CHAPTER CHAPTER. Discrete Dynamical Systems. 9.1 Iterative Equations. First-Order Iterative Equations. (b)

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1 CHAPTER CHAPTER 9 Discrt Damical Sstms 9. Itrativ Equatios First-Ordr Itrativ Equatios For Problms - w us th fact that th solutio of = + a + b is a = a + b. a. = + + = (a),(c) Bcaus a=, b=, =, = + = As icrass, th orbit or solutio approachs 6 from blow. Thus 6 =. 6 = + +, =. = + + = (a),(c) Bcaus a=, b=, =, = + = As icrass, th orbit or solutio approachs 6 from blow. Thus 6 =. 7 = + +, = 85

2 86 CHAPTER 9 Discrt Damical Sstms. = + + = (a),(c) Bcaus a=, b=,, = ( ) = + = + As icrass, th orbit approachs, with succssiv altrativl abov ad blow this valu. Thus =. = +, = + 4. = + + = (a),(c) Bcaus a=, b=,, = = + = + As icrass, th orbit approachs, with succssiv altrativl abov ad blow this valu. Thus =. 4 = + +, = 5. = + + = (a),(c) Bcaus a=, b=, =, = + = 4 As icrass, th solutio grows without boud. 5 = + +, =

3 SECTION 9. Itrativ Equatios = + + = (a),(c) Bcaus a=, b=, =, = + = As icrass, th solutio grows without boud. 5 = + +, = 7. = + + = (a),(c) Bcaus a=, b=, =, ( ) = ( ) + = for all Th orbit starts at ad rmais thr, bcaus this is a fid poit. It is, howvr, a rpllig fid poit, so at a othr valu for th solutio is uboudd. S Problm 8. = + +, =.5 8. = + + = (a),(c) Bcaus a=, b=, =, ( ) = ( ) + = ( ) + As icrass, th orbit displas largr ad largr oscillatios, with succssiv altratl abov ad blow zro. 5 5 = +, = +

4 88 CHAPTER 9 Discrt Damical Sstms 9. = + + = (a),(c) Bcaus a=, b=, =, = + b = + As icrass, th solutio grows without boud. 5 = + +, =. = + + = (a),(c) Bcaus a=, b=, =, = + b = + As icrass, th solutio grows without boud. 5 = + +, =. = + + = (a),(c) Bcaus a=, b=, =, ( ) = ( ) + = ( ) + As icrass, th solutio oscillats btw ad for all tim. thr is o stad stat or quilibrium, but rathr a ccl of priod. = + +, =.5

5 SECTION 9. Itrativ Equatios 89. = + + = (a),(c) Bcaus a=, b=, =, ( ) = ( ) + 5 = ( ) + As icrass, th solutio oscillats btw - ad 4 for all tim. Thr is o stad stat or quilibrium, but rathr a ccl of priod. 5 + = +, = Fishig s Ed. (a) =.4 8,, =,, has solutio + (c) (.4) =,, (.4) 8,.4 =,, (.4),, ((.4) ) =,,.4 Th gativ trm will grow util it cds th positiv trm, so tictio will occur. W ot from Figur 9.. that th orbit,, curvs slightl dowward, so w pct tictio to occur bfor =. Plottig th t poits of th orbit shows that tictio occurs btw = 7 ad = 8 (s figur). From th solutio i part (a), tictio will occur wh.4 =, or wh l = l.4 This is cosistt with th graphical stimat i part. = +, = +

6 8 CHAPTER 9 Discrt Damical Sstms Lab Problm: Spradsht Prdictios I problms 4-, wh thr is a quilibrium or stad stat solutio, its act valu is calculatd b sttig. = + 4. =. + =. (a) A simpl spradsht calculatio ilds th followig first itrats ad graph =., =. Th solutio dcrass mootoicall ad approachs a stad stat ar -.4. Calculatio givs.7 = -, or = (c) Th cofficit of is small so w ar ot surprisd to s thr is littl ffct as bcoms larg. Th costat - cotributs to th lvl of th stad stat. Th iitial valu. > causs th orbit to approach th quilibrium from abov =. - =.6 (a) A simpl spradsht calculatio ilds th followig first itrats ad graph =., = 6. Th solutio dcrass mootoicall ad approachs a stad stat ar -.4. Calculatio givs.7 = -, or = (c) Th stad stat is th sam as i Problm 4; th chag i iitial coditios to

7 SECTION 9. Itrativ Equatios = -. - =. (a) A simpl spradsht calculatio ilds th followig first itrats ad graph =., =. (c) Th solutio hibits a dampd oscillatio that quickl approachs a stad stat ar -.7. Calculatio givs. = -, or = Th small cofficit of prdicts stabilit; th gativ sig with this cofficit prdicts oscillatio about th stad stat, which dpds somwhat o th costat trm =. +.5 =. (a) A simpl spradsht calculatio ilds th followig first itrats ad graph..8 + =. + 5., =. Th solutio dcrass mootoicall ad approachs a stad stat ar.7. Calculatio givs.7 =.5, or = (c) As i Problm 4 th cofficit of is small so w ar ot surprisd to s littl ffct as bcoms larg. Th costat.5 cotributs to th quilibrium lvl. Th iitial valu =. < causs th solutio to approach from blow.

8 8 CHAPTER 9 Discrt Damical Sstms 8. + =. - =. (a) A simpl spradsht calculatio ilds th followig first itrats ad graph. 5 + =., =. (c) Th solutio dcrass mootoicall ad approachs ifiit rathr tha a stad stat. Thr is a quilibrium for this quatio, bcaus calculatio givs -. = -, or =... ; it is a ustabl quilibrium that ca ol b rachd b itratig backward with gativ. Th cofficit of is multiplid at ach stp b., th - is subtractd. Hc oc th solutio bcoms gativ it bcoms mor ad mor gativ ad th orbit gos to mius ifiit = -. - =. (a) A simpl spradsht calculatio ilds th followig first itrats ad graph =., =. (c) Th solutio oscillats with largr ad largr amplitud, ot rachig a stad stat (ulss itratio gos backward with gativ as i Problm 8). Equilibrium calculatio givs =., or = Th gativ cofficit of causs th solutio to oscillat o ach itratio ad for th amplituds to gt largr ad largr i absolut valu. Aftr a whil th costat of - that is subtractd o ach itratio dos ot caus much ffct.

9 SECTION 9. Itrativ Equatios 8. + =. +.5 =. (a) A simpl spradsht calculatio ilds th followig first itrats ad graph. 5 + =. + 5., =. (c) Th solutio icrass mootoicall without boud. Calculatio givs. +.5 =, or = Th positiv multiplir. causs to icras as soo as it bcoms positiv, ad th positiv costat.5 maks that happ o th first stp.. + = -. - =. (a) A simpl spradsht calculatio ilds th followig first itrats ad graph. 5 + =., =. (c) Eactl as i Problm 9 th solutio oscillats with largr ad largr amplitud, ot rachig a stad stat (ulss itratio gos backward with gativ ). Equilibrium calculatio =. givs = Th ol diffrc from Problm 9 is that th costat trm is ow, so th quilibrium has movd; th log-trm bhavior is sstiall th sam.

