essayiii

Chaosophy

(Period 3 implies Chaos)

We are going to explore the logistic map fa (x) =ax(1 - x) for a>3. Similar to the extrapolations in essay two. We have a fixed point for x=(a-1)/a, except this is unstable due to the derivative being greater then 1.

Computer experiment 1.3 (see reference A)

Useing the logistic map program (available as an excel download). Investaigate the behaviour of fa (x) for a near 3.45.

We start with a=3.43

IC =.1	IC = .5	IC = .7
3.43	3.43	3.43
0.3087	0.8575	0.7203
0.7319767833	0.4191245625	0.6910349313
0.672920627993562	0.8350649311795	0.732324396729591
0.754937705501091	0.472419017195313	0.672367035166915
0.634573208456468	0.85489076259922	0.755593345794043
0.795382876870461	0.425500234912583	0.633425102639317
0.558228919257434	0.838462762556585	0.796438255010909
0.845870217860041	0.464569347183598	0.55608655811368
0.447182107923245	0.853194214124642	0.846710242136436
0.847931227048629	0.429620695583446	0.445186587432091
0.442277444068413	0.840510364538758	0.847194530023125
0.846071605420966	0.459800732330969	0.444033937044534
0.446704142650248	0.851957184754537	0.846756553304619
0.847757261951487	0.432612660541837	0.445076432099233
0.442692561586816	0.841924186428974	0.847153067793441
0.846235391233641	0.456491328020654	0.444132683999096
0.446315214730497	0.851006994436696	0.846794431499793

Notice the rapid rate of convergence among the period 2 orbits. Let's examine the following data sheets for

a=3.44 and a=3.45:

IC = .1	IC = .5	IC = .7
3.44	3.44	3.44

0.851944344387568	0.851850343639091
0.433905013880401	0.434132594742824
0.844972197665884	0.845075508140923
0.450619988961242	0.450374754263617
0.851611953913754	0.851528432350476
0.434709476495436	0.434911498697305
0.84533578753932	0.845426395273702
0.449756586825416	0.449541282741864
0.851316062048034	0.851241477413162
0.435424964441893	0.435605220433489
0.845655422852382	0.845735394534157
0.448996810565429	0.44880660716411
0.851051440856204	0.850984573663029
0.436064327402373	0.436225811934002
0.845938070407042	0.846009014101638
0.4483243689641	0.448155501831819

on the 40-55 iteration we see convergence to our period 2 orbit (much more than when a = 3.43)

IC =.1	IC = .5	IC = .7
3.45	3.45	3.45

0.457527376886553	0.842984970858901	0.842506468247074
0.856276463186228	0.45664651868897	0.457778151270761
0.424581432139318	0.856015641020839	0.856349738440104
0.842876536695361	0.425222378552398	0.424401280502159
0.456903098014093	0.843208660290705	0.84278267595549
0.856092166785299	0.456116963480035	0.457125127873539
0.42503437220182	0.855856262914967	0.856158021423482
0.843111533538445	0.42561380449747	0.424872550026202
0.456346940657492	0.843410094019375	0.843027738598499
0.855925715914636	0.455639800273334	0.456545298404399
0.425443252389755	0.855710995746726	0.855985326736869
0.843322455280949	0.425970421891811	0.425296842660755
0.45584743636452	0.843592694399676	0.843247062078241
0.855774401379231	0.455207008225338	0.456026017592565
0.425814784865211	0.855577878213169	0.855828696605746
0.843513110801147	0.426297085212701	0.425681488417904

What has happend here? The above results represent the 50-65 iteration and we can see jumping values from .85 to .42 to .84 to .45 (in the respective colors above). It seems there are new periodic orbits coming into being (xepher).

Let's look at the following for various a values which are greater than 3.45:

IC = .1	IC = .5	IC = .7
3.5	3.5	3.5
0.315	0.875	0.735
0.7552125	0.3828125	0.6817125
0.647033029453125	0.826934814453125	0.759431985703125
0.799334508874428	0.500897694844753	0.639432656779467
0.561395981289168	0.87499717950388	0.806954869781968
0.861806867185391	0.382819903774472	0.545225477709974
0.416835268001226	0.826940887670016	0.867841296580666
0.850792695730502	0.500883795893397	0.40142473185449
0.444305696177445	0.874997266166866	0.840990207785143
0.864143505826023	0.382819676285819	0.468039873681258
0.410898275076565	0.826940701069839	0.871424926139915
0.84721308915484	0.500884222943868	0.392152334846855
0.453050747518436	0.874997263524249	0.834291083923553
0.867285186919978	0.382819683222636	0.483873149232256
0.402855570142048	0.826940706759849	0.874089736395102
0.841970359116507	0.500884209921797	0.38519904193345
0.465696957200045	0.87499726360485	0.828872590094508
0.870881554391325	0.382819683011062	0.496449868195855
0.39356405414296	0.826940706586302	0.874955887974606
0.835349863003311	0.500884210318973	0.382928287256111
0.481391642842813	0.874997263602391	0.827029749263243
0.873788051653687	0.382819683017515	0.500680400838871

