Math 312 Lecture Notes One Dimensional Maps

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1 Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example, a populatio icreases by 15 percet each year. Let p be the populatio at the ed of year, ad assume that p is give. The a icrease of 15 percet each year gives p +1 = 1.15p. (1) If p = 1, the p 1 = 1.15(1) = 115, p 2 = 1.15(115) = 132.3, p 3 = 1.15(132.3) = 152.1, ad so o. Table 1 shows the first ie iteratios of this process. The data is plotted i Figure 1. We derive a formula for p by otig that p p Table 1: Iteratio of the map (1), with p = 1. p 1 = 1.15p p 2 = 1.15p 1 = (1.15) 2 p p 3 = 1.15p 2 = (1.15) 3 p. (2) p = (1.15) p Thus we have the familiar expoetial growth of the populatio. More geerally, the same argumet shows that the solutio to p +1 = ap, (3) 1

2 p Figure 1: Plot of the data for the simple populatio growth model (1). The data is i Table 1. give p, is p = a k p. (4) Let s compare the discrete time map to the cotiuous time model. Recall that a first order differetial equatio that leads to expoetial growth (or decay) is which has the solutio dp dt = rp, p() = p, (5) p(t) = p e rt = (e r ) t p. (6) I ve writte the solutio as (e r ) t p to make clear the aalogy betwee the cotiuous solutio ad the solutio to the discrete model i (4). The term e r is aalogous to a, ad t is aalogous to. Let s rewrite (1) as p +1 = (1 +.15)p = p +.15p (7) or p +1 p =.15p. (8) This form of the equatio expresses the discrete chage i p as a fuctio of p. (Such a equatio is ofte called a differece equatio.) Compare this to equatio (5), which gives the istataeous rate of chage of p as a fuctio of p. Oe-dimesioal Maps The geeral oe-dimesioal map has the form +1 = f( ), give x. (9) 2

3 Table 2: Iteratio of the logistic map (1), with r = 2.8 ad x = Figure 2: The first several iterates of the logistic map (1) with r = 2.8 ad x =.1. This is a plot of the data show i Table 2. We ll develop several tools for studyig such maps. Cobwebbig : The Graphical Iteratio Procedure There is a simple graphical procedure for geeratig the iteratios of a oe-dimesioal map. As a example, we cosider the logistic map +1 = r (1 ) (1) where r > is a costat. Table 2 shows the first few iterates i the sequece that results whe r = 2.8 ad x =.1. Figure 2 shows the plot of as a fuctio of. The graphical techique for fidig the iteratios of this map begis by plottig the graph of the f(x) = rx(1 x), as i Figure 3 (where r = 2.8). Give x, we draw a lie from the x = x o the x axis up the the graph of x to obtai x 1. To obtai x 2, we eed to go to x = x 1 o the x axis. This ca be doe by drawig a lie horizotally from (x, x 1 ) to the lie y = x. See the upper left plot of Figure 4. To fid x 2, we agai draw a lie vertically at x = x 1 to the graph, ad from the graph we draw a lie horizotally to y = x. See the upper right plot of Figure 4. We cotiue this process to geerate further iteratios, as i the lower left ad right plots of Figure 4. I these plots, I ve icluded vertical lies from the x axis up to the graph, but we do t really eed these lies. I practice, the procedure is: draw a lie vertically to the graph, the draw a lie horizotally to y = x, ad the repeat the process. A example is show i Figure 5. 3

4 Figure 3: A graph of the logistic map rx(1 x) for r = x x x x 1.5 x x x 1.5 x 2 x 3 1 x x 1.5 x 2 x 4 x 3 1 Figure 4: A sequece of plots that illustrate cobwebbig. The iitial poit is x =.1. 4

5 Figure 5: Whe cobwebbig, we usually do t bother droppig lies dow to the horizotal axis. After the first vertical lie is draw from (x, x ) to the graph, we draw a lie horizotally to the lie y = x, ad the aother lie vertically to graph, ad repeat. This cobweb plot shows the same data listed i Table 2 ad plotted i Figure 2. Liear Maps A liear oe-dimesioal map is +1 = a (11) where a is a costat. (The populatio model (1) is a example.) The solutio is Let s look at how the behavior of the solutio depeds o a. = a x. (12) 1. If a > 1, the a >, ad a icreases without boud as icreases. So grows expoetially. 2. If a = 1, the a = 1 for all, so = x. 3. If < a < 1, the a >, ad a approaches zero as icreases; decays to zero mootoically. 4. If a =, the = for all >. 5. If 1 < a <, the a alterates sig, ad a approaches zero as icreases; decays to zero while alteratig sig. 6. If a = 1, the a = ( 1), which is 1 if is eve ad 1 if is odd. Thus alterates betwee x ad x. 7. If a < 1, the a alterates sig, ad a icreases without boud as icreases. So alterates sig while grows expoetially. Overall, the magitude of grows if a > 1, ad decays if a < 1. Figures 6 ad 7 show sample plots of ad cobweb diagrams for the various cases. 5