10 84 CHAPTER 9 Discrt Damical Sstms Closd-Form Sums. S = (a) If S is th sum of th first positiv itgrs, this prssio rquirs S = for th giv prssio, which ilds th followig pattr: S = S S S = + = S + = + + = S + = = S S = = S + ad so o. Hc, th itrativ quatio for S is S = S + + ( + ), with S =. To show that th closd form prssio, ( + ) S = satisfis th prcdig itrativ quatio, w ot that it givs S =, ad also that as prdictd. ( + )( + ) ( + ) ( + ) S+ S = = = +. S = (a) This prssio rquirs that S =, ad w obsrv th gral pattr S = + = S + S S S = + + = S + = = S + Hc, th itrativ quatio for S is S = = + S = + S +, with S =. + + To show that th closd form prssio S = satisfis th prcdig itrativ quatio w ot that it givs S =, ad that as prdictd. S S = + ( ) + ( ) = =,

11 SECTION 9. Itrativ Equatios S = (a) This prssio rquirs S =, ad w obsrv th gral pattr S = S + = + S S = S + = + + Hc, th itrativ quatio for S is S = = + with S =. S = S + + +, To show that th closd form prssio S ( + ) = satisfis th prcdig itrativ quatio, w ot that it satisfis S = ad that S S = ( ) ( ) ( ) = =, as prdictd. Nohomogous Structur 5. For th gral liar itrativ quatio = a + + b, (a) Th fuctio u = a is a solutio of th corrspodig homogous quatio = + a bcaus (c) u au a a a = + = +. Th fuctio a p = b a satisfis th ohomogous quatio = a + + b bcaus + a a( a ) + ( a ) a a + b = ab + b = b = b = +. a a a From part (a), u is a solutio of th homogous quatio, ad from part, p is a particular solutio of th ohomogous quatio. Hc th gral solutio to th ohomogous quatio is = u + p or a = +. a b a

12 86 CHAPTER 9 Discrt Damical Sstms Whr It Coms From 6. I Problm w foud th corrspodig itrativ IVP for th giv squc to b S S = + +, with S =. (a) Th costat squc h S S = +, bcaus h h = + c c =. = c satisfis th corrspodig homogous quatio To fid a particular solutio of th ohomogous quatio S+ S = + w tr P = A + B+ C. Substitutig this ito th quatio ilds A ( + ) + B ( + ) + C A + B+ C = +. Multiplig out th lft sid of th quatio ad collctig lik powrs of ilds A ( + + ) + B+ B+ C A B C= + ( A A) + ( A+ B B) + ( A+ B+ C C) = + A + ( A + B) = +. (c) Equatig cofficits of,, ad w gt A =, A+ B=. Hc ad thr is o coditio o C. Howvr, S = is giv, so Hc, C =, ad so our particular solutio is p = +. A =, B =, p = + + C =. Usig th homogous solutios from part (a) ad a particular solutio from part ilds th gral solutio of th ohomogous quatio, s = c+ + whr c is a arbitrar costat. Substitutig ito this prssio th iitial coditio S = w gt c =. Hc, th solutio to th giv IVP is ( + ) S = h + p = + = ( + ) =.

13 SECTION 9. Itrativ Equatios 87 Ecptioal Cas 7. Wh o igvalu of a b c = is zro, th charactristic quatio is aλ + bλ =, or λ( aλ + b) =, ad hc th solutio is b. = c + c a For ampl, th quatio + + = has th charactristic quatio λ λ = ad igvalus ad. Hc, = c+ c. Noral Eigvalus 8. I cas th roots λ ad λ of th charactristic quatio ar compl, λ ad λ ± i = r θ, which ilds two solutios i λ = ( r θ ) ad λ = ( r iθ ). Rwritig th first solutio i Cartsia form usig Eulr s quatio ilds iθ iθ ( r ) = r = r (cosθ + isi θ ). Rcall from Chaptr 6. that ral ad compl parts of a compl solutio ar also solutios; thrfor, r cos θ ad r si θ ar solutios. Hc, th gral solutio for compl i igvalus r ± θ is = cr cos θ + c r si θ. Scod-Ordr Liar Itrativ Equatios = Th charactristic quatio is r 5r+ 6=, which has roots r = ad r =. Hc, th gral solutio is = c + c = Th charactristic quatio is r 4r+ 4=, which has a doubl root r = r =. Hc, th gral solutio is = c + c.

14 88 CHAPTER 9 Discrt Damical Sstms. + = + + Th charactristic quatio is r + r =, which has roots r = ad r =. Hc, th gral solutio is = c + c = Th charactristic quatio is r 4r 4=, which has roots r = + ad r =. Hc, th gral solutio is = c ( + ) + c ( ) c (4.88) + c (.88). Lab Problms: Mor Spradshts Th radr should primt with a spradsht ad th chck th rsults with th followig solutios.. + =, =, = (a) Th followig itrats wr foud usig Ecl =, =, = It is clar from th rsults of th spradsht ad th itrativ quatio itslf that th solutio will cotiu to oscillat btw ad.

15 =, =, = SECTION 9. Itrativ Equatios 89 (a) Th log-trm bhavior is idicatd b th first itrats of th solutio =, =, = W ca solv th IVP to ascrtai th log-trm bhavior of th solutio. Th charactristic quatio is r + =, which has compl roots p ± iq =± i, ad so p = ad q = as dotd i th tt. Hc, r p q = + = + = q π ta θ = = θ =. p Hc π π = r ( ccosθ + csi θ) = ccos + csi. Substitutig th iitial coditios = ad = ilds c = ad c =.Hc, th solutio of th IVP is π = si, =,,,.

16 84 CHAPTER 9 Discrt Damical Sstms Epidmic Modl 5. Sttig th solutio = 5. = 5, ad solvig for (umbr of ars) ilds 5, (.) =. 5 Hc, log(.) = log( ) or log = 84 ars. log. Rabbits Agai 6. W rwrit th Fiboacci quatio as =. + + Th charactristic quatio is λ λ =, which has solutios 5 λ, λ = ±. Th gral solutio ca th b writt as 5 5 = c + + c. Usig th coditios = ad = ilds c c = 5 + ad = Thus, th solutio bcoms th Bit formula =. 5

17 SECTION 9. Itrativ Equatios 84 Gralizd Fiboacci Squc: Mor ad Mor Rabbits 7. (a) If ach adult rabbit pair has two rabbit pairs vr moth, th th umbr of rabbit pairs aftr moth + will satisf th discrt iitial-valu problm = +, =, ad =. + + Th charactristic quatio for th itrativ quatio i part (a) is λ λ =, which has roots ad. Hc, th gral solutio of th quatio is = c + c. Usig th iitial coditios = ad = ilds c = ad umbr of rabbit pairs aftr moth will b + = ( ) +. c =. Hc, th To giv som maig to this algbraic solutio, w compar th rgular Fiboacci squc (s third colum) whr thr is o rabbit pair bor ach moth to th cas whr ach adult has two pairs of offsprig ach moth. Not that i this cas thr will b 7 rabbit pairs at th d of th ar (aftr moths) compard with i th cas of th Fiboacci squc. Moth pairs pair Moth pairs pair Two ad o rabbit pairs bor pr moth (c) If ach rabbit pair has k rabbit pairs th IVP bcoms k =, =, ad =. + + Its charactristic quatio is λ λ k =, which has th two ral roots + 4k + 4k λ = + ad λ =. Th gral solutio ca th b writt as 4k 4k c + c + = + +. It is lft to th radr to valuat th costats c ad c that satisf th IC.

18 84 CHAPTER 9 Discrt Damical Sstms Probabilistic Fiboacci Squc 8. If ach rabbit pair pcts to hav.5 rabbit pairs ach moth, th th umbr of rabbit pairs aftr th + moth will satisf th quatio + = =, =, =. Th charactristic quatio is λ λ.5 =, which has roots λ = λ = ±. Th gral solutio ca th b writt as = c + + c which usig th coditios = ad = ilds c = ( + ) ad c = ( ). Thus, th solutio of th IVP is + = To giv som maig to this awkward formula, w compar th rgular Fiboacci squc (s third colum) whr thr is o rabbit pair bor ach moth to th cas whr ach adult has o avrag.5 pair of offsprig ach moth. Not that i this cas thr will b rabbit pairs at th d of th ar (aftr moths) compard with i th cas of th Fiboacci squc. +. Moth.5 pairs pair Moth.5 pairs pair O ad o half rabbit pairs bor pr moth Not that th umbrs i th scod colum ar ot itgrs, th rprst th pctd (avrag) umbr of pairs.