IC =.1	IC = .5	IC = .7
3.7	3.7	3.7
0.333	0.925	0.777
0.8218107	0.2566875	0.6411027
0.541820131452386	0.705956401171875	0.851333103795027
0.918528983439629	0.76805325502042	0.468290685657686
0.276883913077666	0.659145574149943	0.921279721720579
0.740811083498922	0.831288939045395	0.268336565448016
0.710437081637078	0.518916263804854	0.726428596438842
0.761150068286658	0.923676047365561	0.735301335644524
0.672662374785333	0.260844845488171	0.720143141342442
0.81469450603391	0.713377804660565	0.745686890083563
0.55857926112667	0.75653867616948	0.701660422551949
0.912303339613657	0.68149525822808	0.774532373711826
0.296022037730673	0.803120043590673	0.646138310401356
0.771054066361276	0.585037484942277	0.845981298662237
0.653159864503363	0.898243916772361	0.48209868161117
0.838205606849653	0.338186596189094	0.923814308359785
0.501782779733195	0.828120762684376	0.260411298509215
0.924988240276765	0.526646030853067	0.71260984023645
0.256724483817898	0.922372959447176	0.757749106588136
0.706022985937427	0.264924007572985	0.679191972796166
0.767951758282079	0.72053532780248	0.806193876476047
0.659346864364107	0.745047426006895	0.578107647031713

Not to bore you with data sets, but you can see that we have given birth to new periodic orbits!. As a matter of fact, as one increases a from 1 to 4 our picture bi-furcates. What does this mean? Well, from 1 to 3 we have 1 periodic orbit, then at around 3, this splits (or bifurcates) into more complicated orbits. As we increase the parameter a, we will experience more bi-furcations. An amazing amount of them as a matter of fact. What follows is a picture for the bifurcation diagram of fa(x) = ax (1-x) generated using the alogrithm which follows:

Bifurcation algorithm:

Pick a value of a (start at a=1).

Choose x at random (our IC) in [0, 1].

Calculate the orbit of x under fa(x).

Iterate about 100 times (ignore these), increase a and then start over

Each vertical slice (if you can imagine it) corresponds to a period n sink (orbit). For example, a vertical line which intersects twice, is hitting at 2 period 2 sinks. Now, here is the really cool part, there is an entire sequence of periodic sinks (corresponding to each period, for n=1, 2, 3, ...). This sequence is termed a period-doubling cascade. For the more "static" resembling parts of the graph, we encounter chaotic attractors. The following is a cobweb plotting for a "slice" at around 3.8

A characteristic of chaotic attractors is that, well, they are chaotic. They possess strange properties and may disappear and then resurface again, or otherwise spontaneously mutate. At a>4 there is no attracting set. Take a look at the following mess...

IC =.1	IC = .5	IC = .7
4.1	4.2	4.5
0.369	1.05	0.945
0.9546399	-0.2205	0.2338875
0.177540501444759	-1.13030505	0.806328618046874
0.59868147434516	-10.1131571354331	0.702732499944715
0.985074063146368	-472.034238401859	0.940047900597748
0.0602829283769595	-937811.097140506	0.253610303307815
0.232260477385357	-3693860485270.11	0.851814528137698
0.731093746923341	-5.73073421955031E+25	0.568019420060619
0.806042288545017	-1.37933521719525E+52	1.10418011322578
0.640986282250728	-7.99077569385925E+104	-0.517651241478804
0.943503759690185	-2.68180483996201E+210	-3.53526322177508
0.218548102076751	#NUM!	-72.1500717105478
0.700217799537145	#NUM!	-23750.0231379648
0.860642614268865	#NUM!	-2538393070.84651

What is even more fascinating is that at we go further and further into chaos, there appear to be periodic windows, of small period sinks. This is visible on the graph as the areas of open space.

There is a concept called, sensitive dependence on initial conditions. This is when very small changes in an initial condition, cause dramatic (Joan Crawford level) changes in a points orbit. A chaotic orbit is a non-periodic orbit, which displays sensitive dependence on it's initial conditions.

The precedeing three essays have served as an introduction to Chaos theory. There is a wealth of unexplored territory in this field, and a wealth more to study by oneself. I hope that I have sparked your interest for further study in this subject. I know that I have learned how much I do not know.

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