### Abstract

This chapter is devoted to transient chaos in higher-dimensional dynamical systems. The defining characteristic of high-dimensional transient chaos is that the underlying chaotic set has unstable dimension more than one, in contrast to most situations discussed in previous chapters, where chaotic sets have one unstable dimension. We shall call nonattracting chaotic sets with one unstable dimension low-dimensional, while those having unstable dimension greater than one high-dimensional. The increase in the unstable dimension from one represents a highly nontrivial extension in terms of what has been discussed so far about transient chaos. For instance, the PIM-triple algorithm, which is effective for finding an approximate continuous trajectory on a low-dimensional chaotic saddle, is generally not applicable to high-dimensional chaotic saddles. In a scattering experiment in high-dimensional phase space, the presence of a chaotic saddle cannot guarantee that chaos can be physically observed. In particular, if the box-counting dimension of the chaotic saddle is low, its stable manifold may not intersect a set of initial conditions prepared in the corresponding physical space; only when the dimension is high enough can chaotic scattering be observed.

Original language | English |
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Title of host publication | Applied Mathematical Sciences (Switzerland) |

Publisher | Springer |

Pages | 265-310 |

Number of pages | 46 |

DOIs | |

Publication status | Published - Jan 1 2011 |

### Publication series

Name | Applied Mathematical Sciences (Switzerland) |
---|---|

Volume | 173 |

ISSN (Print) | 0066-5452 |

ISSN (Electronic) | 2196-968X |

### Fingerprint

### Keywords

- Chaotic Attractor
- Invariant Subspace
- Lyapunov Exponent
- Stable Manifold
- Unstable Manifold

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Applied Mathematical Sciences (Switzerland)*(pp. 265-310). (Applied Mathematical Sciences (Switzerland); Vol. 173). Springer. https://doi.org/10.1007/978-1-4419-6987-3_8

**Transient Chaos in Higher Dimensions.** / Lai, Ying Cheng; Tél, T.

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

*Applied Mathematical Sciences (Switzerland).*Applied Mathematical Sciences (Switzerland), vol. 173, Springer, pp. 265-310. https://doi.org/10.1007/978-1-4419-6987-3_8

}

TY - CHAP

T1 - Transient Chaos in Higher Dimensions

AU - Lai, Ying Cheng

AU - Tél, T.

PY - 2011/1/1

Y1 - 2011/1/1

N2 - This chapter is devoted to transient chaos in higher-dimensional dynamical systems. The defining characteristic of high-dimensional transient chaos is that the underlying chaotic set has unstable dimension more than one, in contrast to most situations discussed in previous chapters, where chaotic sets have one unstable dimension. We shall call nonattracting chaotic sets with one unstable dimension low-dimensional, while those having unstable dimension greater than one high-dimensional. The increase in the unstable dimension from one represents a highly nontrivial extension in terms of what has been discussed so far about transient chaos. For instance, the PIM-triple algorithm, which is effective for finding an approximate continuous trajectory on a low-dimensional chaotic saddle, is generally not applicable to high-dimensional chaotic saddles. In a scattering experiment in high-dimensional phase space, the presence of a chaotic saddle cannot guarantee that chaos can be physically observed. In particular, if the box-counting dimension of the chaotic saddle is low, its stable manifold may not intersect a set of initial conditions prepared in the corresponding physical space; only when the dimension is high enough can chaotic scattering be observed.

AB - This chapter is devoted to transient chaos in higher-dimensional dynamical systems. The defining characteristic of high-dimensional transient chaos is that the underlying chaotic set has unstable dimension more than one, in contrast to most situations discussed in previous chapters, where chaotic sets have one unstable dimension. We shall call nonattracting chaotic sets with one unstable dimension low-dimensional, while those having unstable dimension greater than one high-dimensional. The increase in the unstable dimension from one represents a highly nontrivial extension in terms of what has been discussed so far about transient chaos. For instance, the PIM-triple algorithm, which is effective for finding an approximate continuous trajectory on a low-dimensional chaotic saddle, is generally not applicable to high-dimensional chaotic saddles. In a scattering experiment in high-dimensional phase space, the presence of a chaotic saddle cannot guarantee that chaos can be physically observed. In particular, if the box-counting dimension of the chaotic saddle is low, its stable manifold may not intersect a set of initial conditions prepared in the corresponding physical space; only when the dimension is high enough can chaotic scattering be observed.

KW - Chaotic Attractor

KW - Invariant Subspace

KW - Lyapunov Exponent

KW - Stable Manifold

KW - Unstable Manifold

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U2 - 10.1007/978-1-4419-6987-3_8

DO - 10.1007/978-1-4419-6987-3_8

M3 - Chapter

AN - SCOPUS:85067928319

T3 - Applied Mathematical Sciences (Switzerland)

SP - 265

EP - 310

BT - Applied Mathematical Sciences (Switzerland)

PB - Springer

ER -