19 SECTION 9. Itrativ Equatios 84 Chck This with Your Bakr 9. W us th formula for compoud itrst. Bcaus itrst is compoudd dail, w hav.8 r =.9. Hc, if th iitial dposit is = $, th th valu i th accout 65 will b as follows. da: da: ar:.8 = $ + = $(.9) = $ $ $(.9) $.9 = + = = $ $(.9) $8.8 = + = = 65 How Much Mo Is Eough? 4. (a) As our accout ars 8% itrst ach ar ad $, is withdraw ach ar, th iitial amout is = $,. Th, aftr ars, ou will hav =.8 +,, = $,. Th solutio of th iitial-valu problm i part (a) is (.8) = (.8) $, = $,(.8) $75, (.8).8 = $75, $75,(.8). Clarl, this squc gts smallr ad smallr util ou ru out of mo. To fid out wh, st = ad solv for. Doig this ilds 75,.8 = 75, which has th solutio 9.9 ars. How to Rtir a Millioair 4. (a) W ar giv r =.8, =, ad d is uspcifid. Thus th itrativ IVP is = d, = which has th solutio (.8) = d =.5 d (.8)..8 I ordr for Shrl to b worth a millio dollars i 5 ars, solv 5.5d.8 = $,, ildig d $74.86 as hr aual dposit.

20 844 CHAPTER 9 Discrt Damical Sstms Amazig But Tru 4. (a) If wkl itrst is.8.54 (.54%), ad wkl dposits ar $5, th th 5 amout of mo Wi Ch will hav i th bak aftr wks satisfis th itrativ IVP = , =. Th solutio of th IVP i part (a) is 5 = (.54) = $6,7 (.54).54. (c) Substitutig = 8 (4 ars) ito th solutio foud i part ilds 8 = $6.94. Amortizatio Problm 4. (a) W rwrit quatio p = + r p d i th form of a ohomogous liar itrativ + p + + r p = d. W solv it b first solvig th corrspodig homogous quatio p + + r p = whos charactristic quatio is ( r) λ + = which ilds λ = + r. Th solutio of th homogous quatio is th ( ) p = c + r. p + r p = d, w p =. This givs p p + A + r A= d, or d A =. Th gral solutio is th giv b r d p = c( + r) +. r d d Sttig p = S ilds S = c+ or c= S. Hc, w fiall hav as th IVP r r solutio d d ( + r) P = S ( + r) + = S( + r) d. r r r To fid a particular solutio of th ohomogous quatio + ( ) tr a costat A =, so substitutio givs ( )

21 SECTION 9. Itrativ Equatios 845 Th valu p rprsts th outstadig pricipal, ad to hav th outstadig pricipal qual aftr N priods, st p N =, which ilds Solvig this quatio for d, ilds ( r) N N + = S( + r) d r d = Sr ( + r) N ( r) Dividig th top ad bottom b ( r) N +. N r d = S + + ilds N r (c) Th paramtrs costat i th quatio ar S = $,, r =., ad N = 6. Substitutig ths valus ito th prvious prssio ilds th mothl pamt of d = $8.6.. Fishris Maagmt 44. (a) Assum th haddock toag icrass b % pr ar, but dimiishs tos pr ar du to fishig. If th iitial stock is, tos of haddock, th th futur toag (i thousads of tos) of haddock will satisf th IVP =. +, =. W us biological growth with priodic dpltio, as i th tt Eampl, to fid th solutio (.) = (.) = 5 (.) +.. (c) If th toag of haddock caught vr ar is chagd to d = 5 ( 5, tos), th th solutio chags to (.) = (.) 5 = 5 5(.).. Hc, th toag dcrass ad rachs wh = 5.8 ars.

22 846 CHAPTER 9 Discrt Damical Sstms Dr Populatio 45. (a) Masurig dr i th thousads with r =., d = 5, ad =, ilds =. + 5 with solutio 5 = (.) (.) = 5(.) + 5. thousads of dr. I ordr for th populatio to b costat, simpl harvst th w populatio. If iitiall thr ar, ad th grow b % pr ar, harvst (hut), dr pr ar. This will kp th populatio at, dr. Sav th Whals 46. (a) If th iitial populatio is = whals, th populatio is icrasig at a aual rat of r =.5 (5%). If d = whals ar harvstd ach ar, th.5 + ad aftr ars th umbr of whals will b = (.5) (.5) = (.5) + whals..5 Th populatio is clarl dcrasig. Sttig = ilds 8.. If this cotius, th whals will b tict i 8 ars. Drug Thrap 47. (a) W hav = ad =, which has th solutio (.75) = = 4 (.75).75 grams of isuli. As icrass,.75 tds to, ad so th log-trm amout of isuli i hr bod tds to 4 grams.

23 SECTION 9. Itrativ Equatios 847 Cosquc of Priodic Drug Thrap 48. (a) W assum Kashkooli has o drug i his sstm o da zro. Howvr, is th amout of drug i his bod immdiatl aftr h taks his dos of mg. Thrfor, = mg. Furthr, bcaus h loss 5% pr da ad gais mg, th itrativ quatio that dscribs th umbr of mg of drug h has i his bod o da immdiatl aftr takig th drug is = Thus, = , =. Th itrativ quatio i part (a) ca b writt.75 = + ad has charactristic quatio r.75 =. Hc, th homogous solutio is = c.75. Fidig a particular solutio of th ohomogous quatio ilds = 4 mas th gral solutio is = c , which Substitutig th iitial coditio = ilds c =. Thus, th amout of drug h has i his bod aftr das is 4.75 =. (c) As icrass (.75) gos to zro, ad hc, tds to 4 mg. (d) Th homogous solutio c (.75) alwas tds to, as icrass o mattr what th dosag. Thus, th particular solutio is = A for +.75 = d, whr d is th dail dosag, which is A= 4d. I othr words, if th limitig amout of th drug is A, th th dail dosag should b d =.5 A mg. If Kashkooli wats th limitig amout of th drug i his bod to b A = 8 mg, th his dail dosag should b d =.5( 8) = mg.

24 848 CHAPTER 9 Discrt Damical Sstms Gral Growth Problm 49. W ar giv th quatio + r( ) which ca b rwritt as + ( r ) r This has th charactristic quatio λ ( r ) λ ( r ) =, + + =. + + r = ad roots r+ + 4r r+ r λ, λ = ± = ± = r,. Hc, th gral solutio is c c r = +. Substitutig th iitial coditios ilds c+ c = ad c+ cr =, which givs r c = ad c =. r r Thus, th populatio is = r + r r. Chims i a Da 5. Th umbr of chims + o th =. = ad + hour is giv b Th charactristic quatio of this quatio is λ =. Hc, th homogous solutio is a costat squc = c = c. To fid a particular solutio, w ormall sk a solutio of th form = A+ B. Howvr, th homogous solutio has th costat solutio, so w multipl b. This ilds th trial solutio = A + B. Substitutig this ito th itrativ quatio ilds A + + B + = A + B + +. Comparig cofficits of,, ad ilds A= B=. Hc, th gral solutio is ( + ) = + c. Substitutig th iitial coditio givs c =, ad so w gt ( + ) =. Thus, th total umbr of chims ovr a 4-hour priod is = = 56.

