Mathematical Foundation of System Dynamics

MathematicalFoundationofSystemDynamics

PUGACHEVAELENA,AssociateProfessor

Visiting Academic ScholarRadboud University Nijmegen

(The Netherlands)

• Softwareofsystemdynamics(Stella,VensimandPowerSim)makesitpossibletobuildacomplicateddynamicmodelwithoutknowingsystemdynamicsanddifferentialequations

FromDynamicstoSystemDynamicsTheaimistointroducesystemdynamics

fromadynamicsviewpointWhy?

• Dynamicanalysisisafoundationofsystemdynamics

• Dynamicanalysishelpstounderstandthemechanismsthroughwhichunpredictable,unknownandemergentchangehappens

Twodifferentconceptsoftime

• 1.Timeasamomentoftimeorapointintime,denotedhereasτ (τ =1,2,3,....);

• 2.Timeasaperiodoftimeoranintervaloftime,denotedhereast,suchthatt=1st,2nd,3rd,...,ormorelooselyt=1,2,3,…

• Unitsoftheperiodcouldbeasecond,aminute,anhour,aweek,amonth,aquarter,ayear,adecade,acentury,amillennium,etc.,dependingonthenatureofthedynamicsinquestion.

Stocks

• Stocksareaccumulations• Stocksholdthecurrentstateofthesystem• Stocksfullydescribetheconditionofthesystematanypointintime

• So,stockistheamountthatexistsataspecificpointintime.

• Letxbesuchanamountofstockataspecificpointintimeτ .Thenstockcanbedefinedasx(τ)whereτ canbeanyrealnumber.

Flows

• Flowsdothechanging• Flowsincreaseanddecreasestocksnotjustonce,buteveryunitoftime

• Flowisdefinedaschangeofstockduringaunitinterval,anddenotedhereby𝑓(𝑡).

Stock-flowrelation

𝑥(𝜏 + 1) = 𝑥(𝜏) + 𝑓(𝑡)𝜏𝑎𝑛𝑑𝑡 = 0; 1; 2; 3;…

τ+1τ

𝑥(𝑡 + 1)

𝑥(𝑡) 𝑓(𝑡)

x(t+1)=x(t)+f(t)t=0;1;2;3;…

𝑥 𝑡 = 𝑥 0 +3𝑓(𝑖)567

89:

StockFlow

Definedataperiodoftime

Definedatamomentoftime

ContinuousFlow

• Theinfinitesimalamountofflowthatisaddedtostockataninstantaneouslysmallperiodintimecanbewrittenas

dx = f(t)dtContinuous flow and stock are transformed todifferential equation, and the amount of stock at𝑡is obtained by solving the differential equation.

;<;5

=f(t)

𝑥 𝑡 = 𝑥 0 + = 𝑓 𝑢 𝑑𝑢5

:

ConstantFlow𝑓(𝑡) = 𝑎

Discreteinterpretation:𝑥(𝑡 + 1) = 𝑥(𝑡) + 𝑎

𝑥(𝑡) = 𝑥(0) + 𝑎𝑡

Continuousinterpretation𝑑𝑥𝑑𝑡

= 𝑎

𝑥 𝑡 = 𝑥 𝑜 + ∫ 𝑎𝑑𝑢5: = 𝑥(0) + 𝑎𝑡

Stock2000

1500

1000

500

00 12 24 36 48 60 72 84 96 108 120

Time (Month)

custo

mer

s

Stock : Current

Changestakeplaceata“constant”rate

CustomersCustomer

Acquisition Rate

Constant growthper month

Feedback

• Flowbecomesafunctionofstock

BankBalanceInterest Payments

Interest Rate

𝑥 𝑡 + 1 = 𝑥 𝑡 + 𝑓 𝑥 𝑡 , 𝑡 = 0; 1; 2; …𝑑𝑥/𝑑𝑡 = 𝑓(𝑥(𝑡))

PositiveFeedback

Let a>0beinflowfraction𝑓(𝑥) = 𝑎𝑥(𝑡)

Bank Balance2000

1500

1000

500

00 5 10 15 20 25 30 35 40 45 50

Time (Year)

$

Bank Balance : Current

𝑥 𝑡 + 1 = 𝑥 𝑡 + 𝑎𝑥 𝑡 ; 𝑡 = 0; 1; 2; …

𝑑𝑥/𝑑𝑡=𝑎𝑥,𝑎 > 0𝑋(𝑡) = 𝑥(0)𝑒S5,e=2.71…

Note:theinitialvalueofthestock𝑥(0)cannotbezero,sincenon-zeroamountofstockisalwaysneededasaninitialcapitaltolaunchagrowthof flow

anincreaseinflow↑→anincreaseinstock↑→anincreaseinflow↑

self-increasingrelation

Stock2000

1500

1000

500

00 5 10 15 20 25 30 35 40 45 50

Time (Month)

$

Stock : Current

StockFlow

Decay Factor

Self-regulatingorbalancingrelation

NegativeFeedback

𝑑𝑥/𝑑𝑡=−𝑏𝑥,𝑏 > 0

anincreaseinflow↑→adecreaseinstock↓→adecreaseinflow↓

Let𝑏 > 0 beoutflowfraction

AddingConstantFlowsInterestRate=10%Payments=$1

Bank Balance80

40

0

-40

-800 5 10 15 20 25 30 35 40 45 50

Time (Year)

$

Bank Balance : Starting capital = 8Bank Balance : Starting capital = 12Bank Balance : Starting capital = 10

Criticalvalue:𝑥(𝑡 + 1) = 1,1𝑥(𝑡)– 1𝑥∗ = 𝑥(𝑡 + 1) = 𝑥(𝑡)

𝑥 ∗= 1,1𝑥∗ − 1𝑥∗ = 1/0,1 = 10

;<;5=0,1𝑥 − 1𝑑𝑥𝑑𝑡

= 0

0,1𝑥 ∗ −1 = 0,𝑥 ∗= 10

BankBalance

Interest Payments

Interest Rate

Payments

CombiningFeedbackStock

Inflow Outflow

Inflow Fraction Outflow Fraction

Stock400

300

200

100

00 1 2 3 4 5 6 7 8 9 10 11 12

Time (Month)

$

Stock : Outflow Fraction DominateStock : Inflow Fraction DominateStock : Inflow Fraction = Outflow fraction

𝑑𝑥𝑑𝑡

= 𝑎 − 𝑏 𝑥

DynamicalSystems

• Dynamicalsystemisasystemthatevolvesintimeaccordingtoawell-definedunchangingrule.

