11— Non-Linear Transformation Studies On Electronic Computers: With P. R. Stein (LADC-5688, 1963)

### Appendix II

1. In this appendix we collect the photographs and tables illustrating the phenomena discussed in section IV and VI. The notation used has been described in sections II, III, IV, and VI. For convenience, some of the transformations are written out explicitly.

2. Modifications of the TransformationTA. In shorthand notation, this transformation is C1= {3, 5, 7, 9, 10} , C2 = {1,2,8} 1

In the S, a coordinates, this reads: 3S2 3 2 +S' = F(S, a) S3-6S2a-3Sa2 + 4a3+ 3Sa- +1, 2 2 a' = G(S, a) -S 3 + 3Sa2 + 2a3+S2-3Sa +a2 . (2) 2 2

The (repellent) fixed point has coordinates: So = 0.5885696, ao = 0.1388662.

The generalized transformations based on TA may be written: S' = (1 -At)S + At{F(S, a) + efrs(S, a)}, a' = (1 - At)a + At{G(S, a) + egs(S, a)} ;

the original transformation is recovered by setting At = 1, e = 0.

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TABLE A Figure number At e r, s Comments A-6, Class IV limit set A-2, Period: k A-2, Period: k A-2, Class I limit set A-2, Class I limit set A-2, Scaled plot of part of A- Scaling factor .(< S< 485, < a < A-2, Scaled plot of part of A- Scaling factor ~(<S< 492, <a < A-2, Pseudo-period A-2, Scaled plot of part of A- Scaling factor ~(<S< 620, <a< A-5, Class IV limit set A-5, Scaled plot of part of A- Scaling factor ~(,<S< 64, < a < A-3, Period: k A-3, Class IV limit set A-5, Period: k A-5, Class IV limit set A-successive iterates of T()p) Initia point: So 7034477, ao Scaling factor ~

Modifications ofTB. In shorthand notation, this transforma- tion is C{2,4,6,7,, C{5,8, . In the S,a coordinates, it takes the form S' F(S, a) -a-S-a- aS a ( a'G(S, a) - -S - Sa - a-S- a.

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The coordinates of the fixed point are So = 0.6887703, ao = 0.1592083.

The only associated T(r,s)e discussed is the case (r,s) = (1,0); for this case, the generalized transformation, written in the form of equation (4), has frs = f0o = (S-a) (8) grs = 910= 0 .

In table B below, all scaled plots show the region: 0.30 <S< 0.65, 0.20 < a < 0.36, i.e., the upper left-hand piece of the complete limit set (shown in figure B-1 for the unmodified transformation). The scale factor is ~ 2.9.

TABLE B Figure number At e r, s Comments B-Class IV limit set (points) B-- Scaled plot of part of figure B-B-Class IV B-Class IV Compare figure B- B-- Class IV B-- Period: k (points in this piece) B-- Class IV B-1, Class IV: separate pieces (shown here) Compare figure B-B-1, Period: k (points shown here) B-1, Class IV B-ll 1, Period: k (points shown here) B-1, Class IV B-1, Period: k (points shown here) B-1, Period: k (points shown here) B-Class IV; shows "transition" from period with k (At

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4.Modifications ofTD andTE. These transformations are given by the schemes: T C1 - {2, 7,8,9, 10}, C2 {= 4,5,6} , C1- = {2,5, 7,8,9} (10) C 2 = {4,6,10} .

In the S, a coordinates, these read explicitly: S'= S3 + S2a+ Sa2+ a3- 6Sa-6a2 + S + a T2 2 2 2 2 2 3 9 2 2 9 32 2 3 a= S3+ S2a- -Sa2 - a-3 + 3a2 + S-a, (11)

with fixed points: So = 0.6525211 (12) ao = 0.3056821; S' = 9S2 a - a3 - 3S2 - 12Sa + 3a2 +3S + 3a (13), a' =-3S2 a + 3a3 + 6Sa - 6a2

with fixed point: So = 0.6444612, (14) ao = 0.3219578 .

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TABLE D Figure number At e r,sComments D-- D-Compare figure D-D-96, Limit set for At superimposed on L(TD) D-- D-1, D-1, Compare figure D- TABLE E Figure number At e r, s Comments E-- E-1, Shows convergence to periodic limit (set k from initial point close to fixed point E-- Compare with E-5, E-E-Compare with E-5, E-E-1, E-1,

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5. Broken Linear Transformations. These are described in section VI. Figures H-1 and H-2 show, respectively, 1000 points in the limit sets of T1 and T2 . The latter are specified as follows: 1 1 Ti l= , Y 1, dl 1 0.95, (15) X2 = 0.6, Y2 = 0.5, d2 = 0.95,

with fixed point: 1 2 xo= 2' Yo= 3 (16) and T x = 0.5, yi = 0.9 , d = (1 ,7 2 2 = 0.3, Y2 = 0.7, d 2 = 0.8 , with fixed point: 80 72 43' o= 143 (18)

The remaining figures, H-3 through H-17, show the limit sets belonging to the one-parameter family T,: T Xi=Y1 = z, d = 11 x2 =2 = 1-z, d2 = 19

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The identification is given by the following table (all figures except H-11 show 1000 points):

TABLE H Figure number z Comments H-For z - 5, see the remarks in section VI, note H-H-Compare figure H-H-H-H-Like "class IV" limit set H-Like "class IV" limit set H-H- Points of H-plus the next consecutive iterates H-H-H-H-H-H-Fixed point becomes attractive below z

11— Non-Linear Transformation Studies On Electronic Computers: With P. R. Stein (LADC-5688, 1963)