25 SECTION 9. Itrativ Equatios 849 Vr Itrstig 5. Oc ou mak a fw sktchs, ou will covic ourslf that th itrativ quatio = is corrct. Hc, w simpl d to solv it. As otd i th hit, th w li divids + rgios ito + rgios, so if thr wr rgios bfor, thr ar ow + + rgios. Th homogous quatio has charactristic quatio λ =, so th homogous solutios ar costats = c. W thrfor sk a particular solutio of th form = A + B. Substitutig this valu ito th ohomogous quatio ilds A + ( A + B) = +. Sttig cofficits qual ilds A= B=. Hc, th gral solutio is ( + ) = c +. Substitutig th iitial coditio = ilds c =,so = + = distict rgios. Platrs Pauts Problm 5. (a) Cout th umbr of was backwards startig at th last lttr (th S i th middl of th bottom row of th pramid), ad cout all th paths othr tha th o that gos straight up. Thr ar T paths. Now add th path that gos straight, to obtai a total of T + paths. Hc, th diffrc quatio T + T + T =. It ilds th iitial coditiot =, as a pramid with o lttr has o path. Th charactristic quatio of th homogous quatio is r =, so th homogous solutio is T = c whr c is a arbitrar costat. To fid a particular solutio, tr = A, which ilds A =. Hc, th gral solutio is T = c. Pluggig this ito th iitial coditio ilds c =, so th solutio ist =. Bcaus PLANTERS PEANUTS has 5 lttrs, thr is a total of 5 T 5 = =,767 was to spll th word. Suggstd Joural Etr 5. Studt Projct

26 85 CHAPTER 9 Discrt Damical Sstms 9. Liar Itrativ Equatios Sstm Classificatio = Comput th igvalus, λ =.75 ad λ =.5, or comput TrA = ad A =.875 Bcaus both igvalus ar positiv ad lss tha, th quilibrium poit is a sik. Usig Ecl, w plot th first fw poits, =,. startig at th poit.7 = + +. Comput th igvalus, λ =.7 ad λ =., or comput TrA = ad A =.. Bcaus both igvalus ar positiv ad lss tha, th quilibrium poit is a sik. Usig Ecl, w plot th first, =,. fw poits startig at th poit = + + Comput th igvalus, λ = i ad λ = i, or comput TrA = ad A =. From th tracdtrmiat pla i Figur 9..7 th poit (, ) for this matri A lis i th ctr rgio. Hc, th quilibrium is a ctr, ad ach itrat rmais at th sam distac from th origi. I this cas th orbit ccls aroud four poits (s figur). Usig Ecl, w plot th first fw poits startig at th poit, =, Sik Sik 4-4 Ctr = + +. Comput th igvalus, λ =.9 ad λ =., or comput TrA = ad A =.9. Bcaus both igvalus ar positiv ad gratr tha, th quilibrium poit is a sourc. Usig Ecl, w plot th, =,. first fw poits startig at th poit 5 Sourc

27 SECTION 9. Liar Itrativ Equatios = Comput th igvalus, λ =.9 ad λ =., or comput TrA = ad A =.9. Both igvalus hav a absolut valu gratr tha, so th quilibrium poit is a sourc. Howvr, both igvalus ar gativ, so it is a doubl-flip sourc. Usig Ecl, w plot th first fw poits startig at th poit, =, Doubl-flip sourc = Comput th igvalus, λ =.5 ad λ =.5, or comput TrA = ad A =.75. O igvalu has a absolut valu gratr tha, ad th othr is lss tha, so th quilibrium is a saddl. Bcaus ol o igvalu is gativ, it is a flip saddl. Usig Ecl, w plot th first fw poits startig at th poit, =,. -.5 Flip saddl = Comput th igvalus, λ =.5 ad λ =.5, or comput TrA = ad A =.5. Both igvalus hav a absolut valu gratr tha, so th quilibrium poit is a sourc. Bcaus ol o igvalu is - gativ, it is a flip sourc. Usig Ecl, w plot th first, =,. Flip sourc fw poits startig at th poit = Comput th igvalus, λ =.5 ad λ =., or comput TrA =.4 ad A =.65. Both igvalus hav a absolut valu gratr tha, so th quilibrium poit is a sourc. Bcaus ol o igvalu is gativ it is a flip sourc. Usig Ecl, w plot th first -4, =,. Flip sourc fw poits startig at th poit 4

28 85 CHAPTER 9 Discrt Damical Sstms 9..9 = Comput th igvalus, λ =.i ad λ =.i, or -5 comput TrA = ad A =.44. W s that from th trac-dtrmiat pla i Figur 9..7 th poit (,.44) of th matri A lis i th spiral sourc rgio. -5 Usig Ecl, w plot th first fw poits startig at th, =,. Spiral sourc poit 5..6 = Comput th igvalus, λ =.6 ad λ =.4, or comput TrA = ad A =.84. O igvalu has a absolut valu gratr tha, ad th othr is lss tha, so th quilibrium is a saddl. Bcaus both igvalus ar gativ, it is a doubl-flip saddl. Usig Ecl, w plot th first fw poits startig at th, =,. poit - -5 Doubl-flip saddl 4..8 = Comput th igvalus, λ =.8 ad λ =.6, or comput TrA =.4 ad A =.48. Both igvalus - hav a absolut valu lss tha, so th quilibrium poit is a sik. Bcaus both igvalus ar gativ it -.5 is a doubl-flip sik. Usig Ecl, w plot th first fw, =,. Doubl-flip sik poits startig at th poit = Comput th igvalus, λ =.8i ad λ =.8i, or comput TrA = ad A =.64. W s that from th - trac-dtrmiat pla i Figur 9..7 th poit (,.64) for th matri A lis i th spiral sik rgio. Usig Ecl, w plot th first fw poits startig at th -.5, =,. Spiral sik poit.5 4

29 (a) λ =.5, λ =. Th quilibrium (at th origi) is a sik, bcaus both igvalus ar lss tha. = (.5) = (.) (c) A fw itrats ar show (s figur) startig at (, ). (a) λ =.5, λ =.5 Th quilibrium (at th origi) is a flip sik, bcaus both igvalus hav absolut valus lss tha ad ol o igvalu is gativ. = (.5) = (.5) (c) A fw itrats ar show (s figur) startig at (, ). (a) λ =.5, λ =. Th quilibrium (at th origi) is a doubl-flip sik, bcaus both igvalus ar gativ ad hav absolut valus lss tha. = (.5) = (.) (c) A fw itrats ar show (s figur) startig at (, ). (a) λ =.5, λ = Th quilibrium (at th origi) is a sourc, bcaus both igvalus ar positiv ad gratr tha. = (.5) = (c) A fw itrats ar show (s figur) startig at (, ). SECTION 9. Liar Itrativ Equatios 85.5 Š Š Š..5

30 854 CHAPTER 9 Discrt Damical Sstms (a) λ =.5, λ = Th quilibrium (at th origi) is a flip sourc, bcaus both igvalus hav absolut valus gratr tha ad ol o igvalu is gativ. = (.5) = (c) A fw itrats ar show (s figur) startig at (, ). (a) λ =.5, λ = Th quilibrium (at th origi) is a doubl-flip sourc, bcaus both igvalus ar gativ ad hav absolut valus gratr tha. = (.5) (c) = ( ) A fw itrats ar show (s figur) startig at (, ). Š4 Š Š (a) λ =.5, λ = (c) Th quilibrium (at th origi) is a saddl, bcaus both igvalus ar positiv ad ol o igvalu is gratr tha. = = (.5) A fw itrats ar show (s figur) startig at (, ) (a) λ ] =.5, λ = (c) Th quilibrium (at th origi) is a flip saddl, bcaus o igvalu is positiv ad lss tha, whil th othr igvalu is gativ ad has a absolut valu gratr tha. = (.5) = A fw itrats ar show (s figur) startig at (, ). Š

