Transient Chaos in Higher Dimensions

Ying Cheng Lai, T. Tél

Research output: Chapter in Book/Report/Conference proceedingChapter

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 languageEnglish
Title of host publicationApplied Mathematical Sciences (Switzerland)
PublisherSpringer
Pages265-310
Number of pages46
DOIs
Publication statusPublished - Jan 1 2011

Publication series

NameApplied Mathematical Sciences (Switzerland)
Volume173
ISSN (Print)0066-5452
ISSN (Electronic)2196-968X

Fingerprint

Chaos theory
Higher Dimensions
Chaos
Saddle
High-dimensional
Unstable
Scattering
Dynamical systems
Trajectories
Box-counting Dimension
Stable Manifold
Intersect
Phase Space
Initial conditions
Dynamical system
Experiments
Trajectory
Experiment

Keywords

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

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

Lai, Y. C., & Tél, T. (2011). Transient Chaos in Higher Dimensions. In 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.

Applied Mathematical Sciences (Switzerland). Springer, 2011. p. 265-310 (Applied Mathematical Sciences (Switzerland); Vol. 173).

Research output: Chapter in Book/Report/Conference proceedingChapter

Lai, YC & Tél, T 2011, Transient Chaos in Higher Dimensions. in 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
Lai YC, Tél T. Transient Chaos in Higher Dimensions. In Applied Mathematical Sciences (Switzerland). Springer. 2011. p. 265-310. (Applied Mathematical Sciences (Switzerland)). https://doi.org/10.1007/978-1-4419-6987-3_8
Lai, Ying Cheng ; Tél, T. / Transient Chaos in Higher Dimensions. Applied Mathematical Sciences (Switzerland). Springer, 2011. pp. 265-310 (Applied Mathematical Sciences (Switzerland)).
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