Python Programming: An Introduction to Computer Science

10 CHAPTER1. COMPUTERSANDPROGRAMS

print x
x = 3.9 x (1 - x)
print x

Obviouslyusingtheloopinsteadsavestheprogrammera lotoftrouble.
Butwhatexactlydothesestatementsdo?Thefirstoneperformsa calculation.

x = 3.9 x (1 - x)

Thisiscalledanassignmentstatement.Thepartontherightsideofthe=isa mathematicalexpression.
Pythonusesthe*charactertoindicatemultiplication.Recallthatthevalueofxis 0 25 (fromtheinput
statement).Thecomputedvalueis 3 9

0  25 

1  0 25
or 0 73125.Oncethevalueontherighthandsideis
computed,it is storedback(orassigned) intothevariablethatappearsonthelefthandsideofthe=, inthis
casex. Thenew valueofx(073125)replacestheoldvalue(025).
Thesecondlineintheloopbodyis a typeofstatementwehave encounteredbefore,aprintstatement.

print x

WhenPythonexecutesthisstatementthecurrentvalueofxis displayedonthescreen.So,thefirstnumber
ofoutputis 0.73125.
Remembertheloopexecutes 10 times.Afterprintingthevalueofx, thetwo statementsoftheloopare
executedagain.

x = 3.9 x (1 - x)
print x

Ofcourse,nowxhasthevalue 0 73125,sotheformulacomputesa newvalueofxas 3 9

0  73125 

1 
0 73125
, whichis 076644140625.
Canyouseehow thecurrentvalueofxis usedtocomputea new valueeachtimearoundtheloop?That’s
wherethenumbersintheexampleruncamefrom.Youmighttryworkingthroughthestepsoftheprogram
yourselffora differentinputvalue(say 0 5).ThenruntheprogramusingPythonandseehow wellyoudid
impersonatinga computer.

1.8 ChaosandComputers.

I saidabove thatthechaosprogramillustratesaninterestingphenomenon.Whatcouldbeinterestingabout
a screenfullofnumbers?If youtryouttheprogramforyourself,you’ll findthat,nomatterwhatnumber
youstartwith,theresultsarealwayssimilar:theprogramspitsback 10 seeminglyrandomnumbersbetween
0 and1.Astheprogramruns,thevalueofxseemstojumparound,well,chaotically.
Thefunctioncomputedbythisprogramhasthegeneralform:k

x

1  x
, wherekinthiscaseis3.9.
Thisis calleda logisticfunction.It modelscertainkindsofunstableelectroniccircuitsandis alsosometimes
usedtopredictpopulationunderlimitingconditions.Repeatedapplicationofthelogisticfunctioncanpro-
ducechaos.Althoughourprogramhasa welldefinedunderlyingbehavior, theoutputseemsunpredictable.
Aninterestingpropertyofchaoticfunctionsis thatverysmalldifferencesintheinitialvaluecanleadto
largedifferencesintheresultastheformulais repeatedlyapplied.Youcanseethisinthechaosprogramby
enteringnumbersthatdifferbyonlya smallamount.Hereis theoutputfroma modifiedprogramthatshows
theresultsforinitialvaluesof 0 25 and 0 26 sidebyside.

input 0.25 0.

0.731250 0.
0.766441 0.
0.698135 0.
0.821896 0.
0.570894 0.
0.955399 0.