31 SECTION 9. Liar Itrativ Equatios 855. (a) λ =.5, λ = Th quilibrium (at th origi) is a doubl-flip saddl, bcaus both igvalus ar gativ ad ol o igvalu has a absolut valu gratr tha. Š = = (.5) Š (c) A fw itrats ar show (s figur) startig at (, ). Itratio b Rotatio. (a) Th igvalus ar th roots of cosθ λ siθ = cosθ λ + si θ = cos θ λcosθ + λ + si θ siθ cosθ λ which has roots = cos + = λ λ θ cosθ ± 4cos θ 4, cos cos cos isi ± i λ λ = = θ ± θ = θ ± θ =. Th rotatio matri cosθ siθ R = siθ cosθ (which rotats th pla i th coutrclockwis dirctio b a agl θ ) has i igvalus θ. Hc, th rotatio matri for a rotatio through a agl of θ is cos θ si θ si θ cosθ. But a rotatio through a agl θ has th sam matri as th th powr R. θ

32 856 CHAPTER 9 Discrt Damical Sstms Spirals or Circls?. + = 4. π This is a pur rotatio matri with θ =. Thus coscutiv poits i th solutio ar rotatd i th coutrclockwis dirctio b 9 dgrs with o chag i th distac from th origi. Th orbit is plottd startig at (, ) (s figur). + = W rwrit th cofficit matri as Š.5.5 Š.5.5 Coutrclockwis cclic trajctor =, Š6 Outward spiral, coutrclockwis π which is a scalar two tims a rotatio matri with θ =. Hc, th actio of th itrativ 4 π sstm is a coutrclockwis rotatio of at ach itratio plus a pasio awa from th 4 origi b a factor of. S figur for a fw poits of th orbit startig at (, ). Š =.5 W rwrit th quatio as. Š =.5, Š. Spiralig iward, coutrclockwis π which is th scalar (.5) tims a rotatio matri with θ =. Hc, th actio of th itrativ 6 π sstm is a coutrclockwis rotatio of at ach itratio plus a cotractio closr to th origi 6 b a factor of.5. S th figur for a fw poits of th orbit startig at (, ).

33 SECTION 9. Liar Itrativ Equatios 857 Moos Etictio For Problms 6-8 th itrativ quatios (of th itroductor ampl) ar w =.7w +.4 m ad m =.6w +.8m (a) w = 7, m = 5 W ot that th moos populatio bcoms tict. S figur for Ecl graph. 8 w m w = 7, m = It is itrstig to s that if th iitial moos populatio wr icrasd, th rsults ar much diffrt. Th moos ca ow hold o. 5 Moos tictios w m Wolf Etictio 7. w =, m = If thr ar o moos prst ad th wolf populatio is iitiall w, w will ob th IVP w = +.7w ad w =, whos solutio is w =.7. Without a rplacmt food suppl, th wolf populatio dis out withi 5 ars (s figur). Moos ad Wolvs Togthr 8. w =, m = W usd Ecl obtai th curv show i th figur. If = corrspods to 99, th th 6th valu corrspods to 995 ad th th valu to. Th -ais is th wolf populatio ad th -ais is th moos populatio. I th ar (last poit at th uppr right) th wolf populatio is roughl ad th moos populatio is roughl. 5 Wolf ad moos both surviv w 5 m 5 Wolf tictio 5 w Moos vrsus wolf populatio

34 858 CHAPTER 9 Discrt Damical Sstms Sstm Aalsis A =.6.98 (a) Th igvalus ar λ =.8 ad λ =. with corrspodig igvctors v =, ad v =,. [ ] [ ] v TrA =.9, A =.88 (c) Th quilibrium (at th origi) is a doubl-flip saddl, bcaus both igvalus ar gativ ad ol o igvalu has a absolut valu gratr th. Th li with gativ slop is th igvctor corrspodig to λ =.8, ad th li with positiv slop is th igvctor corrspodig to λ =.. 5, gts closr ad closr to v ad that th poits flip back ad forth about v whil movig furthr ad furthr awa from th origi. Th figur shows that th solutio startig at - - Tpical doubl-flip saddl solutio v. c c (d) = (.8) + (.) Th first trm i th solutio gos to zro as a rsult of th factor (.8) whil th scod gts largr du to th factor (.). Hc, th orbit gts closr ad closr to v ad th gativ valu of λ causs th solutio to oscillat aroud v with smallr ad smallr amplituds; at th sam tim th gativ valu of λ causs th orbit to flip back ad forth about th origi with largr ad largr amplitud..4.8 A =.8.4 (a) Th igvalus ar computd as λ =.8 ad λ =. with corrspodig igvctors v =, ad v =,. [ ] [ ] v TrA =., A =.88 (c) Th li with gativ slop is th igvctor corrspodig to λ =.8 ad th li with positiv slop is th igvctor corrspodig to λ =.. - v - Sigl-flip saddl solutio

35 Th scod figur shows th solutio startig from = 5, =, which gts closr ad closr to th igvctor v. Although itrats flip back ad forth aroud v (du to th gativ igvalu) for small, th will just mov out alog th igvctor goig awa from th origi. c c (d) = (.8) + (.) SECTION 9. Liar Itrativ Equatios 859 Th first trm i th solutio gos to zro as a rsult of th factor ( ).8. Hc, th orbit gts closr ad closr to v ad oscillats aroud this igvctor with smallr ad smallr amplituds as a rsult of th gativ valu of λ. Mawhil th factor (.) causs th orbit to mov stadil awa from th origi. Owls ad Rats. O+ =.5O +.4R R =.O +.R (a) (c) + Th igvalus ad igvctors of th matri ar.5.4 A =.. λ =.58 = [ 5.6, ] v λ =. v =.764,. [ ] Hc th solutio O c(.58) c(.) R = +. W start with rats ad 5 owls; th solutio is show i th figur. Bcaus TrA=.6; A =.59, w ca s from Figur 9..7 that th origi is a saddl poit, which is what th trajctor idicats. W s that ot ol do both spcis surviv th both thriv! Th log-trm ratio will b alog v, so owls.764 rats = which fits th slop of th igvctor from th right d of th solutio through th origi. rats i thousads 5 v v 5 owls i hudrds Rats vrsus owls

36 86 CHAPTER 9 Discrt Damical Sstms Diabts Mod. (a) Th cofficit matri A =.4.99 has igvalus λ, λ =.98,.99, which tlls us that all solutios approach th origi. Corrspodig igvctors ar v = [, ] ad v = [, ]. Th gral solutio is c(.98) = + c(.99). Substitutig = ad = ilds =.95c +.6c =.c.6c which has solutio c = 7 ad c =. So th solutio of th IVP is = 7(.98) (.99). Th solutio will approach zro (, ), but vr slowl. (c) W plot th solutio usig Ecl ad s from th spradsht that th glucos lvl bcoms gativ wh = 77 miuts (arl hours). glucos lvl Solutio to Diabts Problm

37 Covrsio Job. Usig th quatio umbrs i th tt, th two quatios ar SECTION 9. Liar Itrativ Equatios 86 = a + + b () = c + + d. () Rwrit Equatio () as = c + d () ad th work with quatios () ad () to gt all th trms prssd i s. E.g., multipl Equatio () b c, to gt c = ca + + cb. (4) Now solv, rspctivl, Equatio () for () for c +, w gt c = + d c = d Substitutig ths valus i Equatio (4) ilds or Dcompositio Job 4. Giv th quatio d = a ad + cb a + d + ad bc =. + + p + q + r =, + + w lt = +. Th, r q + = + =. p p c ad Equatio W ca, thrfor, writ th two first-ordr quatios as + = r q = p p +.