6 3 a = a = a = 1. 3 a = a =.75 3 a = a =. 3 a = Figure 6: Examples of liear maps, a. I all cases, x = 1. 6

7 3 2 a = a = a = a = a = a = Figure 7: Examples of liear maps, a <. I all cases, x = 1. 7

8 Fixed Poits, Stability, Liearizatio A fixed poit of the map (9) is a poit x where f(x ) = x. If x = x, the = x for all >, so the fixed poits of f are the costat solutios to (9). I the logistic map show i Figure 3, we see there are two poits where f(x) = x. These are the poits where the graph of f crosses the lie y = x. For the example show i Figure 3, the graph is f(x) = 2.8x(1 x), so to fid the fixed poits, we must solve 2.8x(1 x) = x = 2.8x x = = x = or x = (13) So the two fixed poits are x = ad x = Defiitio. The fixed poit x is a sik or attractor if there is a eighborhood N of x such that x for all x i N. We also say x is asymptotically stable. Defiitio. The fixed poit x is a source or repellor if there is a eighborhood N of x such that if x is i N, the evetually leaves N. Defiitio. The fixed poit x is ustable if for every eighborhood N of x, there are poits arbitrarily close to x whose iterates leave N. Note that a source is ustable, but a ustable fixed poit is ot ecessarily a source. See, for example, Figure 1. Behavior ear a fixed poit: the liearizatio. Let x be a fixed poit of (9). If x is close to x, we ca approximate f(x) with the taget lie at x : f(x) f(x ) + f (x )(x x ). (14) Let u = x (i.e. = x + u ). The, by replacig with x + u i (9), we obtai x + u +1 = f(x + u ) f(x ) + f (x )u. (15) Sice f(x ) = x, we ca cacel x o the left ad f(x ) o the right. We are left with the liearizatio of the map at x : u +1 = f (x )u (16) This is a liear map. We have already see how the solutio to the liear map depeds o a = f (x ); see Figures 6 ad 7. However, the liearizatio is just a approximatio to the actual map (9). The followig theorem tells us whe the liear approximatio is good eough to classify the stability of the fixed poit x of (9). Theorem 1. (i) If f (x ) < 1, the x is a sik. (ii) If f (x ) > 1, the x is a source. 8

9 Figure 8: A example that shows why we ca ot determie the stability of a fixed poit based o the liearizatio whe f (x ) = 1. The fixed poit is x =, ad f (x ) = 1. I this case, x is a source. Example. Cosider agai the logistic map (1) with r = 2.8. The graph is show i Figure 3, ad earlier we foud the fixed poits to be x 1 = ad x 2 = We fid We use Theorem 1 to determie the stability of the fixed poits. f (x) = 2.8(1 2x). (17) At x 1, we have f () = 2.8 > 1, so by Theorem 1, x 1 is a source. At x 2, we have f (1.8/2.8) =.8, so by Theorem 1, x 2 is a sik. If f (x ) = 1, we ca ot make ay coclusios about the stability of the fixed poit. Examples demostratig why this is the case are show i Figures 8, 9 ad 1. 9

10 Figure 9: A example that shows why we ca ot determie the stability of a fixed poit based o the liearizatio whe f (x ) = 1. The fixed poit is x =, ad f (x ) = 1. I this case, x is a sik Figure 1: Aother example that shows why we ca ot determie the stability of a fixed poit based o the liearizatio whe f (x ) = 1. I this case, x = 1/2 is a fixed poit, ad f (x ) = 1. The iteratios of poits close to but less tha x coverge to x, but iteratios of poits that are greater tha x diverge from x. I this case, x is ustable (but it is ot a source). 1

10.6 ALTERNATING SERIES

10.6 ALTERNATING SERIES 0.6 Alteratig Series Cotemporary Calculus 0.6 ALTERNATING SERIES I the last two sectios we cosidered tests for the covergece of series whose terms were all positive. I this sectio we examie series whose