Logistics and Periodicity

Let us now return to the logistic mapping

g(x)=2x(1-x)

We have seen that there are fixed points at x=0 and x=1/2 (at the intersections between g and f(x)=x). Now g '(x) = -4x+2, so g '(0) = 2 and g '(1/2) = 0. Therefore by theorem I we have x=0 to be a source and x=1/2 to be a sink. At the end of the last essay we approached a concept called a basin of the sink. A basin in this case can be thought of as a region which when points pass through it, they get sucked into the sink, like a black hole. Now all iterations on the interval (0,1) are included in the basin of the sink for this mapping. Why? Well, 0 is a fixed point so this acts as a left most boundary, and g(1)=0 which causes any other iterations to be 0. Below is a cobweb plotting to see this visually.

Now we will do this algebraically. We will first utilize the extended result which results from theorem I (the proof of the first part of theorem I lies in text A from reference).

Extended result

|f^k(x) - p| <=(a)^k |x - p| for a<1, f^k(x) converges exponentially to p as k-->oo

That is, we compare | g(x) - 1/2 | and | x - 1/2 |.

| g(x) - 1/2 | = | 2x(1-x) - 1/2 | = 2 | x - 1/2 || x- 1/2 | (after factoring out a 2).

Now if x is on the open interval (0,1) then 2 |x - 1/2 | will be smaller than 1. And since we are multiplying this will reduce this expression again and again (with successive iterations) to our attractor. Outside of this interval we know that none of the points will converge to the sink. We call this the basin of infinity. The basin of attraction can be difficult thing to discover, especially when the mapping is fairly complicated.

Let us now move onto the next topic, of periodic points and some logistic maps. From essay one we had an extended result of the theorem concerning sinks and sources, repeated here:

Extended result

|f^k(x) - p| <=(a)^k |x - p| for a<1, f^k(x) converges exponentially to p as k-->oo

Notice the constant term a^k. Depending on different values of a we can have different fixed points, not too mention different pictures. We have the following definition:

Definition: Given a function f on R. If f^k (p) = p then p is called a periodic point of period k (also if k is the smallest such integer). The orbit of the point p (consisting of k points; iterates) is termed the periodic orbit of period k. The orbit is the minimum number of iterations required for a point to repeat in value, hence the word periodic.

A not so accurate visual interpretation of a periodic point would be a persons birthday. The earth moves in an orbit and returns to the "birthday position" once per year, after following along a given path.

Let us try the following exercise as presented in (A, pg 14): The map f(x)=2x^2 - 5x on R has fixed points at x=0 and x=3. Find a period-two orbit for f by solving f^2(x) = x for x.

After using a spreadsheet to hone down on values for a period two orbit we come up with the following:

Counter iteration-1 iteration-2
2.414 -0.415207999999998 2.42083536652799
2.4141 -0.41474238 2.41773438353613
2.4142 -0.414276719999998 2.41463400146791
2.4143 -0.413811019999997 2.41153422054686
2.4144 -0.413345279999994 2.40843504099652
2.4145 -0.412879499999994 2.40533646304046
2.4146 -0.412413679999993 2.40223848690224
2.4147 -0.411947819999993 2.39914111280546
2.4148 -0.411481919999991 2.39604434097371

We can see that approximately 2.414 satisfies the definition of a period two orbit. Extrapolating this further we have:

2.41421356237309
-0.414213562373095

Which is 1 + (2^.5), 1- (2^.5)

Alternatively if we opted to find the second iterate we have:

f^1(x)=2x^2 - 5x and f^2(x) = f(f(x)) = 2 (2x^2 - 5x)^2 - 5 (2x^2 - 5x)= 8x^4 - 40x^3 + 40x^2 +25x

We can now pull out an x term --> x(8x^3 - 40x^2 +40x + 25) this obviously has a homogenous root at zero, but zero is a fixed point so we can discount this. We first graph the cubic, then see what roots we get.

Note that removing the x term alters the picture of the graph but not the roots (except at zero).