38 86 CHAPTER 9 Discrt Damical Sstms Th Lilac Bush 5. (a) W call = umbr of w stms = umbr of old stms. Bcaus ach old stm grows two w stms vr ar, w hav = +. Also bcaus vr w stm bcoms a old stm th t ar, th umbr of old stms vr ar will b th sum of th old ad w stms o th prvious ar, or = +. Hc, th itrativ sstm is + = = + = + = or i matri form + + ; = =. + Th igvalus ad igvctors of th cofficit matri ar λ =, v = ; (c) (d) λ =, v =. Hc, th gral solutio is c c = +. Substitutig i th iitial valus = ad = ilds = c + c = = c c = whos solutio is c = ad c =. Hc, th umbr of stms o th plat o ar is = = +. O th 6th ar th umbr of w ad old plats is = + 66 = =. 6 I othr words, th t lilac bush has w stms ad old stms (s figur). Lilac bush

39 Saddl Rgios 6. (a) Th charactristic quatio of th cofficit matri of + = a + b = c + d is + a λ c b d λ = λ λ+ = TrA A. This is a quadratic quatio i λ, so w hav Tr = λ + λ = λ λ A ad A whr λ ad λ ar th igvalus of th cofficit matri. Hc, bcaus th rgio S is dfid b TrA < A < TrA, this iqualit provids th rlatioship btw th igvalus as λ + λ < λλ < λ + λ or quivaltl λλ λ λ + < ad λλ + λ + λ + >. SECTION 9. Liar Itrativ Equatios 86 W ca factor ach of th lft-had sids of th iqualitis i part (a) gttig λλ λ λ + = ( λ )( λ ) ad λλ λ λ ( λ )( λ ) = + +. Thus, th iqualitis bcom ( λ )( λ ) ad ( λ )( λ ) < + + >. But it is as to s that both of th prvious iqualitis hold if ithr of th followig sts of iqualitis hold: < λ < ad λ > < < ad λ >. or λ Th dtails of this simpl vrificatio ar lft for th studt. (c) As w obtaid i part o igvalu is positiv ad gratr tha, ad th othr igvalu has a absolut valu lowr th. O ca s that th origi is a saddl poit.

40 864 CHAPTER 9 Discrt Damical Sstms Sourc ad Sik Bifurcatio 7. O th trac-dtrmiat pla, isid th parabola A = ( TrA ) th igvalus of A ar 4 compl cojugats, λ, λ = α ± βi, so w cosidr th matri Not that α + βi A = α βi. A = α + β, but it is also kow that λ ( α β ) If A =, th = +. Thus, A = λ. λ =, which will sparat th sourcs ( λ > ) from th siks λ <. Hc, A = is idd th li of bifurcatio withi th rgio of compl igvalus. Lab Ercis 8. Studt Projct Suggstd Joural Etr 9. Studt Projct

41 SECTION 9. Liar Itrativ Equatios Chaos Agai Liar Itrativ Equatios Chaos Agai Attractors ad Rpllrs. = + St =, which ilds fid poits =,. =, so w hav two Th cobwb diagram shows that th quilibrium at is attractig ad that at is rpllig.. = + St =, which ilds two fid poits =, so w hav 5 = ±. Th cobwb diagram shows that both fid poits ar rpllig. Not that thr is a attractig ccl that passs btw thm.. = + St =, which ilds fid poits =,,. =, so w hav thr Th cobwb diagram shows that is attractig ad that ad ar rpllig. 4. = + cos St = cos, which ilds cos =. Th roots of this quatio ca b foud umricall usig a computr. Graph = ad = cos to discovr that th itrsct ol oc. This valu is approimatl.74 (radias). Th cobwb diagram shows this to b a attractig fid poit. Š Š + O rpllig ad o attractig fid poit Š Š + Two rpllig fid poits ad o attractig ccl Š Š + O attractig ad two rpllig fid poits Š Š + Attractig fid poit

42 866 CHAPTER 9 Discrt Damical Sstms Hidsight 5. (a) Cobwb diagrams for + = a ar show i th followig figurs for diffrt valus of a ad for diffrt iitial coditios. I ach cas = is a quilibrium poit. (i) a >. Th li + = a has slop >, which mas th orbit gos mootoicall toward ± (divrgs) for vr startig poit. Th cobwb diagram idicats this fact for a =. Š Š4 (ii) a =. Th li + = coicids with th diagoal, which mas that vr solutio is a costat, as idicatd b th cobwb diagram. I this cas all poits ar stabl quilibrium poits. Š Š4 (iii) < a <. Th li + =.5 has slop <, which mas that solutios covrg to zro from all startig poits, as show for a = Š4 4 Š4 (iv) a =. Th li + = is a horizotal li with slop. All iitial poits covrg to zro o th first itratio as idicatd b th cobwb diagram. 4 + Š4 4 Š4

43 SECTION 9. Liar Itrativ Equatios Chaos Agai 867 (v) < a <. Th li + = a has gativ slop lss stp tha. Orbits from all iitial poits covrg to zro, as idicatd b th cobwb diagram for a = Š4 4 Š4 (vi) a =. Th li + = has slop. Th orbit from vr iitial poit ccls btw 4 +,,,,, as idicatd b th cobwb diagram for =. Š4 4 Š4 (vii) a <. Th li + = a has slop stpr tha, which mas th solutio divrgs for vr startig poit, as idicatd b th cobwb diagram for a =. 4 + Š4 4 Š4 Wh a, = is a stabl quilibrium poit. Wh a th = is a ustabl quilibrium poit. (c) Wh a is positiv, itratio is mootoic ad cobwb diagrams look lik stair stps. Wh a is gativ, itratios oscillat i valu, ad cobwbs wid aroud th fid poit. b (d) Th valu of b plas a rol i locatig th fid poit. That is, =. a

44 868 CHAPTER 9 Discrt Damical Sstms Stabilit of Fid Poits 6. + (a) f = + Th slop of f ( ) at th lft-had quilibrium is ustabl, ad th right-had quilibrium is stabl. Not: th tt Figur 9..6(a) shows ol th part of f to th right of =, which is a asmptot for f ( ). If w tr a cobwb from a just to th lft of, w discovr w must graph f to lft of as wll. f = ( ) Eballig th slops of f ( ) at th quilibria shows that th origi is rpllig ad ustabl, but at th right-had fid poit = / th slop appars clos to, so w tak th drivativ to chck it out. f = 6, so f (/) =. As show i Problm 5(a)(vi), a straight li with slop producs a cclic orbit of priod ; howvr th curvatur of f = ( ) spoils th possibilit of a ccl. I fact, if th orbit is tdd ou will s it actuall covrgs, vr vr slowl, to = /. Aalzig th Data Thr ar ma was to show a solutio squc for a itrativ quatio. I problms 7- w chos a tim sris with a tpical sd, usuall =.5. A list of valus, or squcs from othr sds (ot at a quilibrium) should giv th sam iformatio. Ecl is a cllt sourc of all ths optios. 7. ( ) = +.5 (a) Startig at =.5, th fuctio itrats toward a quilibrium of zro, as show i th figur. Th quilibrium poit(s) of this itratio ar th = f =.5 ; i.., wh root(s) of.5 +.5=, which ilds = ad =. f =.5 ilds (c) Bcaus f =.5 =.5 <,.6 = +.5 th origi is asmptoticall stabl. Th othr fid poit = is ustabl bcaus f ( ) =.5( + ) =.5>

45 8. ( ) = +.8 SECTION 9. Liar Itrativ Equatios Chaos Agai 869 (a) Startig at =.5, th valus oscillat about.65, gttig closr ad closr. Th quilibrium poit(s) of this itratio ar th root(s) of or.8 ( ) = f = =,.8.8 which ilds = ad =.64. = ilds (c) Bcaus f.8( ) f =.8 =.8>.8 = +.8 f.64 =.8.64 =.8 <, th origi is ustabl ad.64 is asmptoticall stabl as th graph shows. 9. ( ) = +. (a) Startig at =.5, th fuctio itrats i a ccl, show i gra. Startig at =., th fuctio itrats toward th sam ccl, as show i black. Th quilibrium poit(s) of this itratio ar th root(s) of or. ( ) = f =.. =, which ilds = ad (c) f =.>, ad. = = +. f.. =. 6.4 =. 4.4 =. >... Hc both fid poits ar ustabl or rpllig, with a attractig ccl passig btw thm. S Problm.

