AnnouncementsTopics:

-  section7.7(improperintegrals);sections3.1+3.2(DTDSs)

*Readthesesectionsandstudysolvedexamplesinyourtextbook!

Homework:-  reviewlecturenotesthoroughly-  workonpracticeproblemsfromthetextbookandassignmentsfromthecoursepackasassignedonthecoursewebpage(underthe“SCHEDULE+HOMEWORK”link)

DynamicalSystems

•  Discrete-timedynamicalsystemsdescribeasequenceofmeasurementsmadeatequallyspacedintervals

•  Continuous-timedynamicalsystems,usuallyknownasdifferentialequations,describemeasurementsthatarecollectedcontinuously

Discrete-TimeDynamicalSystems

Adiscrete-timedynamicalsystemconsistsofaninitialvalueandarulethattransformsthesystemfromthepresentstatetoastateonestepintothefuture.

Discrete-TimeDynamicalSystemsandUpdatingFunctions

Letrepresentthemeasurementofsomequantity.Therelationbetweentheinitialmeasurementandthefinalmeasurementisgivenbythediscrete-timedynamicalsystemTheupdatingfunctionacceptstheinitialvalueasinputandreturnsthefinalvalueasoutput.Note:representspresenttimeandrepresentsonetime-stepintothefuture

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Solutions

Definition:Thesequenceofvaluesoffor0,1,2,…isthesolutionofthediscrete-timedynamicalsystemstartingfromtheinitialconditionThegraphofasolutionisadiscretesetofpointswiththetimeonthehorizontalaxisandthemeasurementontheverticalaxis.

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Example:ADiscrete-TimeDynamicalSystemfor

aBacterialPopulation

Colony InitialPopulationbt(millions)

FinalPopulationbt+1(millions)

1 0.47 0.94

2 3.30 6.60

3 0.73 1.46

4 2.80 5.60

5 1.50 3.00

6 0.62 1.24

Data:

Example:ADiscrete-TimeDynamicalSystemfor

aTreeGrowth

Tree InitialHeight,ht(m)

FinalHeight,ht+1(m)

1 23.1 23.9

2 18.7 19.5

3 20.6 21.4

4 16.0 16.8

5 32.5 33.3

6 19.8 20.6

Data:

Example:ADiscrete-TimeDynamicalSystemfor

AbsorptionofPainMedicationApatientisonmethadone,amedicationusedtorelievechronic,severepain(forinstance,aftercertaintypesofsurgery).Itisknownthateveryday,thepatient’sbodyabsorbshalfofthemethadone.Inordertomaintainanappropriatelevelofthedrug,anewdosagecontaining1unitofmethadoneisadministeredattheendofeachday.

BasicSolutions

BasicExponentialDiscrete-timeDynamicalSystemIfwithinitialcondition,thenBasicAdditiveDiscrete-timeDynamicalSystemIfwithinitialcondition,then

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bt = b0rt .

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ht+1 = ht + a

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h0

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ht = h0 + at.

CobwebbingCobwebbingisagraphicaltechniqueusedtodeterminethebehaviourofsolutionstoaDTDSwithoutcalculations.Thistechniqueallowsustosketchthegraphofthesolution(asetofdiscretepoints)directlyfromthegraphoftheupdatingfunction.

CobwebbingAlgorithm:1.  Graphtheupdatingfunctionandthediagonal.

2.  Plottheinitialvaluem0onthehorizontalaxis.Fromthispoint,moveverticallytotheupdatingfunctiontoobtainthenextvalueofthemeasurement.Thecoordinatesofthispointare(m0,m1).

3.  Movehorizontallytothepoint(m1,m1)onthediagonal.Plotthevaluem1onthehorizontalaxis.Thisisthenextvalueofthesolution.

4.  Fromthepoint(m1,m1)onthediagonal,moveverticallytotheupdatingfunctiontoobtainthepoint(m1,m2)andthenhorizontallytothepoint(m2,m2)onthediagonal.Plotthepointm2onthehorizontalaxis.

5.  Continuealternating(or“cobwebbing”)betweentheupdatingfunctionandthediagonaltoobtainasetofsolutionpointsplottedalongthehorizontalaxis.

Cobwebbing

Example:Startingwiththeinitialcondition,sketchthegraphofthesolutiontothesystembycobwebbing3steps. €

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ASolutionFromCobwebbing

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Cobwebbing

Example:ConsidertheDTDSforthemethadoneconcentrationinapatient’sblood:Cobwebfor3stepsstartingfrom(i)  (ii)  (iii) 

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Equilibria

Definition:ApointiscalledanequilibriumoftheDTDSifGeometrically,theequilibriacorrespondtopointswheretheupdatingfunctionintersectsthediagonal.

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SolvingforEquilibria

Algorithm:1.  Writetheequationfortheequilibrium.2.  Solvefor3.  Thinkabouttheresults.

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SolvingforEquilibria

Examples:Findtheequilibria,iftheyexist,foreachofthefollowingsystems.(a) (b)

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Mt+1 =12Mt +1

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xt+1 =axt1+ xt

Cobwebbing

Example:ConsidertheDTDSforapopulationofcodfishwhereisthenumberofcodfishinmillionsandistime.Supposethatinitiallythereare1millioncodfish.Determinetheequilibriaandthebehaviourofthepopulationovertimebycobwebbing.

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Solution:

StabilityofEquilibria

Anequilibriumisstableifsolutionsthatstartneartheequilibriummoveclosertotheequilibrium.

Anequilibriumisunstableifsolutionsthatstartneartheequilibriummoveawayfromtheequilibrium.