Both the graphs of 8x^4 - 40x^3 + 40x^2 +25x and (8x^3 - 40x^2 +40x + 25) are superimposed:

The elimination of the extra x obviously reduces the number of times our graph turns but again does not affect the roots (except at zero). This gives us the roots 1+(2)^.5, 1-(2)^.5

Points near the periodic orbit may be repelled or attracted to the orbit just as in the case for fixed points. Note again that fixed points are just that, fixed. No matter how many times we iterate.

Note that we are iterating functions. This is basically taking succesive compositions. From calculus we have the chain rule for taking derivatives of functional compositions to be:

( f o g)' (x) = f ' (g(x))*g '(x).

The derivative can be interpreted as the rate of change of a function. With our definition of periodic orbit in mind, make note of the following occurence. "The derivative of f^2 at a point of a period 2 orbit is simply the product of the derivatives of f at two points in the orbit." [A; pg 15].

Note, when f=g (from the definition of periodic orbit) the derivative equates to: f '(f(x))*f '(x). Along with the theorem from essay i, we can now look at the behaviour of the above function.

We have f(x)=2x^2-5x and f '(x)= 4x-5 --> f '(2.41421356237309)*f '(-.41421356237309)

=-31. So what do we have?

A periodic sink. The orbit will be periodic, but is being drawn down a blackhole. If the derivative is greater than 1, then we would have a periodic source [see theorem one again, essay i].

The following is an extrapolation of stability for period k orbits based upon period two orbits, a proof does not follow.

If |f '(Pk) *** f '(P1)| < 1, then the orbit {P1, ... , Pk} is a sink.

If |f '(Pk) *** f '(P1)| > 1 then the orbit {P1, ... , Pk} is a source.

Let us now examine the logistic map ga(x) = ax(1 - x) using various values of a.

Note that there is obviously a sink at x = 0.

For 0 < a < 1 under the Initial conditions:

Varios initial conditions

0.1

0.5

0.7

IC =.1

IC = .5

IC = .7

0.1

0.1

0.1

0.009

0.025

0.021

0.0008919

0.0024375

0.0020559

0.000089110451439

0.000243155859375

0.000205167327519
8.91025107664444E-06 0.0000243096734603052 0.0000205125233886719
8.91017168407019E-07 2.43090825000814E-06 2.05121026250561E-06
8.91016374495425E-08 2.43090234069322E-07 2.05120605504207E-07
8.91016295104407E-09 2.4309017497646E-08 2.05120563429744E-08
8.91016287165306E-10 2.43090169067177E-09 2.051205592223E-09
8.91016286371396E-11 2.43090168476249E-10 2.05120558801555E-10
8.91016286292006E-12 2.43090168417156E-11 2.05120558759481E-11
8.91016286284066E-13 2.43090168411247E-12 2.05120558755273E-12
8.91016286283273E-14 2.43090168410656E-13 2.05120558754853E-13
8.91016286283193E-15 2.43090168410597E-14 2.0512055875481E-14
8.91016286283185E-16 2.43090168410591E-15 2.05120558754806E-15
8.91016286283185E-17 2.4309016841059E-16 2.05120558754806E-16
8.91016286283185E-18 2.4309016841059E-17 2.05120558754806E-17
8.91016286283185E-19 2.4309016841059E-18 2.05120558754806E-18
8.91016286283185E-20 2.4309016841059E-19 2.05120558754806E-19
8.91016286283185E-21 2.4309016841059E-20 2.05120558754806E-20
8.91016286283185E-22 2.4309016841059E-21 2.05120558754806E-21
8.91016286283185E-23 2.4309016841059E-22 2.05120558754806E-22
8.91016286283185E-24 2.4309016841059E-23 2.05120558754806E-23
8.91016286283185E-25 2.4309016841059E-24 2.05120558754806E-24
8.91016286283185E-26 2.4309016841059E-25 2.05120558754806E-25
8.91016286283185E-27 2.4309016841059E-26 2.05120558754806E-26
8.91016286283185E-28 2.4309016841059E-27 2.05120558754806E-27
8.91016286283185E-29 2.4309016841059E-28 2.05120558754806E-28
8.91016286283185E-30 2.4309016841059E-29 2.05120558754806E-29