46 87 CHAPTER 9 Discrt Damical Sstms. ( ) = + 4 (a) If =.5, th itrats ar =, = for all dos that alwas happ? No! E.g. startig at =., th motio appars chaotic.. Th quilibrium poit(s) of this itratio ar th root(s) of 4 ( ) = f = or 4 =, = which ilds = ad =.75. =, (c) Bcaus f 4 f = 4 = 4>, ad f.75 = 4.75 = >. Hc, both quilibrium poits ar ustabl. I this cas, thr is o attractig cclic poit btw th two rpllig poits, as thr is i Problms 6 ad 9. This lack of athig to attract is what causs th chaotic motio show i part (a). Oddl ough, howvr, thr ar crtai sds (.g., =,.5, or ) that itrat to zro ad sta thr! But a orbit with th slightst chag from ths iitial valus is chaotic.. = + si (a) Startig at =.5, th itrats quickl rach a ccl. Th quilibrium poit(s) of this itratio ar th root(s) of = f = si, or =. = ilds (c) Bcaus f cos 5 f = cos = >, th fid poit zro is ustabl, rpllig toward a ccl priod as th graph shows. = + si A qustio rmais: How dos o aalticall fid th two-ccl show i th graph? Th aswr will b foud i Problms ad.

47 SECTION 9. Liar Itrativ Equatios Chaos Agai 87. cos = + (a) Startig at =.5, th fuctio itrats i a oscillator fashio to a quilibrium.75. Th quilibrium poit(s) of this itratio ar th root(s) of = f = cos, or =.79. (c) Bcaus f = si ilds (.79) si (.79).674 f = = <, = + cos.79 is a asmptoticall stabl quilibrium, as th graph shows.. 5. = + +. (a) Startig at =.5, th fuctio itrats asmptoticall to a quilibrium just abov.. Th quilibrium poit(s) of this itratio ar th root(s) of = f = +., or.6 +. =, which ilds =. ad =.887. (c) Bcaus f = ilds f. =. =.6 <, f.887 =.887 =.774 >, = is asmptoticall stabl ad.887 is ustabl, as th graph shows. 4. = +. (a) Startig at =.5, th fuctio itrats asmptoticall to a quilibrium of about.. Th quilibrium poit(s) of this itratio ar th = f =., or root(s) of.5. =, which ilds =.9 ad =.9. (c) Bcaus f = ilds = +. f (.9) = (.9) =.84 <, f (.9) = (.9) =.84 >,.9 is asmptoticall stabl ad.9 is ustabl, as th graph shows.. 5

48 87 CHAPTER 9 Discrt Damical Sstms 5. = + (a) Startig at =.5, th fuctio sms to show chaotic itrativ bhavior, ad tdig th orbit dos ot chag that viw. Th quilibrium poit(s) of this itratio ar th root(s) of or = f =, =, which ilds = ad =. (c) Bcaus f = ilds f ( ) = ( ) = >, f = = 4>, = + both fid poits ar ustabl, with o vidc of athig attractig btw. This givs ris to th chaos w s i th tim sris = + (a) Startig at =, th fuctio ccls btw ad -..6 Th quilibrium poit(s) of this itratio ar th root(s) of = f =, or =, which ilds =.68 ad =.68. (c) f (.68) = (.68) >, f (.68) = (.68) >. -. = + Both fid poits ar rpllig, but a attractig ccl ca b s that passs btw thm. Pt Rpats 7. Pt is usig dgrs istad of radia masur. To fid th quilibrium of th squc = π + cos, w must solv th quatio cos π =. 8 8 Usig Mapl ilds = , which was th limitig valu Pt got.

49 SECTION 9. Liar Itrativ Equatios Chaos Agai 87 Rpat Pt s Rpat 8. si = + Startig at =.5, th solutio appars to b covrgig vr so slowl to. As w saw i Problm 5, for a liar itrativ quatio, a quilibrium or fid poit of = + f is asmptoticall stabl if f ( ) < ad ustabl if f ( ) >. Th ol quilibrium poit of this itratio is th root of = f = si or =. Bcaus f = cos ilds f = cos =, so th drivativ tst is icoclusiv..6 tim sris 5 = +. Pt s Paramtr 9. rsi = + A orbit diagram for this fuctio is giv i th tt as Figur 9... You will s that th rag of paramtr valus show thr is r π ; w limitd our primts to that rgio*. W show just a fw of th figurs from thos primts, ad discuss how th rlat to th orbit diagram. (a) r =.65 4-ccl r =.9 chaotic orbit (c) r =.95 -ccl Th ccls show match vrtical widows i th orbit diagram citd; th chaotic orbit lads i a prtt black vrtical bad. * (Apolog: th first pritig of th tt suggstd primtatio for r, but as ou ca vrif, thos valus giv othig but fid poits.)

50 874 CHAPTER 9 Discrt Damical Sstms Pt s Got It Dow Pat. + = A fw sampl orbits ar giv i th tabl, startig at th poits =.5, =.5, ad =. =.5 =.5 = Th cobwb graph shows that th orbit that starts with ths iitial valus approachs o. Th fid poit of this itratio is th sigl root of =, which is =. Us th drivativ tst dscribd i Problm 5 to tst its stabilit. Hr, f =, so f () = <. Thrfor, = is asmptoticall stabl, as th graph shows. Itrats of + = Th Broulli Mappig. <.5 + =.5 Th cobwb diagram is show with th orbit for =.4. Th orbit sms to act lik boucig ball i th diagram ad dos t td to go awhr, but boucs all ovr th diagram idfiitl. This is prcisl th bhavior of chaotic motio. Orbits will b chaotic for a othr tha th thr fid poits,.5 or (.5 is ot rall a fid poit, but a sigl itratio movs it to to sta). Cobwb diagram for Bakr mappig

51 Strtch ad Fold..98siπ = + =, th t poit is SECTION 9. Liar Itrativ Equatios Chaos Agai 875 (a) Start at.5 =.98si.5π =.98, which is th maimum valu for th itrativ fuctio. For >.5, th si fuctio is dcrasig ad hits at =. This mas that th t poits ar foldd ovr.5 ad bcom valus lss tha.5. I othr words, poits ar.5 gt strtchd out to ar ad th foldd back ovr.5. I this cas, oc th poits ar lss tha.5, strtchig ol comprsss th poits, so th bcom smallr ad smallr ad approach. A cobwb graph of = siπ is show, ad w obsrv th sam gral shap as = ( ). Thus, it is pctd that th itratio = + rsiπ will hibit similar proprtis to = r +. For r =.98, th slop at both fid poits has absolut valu >, so both fid poits ar rpllig. Bcaus w fid o attractig cclic bhavior btw thm, th orbit is chaotic. Itratig.98siπ. +.8 <.5 =.8( ).5 (a) This fuctio is commol calld th tt map. Start at =.5, th th t poit is =.8(.5) =.9. Hc, poits ar th ctr of th itrval gt mappd to poits > gt mappd ito poits.8( ), which big a dcrasig ar. Poits.5 fuctio, mas th largr poits gt mappd to poits ar. This ca b itrprtd as a foldig. Poits at <.5 gt mappd to.8, which big a icrasig fuctio with slop.8, mas th poits gt strtchd to th right. Itrats () for tt map

