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Lakes of Wada

November 14, 2023

In mathematics, the lakes of Wada (和田の湖, Wada no mizuumi) are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary. In other words, for any point selected on the boundary of one of the lakes, the other two lakes' boundaries also contain that point.

First five stages of the lakes of Wada

More than two sets with the same boundary are said to have the Wada property; examples include Wada basins in dynamical systems. This property is rare in real-world systems.

The lakes of Wada were introduced by Kunizō Yoneyama (1917, page 60), who credited the discovery to Takeo Wada. His construction is similar to the construction by Brouwer (1910) of an indecomposable continuum, and in fact it is possible for the common boundary of the three sets to be an indecomposable continuum.

Construction of the lakes of Wada edit

 
Animation of digging lakes up to day 5

The Lakes of Wada are formed by starting with a closed unit square of dry land, and then digging 3 lakes according to the following rule:

  • On day n = 1, 2, 3,... extend lake n mod 3 (= 0, 1, 2) so that it is open and connected and passes within a distance 1/n of all remaining dry land. This should be done so that the remaining dry land remains homeomorphic to a closed unit square.

After an infinite number of days, the three lakes are still disjoint connected open sets, and the remaining dry land is the boundary of each of the 3 lakes.

For example, the first five days might be (see the image on the right):

  1. Dig a blue lake of width 1/3 passing within √2/3 of all dry land.
  2. Dig a red lake of width 1/32 passing within √2/32 of all dry land.
  3. Dig a green lake of width 1/33 passing within √2/33 of all dry land.
  4. Extend the blue lake by a channel of width 1/34 passing within √2/34 of all dry land. (The small channel connects the thin blue lake to the thick one, near the middle of the image.)
  5. Extend the red lake by a channel of width 1/35 passing within √2/35 of all dry land. (The tiny channel connects the thin red lake to the thick one, near the top left of the image.)

A variation of this construction can produce a countable infinite number of connected lakes with the same boundary: instead of extending the lakes in the order 1, 2, 0, 1, 2, 0, 1, 2, 0, ...., extend them in the order 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, ... and so on.

Wada basins edit

 
Newton fractal forming Wada basins of attraction for z3 − 1 = 0; all three disconnected open basins have the same boundary

Wada basins are certain special basins of attraction studied in the mathematics of non-linear systems. A basin having the property that every neighborhood of every point on the boundary of that basin intersects at least three basins is called a Wada basin, or said to have the Wada property. Unlike the Lakes of Wada, Wada basins are often disconnected.

An example of Wada basins is given by the Newton fractal describing the basins of attraction of the Newton–Raphson method for finding the roots of a cubic polynomial with distinct roots, such as z3 − 1; see the picture.

Wada basins in chaos theory edit

In chaos theory, Wada basins arise very frequently. Usually, the Wada property can be seen in the basin of attraction of dissipative dynamical systems. But the exit basins of Hamiltonian systems can also show the Wada property. In the context of the chaotic scattering of systems with multiple exits, basins of exits show the Wada property. M. A. F. Sanjuán et al.[1] has shown that in the Hénon–Heiles system the exit basins have this Wada property.

See also edit

References edit

  • Brouwer, L. E. J. (1910), "Zur Analysis Situs" (PDF), Mathematische Annalen, 68 (3): 422–434, doi:10.1007/BF01475781
  • Yoneyama, Kunizô (1917), "Theory of Continuous Set of Points", Tôhoku Mathematical Journal, 12: 43–158
  1. ^ Wada basins and chaotic invariant sets in the Henon-Heiles system, Phys. Rev. E 64, 066208 (2001)

Further reading edit

  • Breban, Romulus; Nusse, H E. (2005), "On the creation of Wada basins in interval maps through fixed point tangent bifurcation", Physica D, 207 (1–2): 52–63, Bibcode:2005PhyD..207...52B, doi:10.1016/j.physd.2005.05.012
  • Coudene, Yves (2006), "Pictures of hyperbolic dynamical systems" (PDF), Notices of the American Mathematical Society, 53 (1): 8–13, ISSN 0002-9920, MR 2189945
  • Gelbaum, Bernard R.; Olmsted, John M. H. (2003), Counterexamples in analysis, Mineola, N.Y.: Dover Publications, ISBN 0-486-42875-3 example 10.13
  • Hocking, J. G.; Young, G. S. (1988), Topology, New York: Dover Publications, p. 144, ISBN 0-486-65676-4
  • Kennedy, J; Yorke, J.A. (1991), "Basins of Wada", Physica D, 51 (1–3): 213–225, Bibcode:1991PhyD...51..213K, doi:10.1016/0167-2789(91)90234-Z
  • Sweet, D.; Ott, E.; Yorke, J. A. (1999), "Complex topology in Chaotic scattering: A Laboratory Observation", Nature, 399 (6734): 315, Bibcode:1999Natur.399..315S, doi:10.1038/20573

External links edit

  • An experimental realization of Wada basins (with photographs), andamooka.org
  • An introduction to Wada basins and the Wada property www-chaos.umd.edu
  • , miqel.com
  • Wada basins: Rendering chaotic scattering, astronomy.swin.edu.au