For only 30 iterations we see arapid convergence to zero. Now observe for the alternate values of a = .5 & .9 under 30 iterates:

IC = .1

IC = .5

IC = .7

0.5

0.5

0.5

0.045

0.125

0.105

0.0214875

0.0546875

0.0469875

0.010512893671875

0.025848388671875

0.022389837421875

0.00520118636925943

0.0125901247374713

0.0109442663010485

0.00258706701480583

0.00621580674828313

0.00541224466809012

0.00129018704953337

0.00308858524737556

0.00269147613787143

0.000644261233455291

0.00153952294427263

0.00134211604703535

0.000321923080459179

0.000768576406688343
 0.000670157385775819
0.000160909722994723

0.000383992848497713
 0.000334854137427054
0.0000804419155278845 0.000191922698995008 0.000167371005066851
0.0000402177223130554 0.000095942932336309 0.000083671496006757
0.0000201080524239337 0.0000479668636450219 0.0000418322475437565
0.0000100538240450807

0.000023982281412507
 0.000020915248803411
5.02686148285138E-06 0.0000119908531313426 0.0000104574056778892
2.51341810675751E-06 5.99535467539189E-06 5.22864816027786E-06
1.25670589474346E-06 2.99765936555711E-06 2.61431041075814E-06
6.28352157716878E-07 1.49882518979772E-06 1.30715178806961E-06
3.14175881445222E-07 7.49411471660383E-07 6.53575039711905E-07
1.57087891369369E-07 3.74705455021415E-07 3.26787306275786E-07
7.85439333463816E-08 1.87352657308618E-07 1.63393599742921E-07
3.92719635886161E-08 9.36763111038001E-08 8.16967865227264E-08
1.96359810231645E-08 4.68381511642744E-08 4.08483899241808E-08
9.81799031879636E-09

2.3419074485231E-08
 2.04241941277949E-08
4.90899511120171E-09

1.1709536968389E-08
 1.02120968553236E-08
2.45449754355174E-09 5.85476841563786E-09 5.10604837551834E-09
1.22724876876359E-09 2.92738419067977E-09 2.5530241747233E-09
6.13624383628726E-10

1.4636920910551E-09
 1.27651208410269E-09
3.06812191626095E-10 7.31846044456352E-10 6.38256041236601E-10

IC =.1

IC = .5

IC = .7

0.9

0.9

0.9

0.081

0.225

0.189

0.0669951

0.1569375

0.1379511
0.056256080918391 0.119077308984375 0.107028534607911
0.047782200850285 0.0944081131224732 0.0860160846488348
0.0409491559189692 0.0769456991692148 0.0707555860474674
0.0353450902936439 0.0639225526937179 0.0591743097814921
0.0306862333972002 0.0538528139557531 0.0501054397590385
0.0267701296293835 0.0458574195463201 0.0428353961990327
0.0234481408101086 0.0393790649569856 0.0369004725283541
0.0206084929523921 0.0340455187800892 0.0319849448899845
0.018165404673561 0.0295977792879754 0.0278657173713322
0.0160518804719461 0.0258495756742761 0.0243802972500419
0.0142147958447344 0.0226632376005624 0.0214073085204374
0.0126114618814442 0.0199346537358206 0.0188541320961134
0.0112071716195914 0.0175835369846282 0.0166487884191141
0.00997341383149236 0.0155469205905645 0.0147344456369607
0.00888655036323434 0.0137746924656437 0.0130656075738589
0.00792682162728844 0.0122264552818086 0.0116054077253264
0.00707758841355983 0.0108692721657455 0.0103236500131698
0.00632474654022727 0.00967601797949917 0.00919536503721782
0.00565626970928624 0.00862415339000362 0.00819972926914512
0.00506184869003584 0.00769479963147838 0.00731924433815206
0.00453260374008752 0.0068720307210988 0.00653910570042344
0.00406085331908058 0.00614232532338037 0.00584671121735592
0.00363992651046134 0.00549413744670192 0.00523127446676709
0.00326400970091381 0.0049175567103768 0.0046835174107984
0.00292802034742753 0.00440403691173933 0.00419542386791503
0.00262750233984532 0.00394617723355743 0.0037600400578352

