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E8 and Catastrophe Theory(Bifurcation) - help

  1. Mar 17, 2009 #1

    MTd2

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    I've been looking to what kind of bifurcation E8 corresponds to, but I didn´t find anythng. The only place I've found anything was on wikipedia, but it doesn't say anything. Can anyone help me?

    http://en.wikipedia.org/wiki/Catastrophe_theory
     
  2. jcsd
  3. Mar 17, 2009 #2

    tiny-tim

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    Hi MTd2! Are you any relation of R2d2? :smile:

    As I expect you found, the A,D,E classification of catastrophes by names beginning with A D and E makes the As correspond to the A series of Lie groups, the Ds to the D series, and the Es to the E series … http://en.wikipedia.org/wiki/ADE_classification

    In turn, the A series of Lie groups correspond to SU groups, and the B series to SL groups.

    There are also C and D series of Lie groups (not connected with catastrophes), and finally the E F and G series, officially known as "exceptional" cases … http://en.wikipedia.org/wiki/Simple_Lie_group#Infinite_series

    so I guess the E8 catastrophe has to be categorised as exceptional, and not corresponding to anything other than itself. :redface:
     
  4. Mar 17, 2009 #3

    MTd2

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    Well, people used to call me R2D2 at college! But that's because I use a wheelchair! :biggrin::biggrin: :rofl: :rofl: :rofl:

    Now, seriously, I am asking somthing a little bit different. According to this list provided in the wiki (http://en.wikipedia.org/wiki/Catastrophe_theory): [Broken]

    Vladimir Arnold gave the catastrophes the ADE classification, due to a deep connection with simple Lie groups.

    * A0 - a non-singular point: V = x.
    * A1 - a local extrema, either a stable minimum or unstable maximum V = \pm x^2 + a x.
    * A2 - the fold
    * A3 - the cusp
    * A4 - the swallowtail
    * A5 - the butterfly
    * Ak - an infinite sequence of one variable forms V=x^{k+1}+\cdots
    * D4- - the elliptical umbilic
    * D4+ - the hyperbolic umbilic
    * D5 - the parabolic umbilic
    * Dk - an infinite sequence of further umbilic forms
    * E6 - the symbolic umbilic V = x3 + y4 + axy2 + bxy + cx + dy + ey2
    * E7
    * E8
    //////


    There is just one exceptional explained, but I couldn't find the other 2. I'd like to know what is the stable polynomial of the other 2 Es as well as its sketched behavior.
     
    Last edited by a moderator: May 4, 2017
  5. Mar 17, 2009 #4
    I don't know too much about this myself, but I found http://www.math.purdue.edu/~agabriel/dynkin.pdf" [Broken] which gives the singularities corresponding to E6, E7, E8 (for analytic functions C^3->C). You probably want to check the source cited there (particularly [1] by Arnold) for more info.
     
    Last edited by a moderator: May 4, 2017
  6. Mar 17, 2009 #5

    MTd2

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    Yes, but I've seen that on several places. Note for example the tetrahedral formula
    x^4 + y^3 + z^2, is more complete on wikipedia, because it has the full stable form with 5 control parameters. I said above that this one was explained, well, sorry. It wasn't, except for expliciting the 5 parameters.

    But there is nothing for the other groups. And I checked Arnold's book, in the end, and it is just explained in a very sketchy way the icosahedron/dodecahedron group. I'd like something more complete.
     
  7. Mar 17, 2009 #6
    The codimension 6 singularity

    [tex]x^3+xy^3+z^2[/tex]

    has a universal unfolding ("stable form")

    [tex]x^3+xy^3+z^2+ax+bx^2+cy+dy^2+exy+fx^2y[/tex].

    The codimension 7 singularity

    [tex]x^3+y^5+z^2[/tex]

    has a universal unfolding

    [tex]x^3+y^5+z^2+ax+by+cy^2+dy^3+exy+fxy^2+gxy^3[/tex].

    Also note that [1] in the link I gave is not Arnold's book but an article in Functional Analysis and Applications.
     
  8. Mar 17, 2009 #7

    MTd2

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    Yes, but I can't get that article.
     
  9. Mar 17, 2009 #8
    This is also contained in http://www.mat.univie.ac.at/~michor/catastrophes.pdf" [Broken] (written by one of my profs in the year I was born). Check out p. 39 and 46.
     
    Last edited by a moderator: May 4, 2017
  10. Mar 17, 2009 #9

    MTd2

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    Hmm, Thank you! :biggrin:
     
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