CHAOS THEORY - CRITICAL POINTS

+Critical points

                       a>0 no critical pts
                     |  |  a=0 degenerate critical pts
             ._--_.  |  |   |     a<0 2 nondegen critical pts 
            /      \ | /   /     /      (saddle function)
           /        \|/   /     /
 ---------/----------+---------/------------
         /     /    /|\       /
        /     /    / | \     /
       /     |    |  |  -___-
      /     /     |  |    ^
     /               |    |
                         Critical point - equilibrium pt = attractor

family f(x) = x^3/3 + ax
       x^3/3 = degenerate critical point 
       ax    = stabilizer (polynomial)

+Local/Global Behavior
   Local Infor @ critical point -> global behavior
                       
                     |  
             @@@@@@  |               Information of behavior near
            /      \ |           /   critical points (@@@@@) Gives  /
           /        \|          /    global behavior of function ( / )
 ---------/----------+---------/------------                      /
         /           |\       /
        /            | \     /
       /             |  @@@@@
                     |   
                     |  
Critical point -> dominant terms of taylor expansion @ x0
Critical point (singularity) attractor/repellor / lorenz shifter

+Whitney's theorem (hassler whitney 1955)
any smooth transformation (w/ no creases taking points
of a plane into another plane , only types of points are regular pt, folds, cusps

++Morse Theorem (marston Morse 1935)
A morse function ( = function w/ no degenerate critical points)
near a critical point of index k (=# dimensions that bend up)
there is a smooth change of coordinates  s.t. the tayl of the
taylor series expansion of f vanishes (diffeomorphism
= smooth equivalence)
    f(x)=a0(xo) + [a1(xo)](x-x0) + [a2(xo)](x-x0)^2 + ...
                   f'(x)  (    )    f''(x) (    )   + tayl
Tayl = higher order terms of first non zero a factor
1. Small perturbation of a morse function = another morse function
2. degenerate ritical pt xform into nondegenerate critical pt w/ small "jiggle " of original function

+Splitting Lemma non-morse function f=p(y)+Q(y) ; p(y) = non morse order 3+, q(y)=morse quadratic

+Thom Classification Theorem

corank
/codim fn fold      name               spatial          temporal
1/1    y^3          fold               boundary         begin/end
1/2    y^4          cusp               pleat,fault      separate/unite,change
1/3    y^5          swallowtail        split,furrow     split;tear
1/4    y^6          butterfly          pocket           give/rcv;fill/empty
2/3    y1^3-3y1y2^2 elliptic umbilic   wave crest,arch  collapse engulf
2/3    y1^3+y2^3    hyperbolic umbilic spike,hair       drill 
2/4    y1^2y2+y2^4  parabolic umbilic  mouth            open/close,eject

+corank = degree of independence = # of vanishings in 2nd order in taylor's expansion
+codimension (morse function ness)minimal # of independent terms to add to make non-morse function into morse
+Catastrophe theory( 1970 thom)
becoming degenerate critical point =
bifurcation = catastrophe pt = lorenz shift pt (equilib shift)=
discoherent / discontinuous jump (stock boom/bust,morpheogenesis,origami fold)

1) Morpheogenesis
2) 11/1940 Tacoma Narrow Bridge

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