52 876 CHAPTER 9 Discrt Damical Sstms Orbit Diagram of th Tt Mappig 4. It is a as mattr to writ a computr program to fid th r <.5 + = r( ).5 for th paramtr.4 r. Th sampl program giv blow producs th figur show. Orbit diagram for tt mappig BASIC Program to Comput Orbit Diagrams REM ORBIT DIAGRAM 4 LET X =.5 REM N = # ITERATIONS FOR EACH R 5 LET R = MINR + (I - )*D REM MINR = MINIMUM R 6 FOR J = TO N 4 REM MAXR = MAXIMUM R 7 IF X >.5 THEN GO TO 5 REM RSTEPS = # OF R VALUES 8 X = *R*X 6 SCREEN 9 GOTO 7 WINDOW (,.4) - (, ) X = *R*( - X) 8 LET N = 5 IF J < THEN GOTO 9 LET MINR =.4 PLOT (R, X) LET MAXR =. NEXT J LET RSTEPS = 4 NEXT I LET D = (MAXR - MINR)/(RSTEPS - ) 5 END FOR I = TO RSTEPS It is as to chag this program to draw th orbit diagram for a othr itratio. To chag th rag of th paramtr r, simpl chag statmts 9 ad. (Not: For som itratios th paramtr is ot calld r. It is suggstd that ou simpl call it R ad ot chag th program to a w am.) To chag th itratio fuctio simpl rplac th statmts 7, 8, 9, ad b th w itratio fuctio. (Not: Most itratio fuctios will ol tak o li; th tt mappig is a coditioal fuctio that rquirs mor tha o li.) Li 4 givs th iitial coditio, which ca b chagd. (For th tt mappig, th diagram coms mor quickl if =.) I li, J < # trasit poits to skip i th plottig. This umbr is adjustabl. (For som fuctios, icludig th tt mappig, th umbr of trasits to skip must b larg to avoid traous pattrs.)

53 Chaotic Numrical Itratios 5. Usig Nwto s formula to approimat a ral root of f = + r+ = ilds th formula + = + r +. + r SECTION 9. Liar Itrativ Equatios Chaos Agai 877 Orbit diagram for Nwto s mthod W adapt th BASIC program i Problm 4 b rplacig statmts 7 b th sigl li X ^+ R X + X = X X ^+ R ad chagig th rag of th paramtr r ad th widow bouds to grat th orbit diagram show i th figur. Th rsultig computr program is giv i th Tabl blow. Not that th orbit diagram hibits ithr cclic or chaotic bhavior, dpdig o r. This shars faturs of orbit diagrams for othr fuctios studid. Furthrmor, th orbit diagram looks lik distortd copis of th logistic bifurcatio diagram, strtchd horizotall ad shruk, but diffrt factors, i th vrtical dirctio. BASIC Program to Comput Orbit Diagrams REM ORBIT DIAGRAM 4 LET X =.5 REM N = ITERATIONS FOR EACH R 5 LET R = MINR + (I )*D REM MINR = MINIMUM R 6 FOR J = TO N 4 REM MAXR = MAXIMUM R 7 LET X = X (X^ + R * X + ) / ( * X^ + R) 5 REM RSTEPS = # OF R VALUES 8 IF J < THEN 6 SCREEN 9 PLOT (R, X) 7 WINDOW (.,.) (.5, ) NEXT J 8 LET N = NEXT I 9 LET MINR =. END LET MAXR =.5 LET RSTEPS = 4 LET D = (MAXR - MINR)/(RSTEPS - ) 5 FOR I = TO RSTEPS Etrmum Problm 6. Th valus of th fuctio f = r( ) ar gratr or qual to zro i th itrval [ ],. It dscribs a upsid-dow parabola that crosss th -ais at =, (thus attais a maimum r valu of f (.5) = at =.5, < f < r. If f ( ) is to rmai i th itrval [, ], th 4 ma r = 4. Hc, f < r 4.

54 878 CHAPTER 9 Discrt Damical Sstms Squtial Aalsis 7. (a) Bcaus + is a mooto dcrasig squc boudd blow b w kow that it covrgs to its gratst lowr boud L. Also th squc + is alwas dcrasig bcaus if r, th r( ) <. Multiplig b ilds ( ) r < or, i othr words, < +. W hav + = r, =,, for ad r, hc th gratst lowr boud L of th squc is gratr tha or qual to zro also. (c) Th gratst lowr boud L of th mooto dcrasig squc { } it will vtuall li blow a positiv umbr. Hc L =. + is iasmuch as Mattr of Siz 8. (a) W hav a fid poit of = + r wh r ( ) =. Solvig for ilds r = =. r r Computig th drivativ of = ( ) f r =. Evaluatig th drivativ at ilds f r( ) r = w fid r r r r+ f ( ) = r( ) = r = r = r r r. Hc for r f lss tha o, th it is stabl. < w hav ( ). If th slop at a fid poit has absolut valu

55 SECTION 9. Liar Itrativ Equatios Chaos Agai 879 Not Quit a Two-Ccl 9. r = (a) ( ) = + Th tim sris for th first itrats looks vr much lik it sttls ito a two-ccl.8 f ( ) has fid poits =,. Th first at is rpllig, but at th drivativ tst f = is icoclusiv bcaus f =. Th scod itrat fuctio f ( f ( )) has ol two fid poits, ad (th root has multiplicit ), so thr is ot i fact (c) a -ccl. Th lack of a -ccl mas th tim sris i part (a) is i fact covrgig, trml slowl, to th fid poit, btw th highs ad lows of th orbit show i part (a). (d) Th trml slow covrgc to th fid poit causs th cobwb to appar solid black i th viciit. Fidig Two-Ccl Valus. If f r( ) =, th ( ) = = ( ) ( ) f f rf f r r r. If w st this valu to w gt..( ).( ) =. This fourth-dgr quatio was solvd with Mapl, ildig th four (approimat) roots,.5,.687, ad.799. Th four roots ca b s i th figur as th itrsctio of th curvs ( ) = f f ad =. Of th four fid poits for th scod itrativ fuctio = f ( f ), ad.687 ar ustabl ad.5 ad.799 ar stabl. Th first two ar th fid poits (rpllig) of f. Th lattr two ar cclic poits (attractig) of priod for f ( ). Graph of = f f

56 88 CHAPTER 9 Discrt Damical Sstms Four-Ccl Valus. (a) Usig Mapl with paramtr r =., fid th 6th ordr polomial for th fourth itrat fuctio: ( ( ( ))) = ( ). ( ).4 ( ) (. ( ) ).768 ( ). ( ).4 ( ). ( ) f f f f ( ) Th graph is o th itrval [, ]. Solvig th quatio ( ( ( ))) ( ( )) Graph of f f f f ( ) for r =. f f f f = ilds four ral roots (to thr placs),.5,.687, ad.799. W s from th graph that th absolut valu of th slop of ( ( ( ))) f f f f is gratr tha at ad.687 ad lss tha at.5 ad.799. Thus,.5 ad.799 ccl with priod four. But bcaus thr ar ol two such poits, th ccl is actuall priod two, as s i Problm. That is, th itrats of = +. will td to oscillat btw ths two valus. With r =.5 w gt ( ( ( ))) = 5.65 ( ).5 ( ).5 ( ) (.5 ( ) ) ( ).5 ( ).5 ( ).5 ( ) f f f f ( ) This graph is o th itrval [, ]. Not: As should b pctd, btw vr pair of attractig fid poits thr is a rpllig fid poit, ad vic vrsa. Cotiud o t pag. ( ( )) Graph of f f f f ( ) for r =.5

57 Cotiud from prvious pag. ( ( )) Solvig SECTION 9. Liar Itrativ Equatios Chaos Agai 88 f f f f = for r =.5 ilds ight ral (approimat) roots,.8,.48,.5,.74,.87,.857, ad.874. From clos ispctio of th graph, w s f f f f is gratr tha at,.48,.74, ad ( ( )) that th absolut valu of.874, whras th absolut valu is lss tha at.8,.5,.87, ad.874. Hc, th poits.8,.87,.5, ad.874 form a four-ccl. Th followig tabl shows th valus (from Ecl with 6 dcimal placs) for itrats 9 through 99 i th squc =.5( + ), startig at =. to illustrat two priods of this four-ccl r =. -ccl r =.5 4-ccl