cannot be modeled deterministically in the sense that no trajectory of the model can be realized by the natural chaotic system that the model is supposed to describe and quantify. We argue that unstable dimension variability can arise for small values of the coupling parameter. Severe modeling difficulties, nonetheless, occur only for reasonable coupling when the unstable dimension variability is appreciable. We speculate about the possible physical consequences in this case.

AB - Systems of coupled chaotic maps and flows arise in many situations of physical and biological interest. The aim of this paper is to analyze and to present numerical evidence for a common type of nonhyperbolic behavior in these systems: unstable dimension variability. We show that unstable periodic orbits embedded in the dynamical invariant set of such a system can typically have different numbers of unstable directions. The consequence of this may be severe: the system cannot be modeled deterministically in the sense that no trajectory of the model can be realized by the natural chaotic system that the model is supposed to describe and quantify. We argue that unstable dimension variability can arise for small values of the coupling parameter. Severe modeling difficulties, nonetheless, occur only for reasonable coupling when the unstable dimension variability is appreciable. We speculate about the possible physical consequences in this case.

UR - http://www.scopus.com/inward/record.url?scp=0001460303&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001460303&partnerID=8YFLogxK

M3 - Article

C2 - 11970417

AN - SCOPUS:0001460303

VL - 60

SP - 5445

EP - 5454

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

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

SN - 1539-3755

IS - 5 A

ER -