3/7 ANSWERS TO THE EXERCISES 



-^. ..-> (100^6,100^). 20: Each is converging to 100. 21: One 

 system is stable; the other shows self-aggravating inflation. 22: 

 (80,120) -^(100,80) -^(90,1 10) ->...^(99f,100i). 24: Yes. 25:3. 



dix d2.x dx , „ 



3/^- ^=^-;^-2^^ + ^" = ^- 

 2: dxidt = y, dyjdt = — ax. 

 ^ dx dy y 2 



^'-jt=>'dt=^^-''%-Mr+y'y 



4/1. 1: Three. 2: Yes. 3: Under i?i it goes c -^ <3^^Z>; then under /?2 it 

 goes 6 -> fl -> Z); so it is at 6. 4: (i) Ri and then Rj would do; (ii) Ru 

 Ri, Rz would do. 5: It would become x' = 4, y' = 4 — y; notice that 

 the equation of the first line, belonging to x, is made actually untrue; 

 the fixing forces the machine to behave differently. 6: Within each 

 column the states must be the same. 



4/2. l:{i)g' = 2g-2h,h' = 2g - 2h;(n) g' = g - h, h' = 2g;(iii)g' =0, 

 h' = 2g + 2h. 2: (i) ft' =j, r = e "; (ii) h' = log (2 + sin h), 

 j' = \ + siny. 3: (i) 0; (ii) 2; (iii) alternately 1 and 2; (iv) a = 1 for 

 90 steps and then a = 10. 5: PF=10; yes, approximately. 6: 

 n' = n + fl2. 7: Yes; each jump is n' — n, and this measures 3a. 



4/3. l:ab=00 01 10 11 20 21 



s' = s s t -s -s+2t 



t' = t 2t t-\ 2t t-2 2t 



2: 3. 3: fl = 9/8, b = 1/8. 4: a - 9/10, b = - 1/10. 5: Four 



(fli = 0, l,2or4). 

 4/4. 1: Putting a and b always equal, i.e. making the transducer effectively 



p' =a(p + q), q' = a(p + q). 

 4/5. 1: The graph must consist of a single chain that passes through all 



states. 2: The sequence (8,4), (6,6). 

 4/7. 1 and 2: (omitting brackets) four basins: 



ai :^ bk dj -> bi ^ ak bj -> ci :^ dk 

 aj — ^ di '^ ck 



t 

 cj 



3:ai-^ck-^di->bk->ci^dk:^bi. 4: Yes. 5: «i«2. 6: n^ 

 7: Each part in succession goes to state 0. 8: The change 

 . . . 0,0,1,2,0,0, . . . occurs in each part in turn, somewhat as an impulse 

 passes along a nerve. 

 4/8. 1: ce 



I 

 ae -> df:^ bf 



\ 3: In Z put all the values of jS 



af-^ cf<- be <- de the same. 



4/9. 1: p,q; r,s,t,u. 2: (1,0,1,0,0). 



4/11. 1: Between six pairs, such as AB, there are 6; around four triples, such 

 as ABC, taken in either direction, there are 8; and around all four 

 iABCD, ABDC, ACBD, ACDB, ADBC, ADCB), there are 6. 2: 

 x' = y + 22, y' = 2z, z' = X — z. 3: Yes; the other transformation is 

 x' = y + z, y' = 2z, z' = X - 1. 4: Yes. 



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