Partial control of chaotic systems

Samuel Zambrano, Miguel A F Sanjuán, James A. Yorke

Research output: Contribution to journalArticle

23 Citations (Scopus)

Abstract

In a region in phase space where there is a chaotic saddle, all initial conditions will escape from it after a transient with the exception of a set of points of zero Lebesgue measure. The action of an external noise makes all trajectories escape faster. Attempting to avoid those escapes by applying a control smaller than noise seems to be an impossible task. Here we show, however, that this goal is indeed possible, based on a geometrical property found typically in this situation: the existence of a horseshoe. The horseshoe implies that there exist what we call safe sets, which assures that there is a general strategy that allows one to keep trajectories inside that region with control smaller than noise. We call this type of control partial control of chaos.

Original languageEnglish
Article number055201
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume77
Issue number5
DOIs
Publication statusPublished - May 6 2008

Fingerprint

Chaotic System
Horseshoe
escape
Partial
Trajectory
Control of Chaos
trajectories
Saddle
Lebesgue Measure
Set of points
Exception
Phase Space
saddles
Initial conditions
chaos
Imply
Zero
Strategy

ASJC Scopus subject areas

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

Cite this

Partial control of chaotic systems. / Zambrano, Samuel; Sanjuán, Miguel A F; Yorke, James A.

In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 77, No. 5, 055201, 06.05.2008.

Research output: Contribution to journalArticle

@article{b31096e65c5c4f338c455d6c8da72fb2,
title = "Partial control of chaotic systems",
abstract = "In a region in phase space where there is a chaotic saddle, all initial conditions will escape from it after a transient with the exception of a set of points of zero Lebesgue measure. The action of an external noise makes all trajectories escape faster. Attempting to avoid those escapes by applying a control smaller than noise seems to be an impossible task. Here we show, however, that this goal is indeed possible, based on a geometrical property found typically in this situation: the existence of a horseshoe. The horseshoe implies that there exist what we call safe sets, which assures that there is a general strategy that allows one to keep trajectories inside that region with control smaller than noise. We call this type of control partial control of chaos.",
author = "Samuel Zambrano and Sanju{\'a}n, {Miguel A F} and Yorke, {James A.}",
year = "2008",
month = "5",
day = "6",
doi = "10.1103/PhysRevE.77.055201",
language = "English",
volume = "77",
journal = "Physical Review E",
issn = "1063-651X",
publisher = "American Physical Society",
number = "5",

}

TY - JOUR

T1 - Partial control of chaotic systems

AU - Zambrano, Samuel

AU - Sanjuán, Miguel A F

AU - Yorke, James A.

PY - 2008/5/6

Y1 - 2008/5/6

N2 - In a region in phase space where there is a chaotic saddle, all initial conditions will escape from it after a transient with the exception of a set of points of zero Lebesgue measure. The action of an external noise makes all trajectories escape faster. Attempting to avoid those escapes by applying a control smaller than noise seems to be an impossible task. Here we show, however, that this goal is indeed possible, based on a geometrical property found typically in this situation: the existence of a horseshoe. The horseshoe implies that there exist what we call safe sets, which assures that there is a general strategy that allows one to keep trajectories inside that region with control smaller than noise. We call this type of control partial control of chaos.

AB - In a region in phase space where there is a chaotic saddle, all initial conditions will escape from it after a transient with the exception of a set of points of zero Lebesgue measure. The action of an external noise makes all trajectories escape faster. Attempting to avoid those escapes by applying a control smaller than noise seems to be an impossible task. Here we show, however, that this goal is indeed possible, based on a geometrical property found typically in this situation: the existence of a horseshoe. The horseshoe implies that there exist what we call safe sets, which assures that there is a general strategy that allows one to keep trajectories inside that region with control smaller than noise. We call this type of control partial control of chaos.

UR - http://www.scopus.com/inward/record.url?scp=43449086684&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43449086684&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.77.055201

DO - 10.1103/PhysRevE.77.055201

M3 - Article

AN - SCOPUS:43449086684

VL - 77

JO - Physical Review E

JF - Physical Review E

SN - 1063-651X

IS - 5

M1 - 055201

ER -