This leads to the following conclusion for every initial condition with 0 < a < 1 for the logistic map. That every initial condition is attracted to the zero sink. Notice the rapid convergence. Please note also that this conclusion was indirectly obtained without strict mathematical proof.

Let us now investigate what happens when 1 < a < 3.

I am not altering the initial conditions which are .1, .5, .9 respectively for the colums left to right.. Let us examine the results under successive iterations:

a = 1.5

1.5

1.5

1.5

0.135

0.375

0.315

0.1751625

0.3515625

0.3236625
0.216720897890625 0.341949462890625 0.328357629140625
0.254629425462159 0.337530041579157 0.330808344788659
0.284689921726455 0.335405268916094 0.33206127571027
0.30546235529076 0.334362861749125 0.332694877325907
0.318232657186472 0.333846507648091 0.33301349389051
0.325440949629764 0.333589525468896 0.333173260166018
0.329293706900762 0.333461330949499 0.333253258314547
0.331289042244476 0.333397307566332 0.333293286205927
0.332304919099821 0.333365314310779 0.333313307363972
0.332817539762823 0.333349322287882 0.333323319747093
0.333075037483567 0.333341327427137 0.333328326389806
0.333204085333331 0.333337330284377 0.333330829823965
0.333268684275764 0.333335331784892 0.333332081569248
0.333301002535298 0.333334332553122 0.33333270744894
0.333317166366395 0.33333383294173 0.333333020390549
0.333325249457808 0.333333583137157 0.333333176861794

This appears to be sinking to .3333 or 1/3

Checking a=2

2

2

2

0.18
 0.5

0.42

0.2952
 0.5

0.4872

0.41611392
 0.5

0.49967232

0.485926251164467
 0.5

0.499999785251635

0.499603859187429
 0.5

0.499999999999908

0.499999686144913
 0.5

0.5

0.499999999999803
 0.5

0.5

0.5
 0.5

0.5

0.5
 0.5

0.5

This is sinking to .5

Last Checking a= 2.5

2.5

2.5

2.5

0.225

0.625

0.525

0.4359375

0.5859375

0.6234375

0.614739990234375

0.606536865234375

0.586907958984375

0.592086836602539

0.596624740865082

0.606117516662925

0.603800036311343

0.60165914863189

0.596847681643234

0.59806388115441

0.599163543748598

0.601551316400827

0.600958688032335

0.60041647897805

0.599218325343148

0.599518358276974

0.599791326874127

0.600389309790253

0.600240240914639

0.600104227701753

0.599804966199591

We have a sink of .6. From this point on can we make any generalities about sinks when 1 < a < 3?

What information do we have again?

a = 1.5, we get a sink of .3333 or 1/3 for 3 distinct initial conditions.

a= 2, we get a sink of .5 or 1/2 for 3 distinct initial conditions.

a = 2.5, we get a sink of .6 for 3 distinct initial conditions.

Note:

(1.5 - 1)/1.5 = .5/1.5 = 1/3 = .3333

(2 - 1)/2 = 1/2 = .5

(2.5 - 1)/2.5 = 1.5/2.5 = .6

Generalization for the logistic mapping when 1 < a < 3, we have a sink at

x=(a-1)/a.

You can download a microsoft excel worksheet to experiment with various iterations by clicking:

Worksheet

This is not to difficult to design yourself, and you can alter the initial conditions and the a values for the logistic map fa(x) = ax(1-x)

These resemble the following graphically:

 

Now what happens when a > 3, one may ask?

This is the subject of essay 3.

 

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