November 14, 2023
lakes, wada, this, article, includes, list, general, references, lacks, sufficient, corresponding, inline, citations, please, help, improve, this, article, introducing, more, precise, citations, february, 2023, learn, when, remove, this, template, message, mat. This article includes a list of general references but it lacks sufficient corresponding inline citations Please help to improve this article by introducing more precise citations February 2023 Learn how and when to remove this template message In mathematics the lakes of Wada 和田の湖 Wada no mizuumi are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary In other words for any point selected on the boundary of one of the lakes the other two lakes boundaries also contain that point First five stages of the lakes of WadaMore than two sets with the same boundary are said to have the Wada property examples include Wada basins in dynamical systems This property is rare in real world systems The lakes of Wada were introduced by Kunizō Yoneyama 1917 page 60 who credited the discovery to Takeo Wada His construction is similar to the construction by Brouwer 1910 of an indecomposable continuum and in fact it is possible for the common boundary of the three sets to be an indecomposable continuum Contents 1 Construction of the lakes of Wada 2 Wada basins 3 Wada basins in chaos theory 4 See also 5 References 6 Further reading 7 External linksConstruction of the lakes of Wada edit nbsp Animation of digging lakes up to day 5The Lakes of Wada are formed by starting with a closed unit square of dry land and then digging 3 lakes according to the following rule On day n 1 2 3 extend lake n mod 3 0 1 2 so that it is open and connected and passes within a distance 1 n of all remaining dry land This should be done so that the remaining dry land remains homeomorphic to a closed unit square After an infinite number of days the three lakes are still disjoint connected open sets and the remaining dry land is the boundary of each of the 3 lakes For example the first five days might be see the image on the right Dig a blue lake of width 1 3 passing within 2 3 of all dry land Dig a red lake of width 1 32 passing within 2 32 of all dry land Dig a green lake of width 1 33 passing within 2 33 of all dry land Extend the blue lake by a channel of width 1 34 passing within 2 34 of all dry land The small channel connects the thin blue lake to the thick one near the middle of the image Extend the red lake by a channel of width 1 35 passing within 2 35 of all dry land The tiny channel connects the thin red lake to the thick one near the top left of the image A variation of this construction can produce a countable infinite number of connected lakes with the same boundary instead of extending the lakes in the order 1 2 0 1 2 0 1 2 0 extend them in the order 0 0 1 0 1 2 0 1 2 3 0 1 2 3 4 and so on Wada basins edit nbsp Newton fractal forming Wada basins of attraction for z3 1 0 all three disconnected open basins have the same boundaryWada basins are certain special basins of attraction studied in the mathematics of non linear systems A basin having the property that every neighborhood of every point on the boundary of that basin intersects at least three basins is called a Wada basin or said to have the Wada property Unlike the Lakes of Wada Wada basins are often disconnected An example of Wada basins is given by the Newton fractal describing the basins of attraction of the Newton Raphson method for finding the roots of a cubic polynomial with distinct roots such as z3 1 see the picture Wada basins in chaos theory editIn chaos theory Wada basins arise very frequently Usually the Wada property can be seen in the basin of attraction of dissipative dynamical systems But the exit basins of Hamiltonian systems can also show the Wada property In the context of the chaotic scattering of systems with multiple exits basins of exits show the Wada property M A F Sanjuan et al 1 has shown that in the Henon Heiles system the exit basins have this Wada property See also editList of topologies List of concrete topologies and topological spacesReferences editBrouwer L E J 1910 Zur Analysis Situs PDF Mathematische Annalen 68 3 422 434 doi 10 1007 BF01475781 Yoneyama Kunizo 1917 Theory of Continuous Set of Points Tohoku Mathematical Journal 12 43 158 Wada basins and chaotic invariant sets in the Henon Heiles system Phys Rev E 64 066208 2001 Further reading editBreban Romulus Nusse H E 2005 On the creation of Wada basins in interval maps through fixed point tangent bifurcation Physica D 207 1 2 52 63 Bibcode 2005PhyD 207 52B doi 10 1016 j physd 2005 05 012 Coudene Yves 2006 Pictures of hyperbolic dynamical systems PDF Notices of the American Mathematical Society 53 1 8 13 ISSN 0002 9920 MR 2189945 Gelbaum Bernard R Olmsted John M H 2003 Counterexamples in analysis Mineola N Y Dover Publications ISBN 0 486 42875 3 example 10 13 Hocking J G Young G S 1988 Topology New York Dover Publications p 144 ISBN 0 486 65676 4 Kennedy J Yorke J A 1991 Basins of Wada Physica D 51 1 3 213 225 Bibcode 1991PhyD 51 213K doi 10 1016 0167 2789 91 90234 Z Sweet D Ott E Yorke J A 1999 Complex topology in Chaotic scattering A Laboratory Observation Nature 399 6734 315 Bibcode 1999Natur 399 315S doi 10 1038 20573External links editAn experimental realization of Wada basins with photographs andamooka org An introduction to Wada basins and the Wada property www chaos umd edu Reflective Spheres of Infinity Wada Basin Fractals miqel com Wada basins Rendering chaotic scattering astronomy swin edu au Retrieved from https en wikipedia org w index php title Lakes of Wada amp oldid 1158052753, wikipedia, wiki, book, books, library,

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