• Oneofthegoalsofthestudyofdynamicalsystemsistoclassifyandcharacterizethesortsofbehaviorsseeninclassesofdynamicalsystems

TwoTypesofDynamicalSystems

DifferenceEquations• Theoutputofonestepis

usedastheinputforthenext.

• 𝑥5X7 = 𝑓(𝑥5)•

• Y𝑥5X7 = 𝑓(𝑥5)𝑥𝑜

• Timeisdiscrete.

DifferentialEquations

• Z;<;5= 𝑓(𝑥)

𝑥 0 = 𝑥:• Ifthefunctionf(x)is

continuousanddifferentiable,thanthesolutionexists andisunique.

• Thederivative;<;5

istheinstantaneousrateofchange𝑥.

• Timeiscontinues.

fX f(X)

FixedPoints

• Apoint𝑥 isfixedpointifitdoesnotchange(equilibriumpoint)

• Iteratedfunction:𝑓(𝑥 ∗) = 𝑥*• Differentialequations:𝑑𝑥/𝑑𝑡 = 0• Afixedpointisstable ifnearbypointsmoveclosertothefixedpointwhentheyareiterated(attractor)

• Afixedpointisunstableifnearbypointsmovefurtherawayfromthefixedpointwhentheyareiterated(repellor)

LogisticEquation

• PopulationModel• Population(nextyear)=

FunctionofPopulation(thisyear)• 𝑃5X7 = 𝑟𝑃5,• wherer– growthrate

(parameter)

Population2

1.5

1

.5

00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1

Time (Month)

rabb

its

Population : r greater than 1Population : r = 1Population : r less than 1

𝑟 > 1 – populationtendstowardinfinity𝑟 = 1 – populationstaysthesame0 < 𝑟 < 1 – populationapproacheszero

LogisticEquation• 𝑓(𝑝) = 𝑟𝑝(1 − 𝑝/𝐴)• 𝑝`X7 = 𝑟𝑝𝑛(1 − 𝑝𝑛/𝐴)• 𝑟 – growthrate• 𝐴- annihilationpopulation

• If𝑝 = 𝐴,𝑓(𝑝) = 0,• If𝑝<< 𝐴,𝑓(𝑝) ≈ 𝑟𝑝

LogisticEquationinaStandardForm

A

𝑥 – populationexpressedasfractionofannihilationpopulation𝑥`X7 = 𝑟𝑥𝑛(1 − 𝑥𝑛)𝑓(𝑥) = 𝑟𝑥(1 − 𝑥)

Experiment

Fixedpoint=0,333

Experiment

Cycle:0,560,76

Experiment

Cycle:0,8750,3830,8270,501

Experiment

SensitiveDependenceonInitialConditions

• Adynamicalsystemhassensitivedependenceoninitialconditions(SDIC)ifarbitrarilysmalldifferencesininitialconditionseventuallyleadtoarbitrarilylargedifferencesintheorbits.

ButterflyEffect

• Averysmallerrorintheinitialconditiongrowsextremelyrapidly

• Long-termpredictionisimpossible• Adeterministic(rule-based)systemcanbehaveunpredictably

DefinitionofChaos

• Adynamicalsystemischaoticif:• 1.Thedynamicalsystemisdeterministic.• 2.Thesystem’sorbitsarebounded.• 3.Thesystem’sorbitsareaperiodic;i.e.,theyneverrepeat.

• 4.Thesystemhassensitivedependenceoninitialconditions.

DifferentialEqs vs.IteratedFunctions

DifferentialEquations• 𝑑𝑃/𝑑𝑡 =𝑟𝑃(1−𝑃/𝐾)• Timeiscontinuous• Piscontinuous

• Cyclesandchaosarenotpossible

Iteratedfunctions• 𝑝5X7 = 𝑟𝑝5(1 − 𝑝5/𝐴)• Timemovesinjumps• Pmovesinjumps

• Cyclesandchaosarepossible

TheLogisticDifferentialEquation

• 𝑑𝑃/𝑑𝑡 = 𝑟𝑃(1 − 𝑃/𝐾)• risagrowthparameter• Kisthecarryingcapacity

• Equilibrium:• 𝑑𝑃/𝑑𝑡 = 0• 𝑃 = 0; 𝑃 = 𝐾

Population100

75

50

25

00 2 4 6 8 10 12 14 16 18 20

Time (Month)

rabb

its

Population : r greater than 1

0 K

LogisticEquationwithHarvest• 𝑑𝑃/𝑑𝑡 = 𝑟𝑃(1 − 𝑃/𝐾)– ℎ• r- growthparameter• K- carryingcapacity• h- harvestrate

r=3K=100

𝑃7 =dX de6fdg/h�

jd/h

𝑃j =d6 de6fdg/h�

jd/h

Bifurcation:ℎ = 𝑟𝐾/4

ImportantLessonsforManagers

• Sometimespropertiesofcontinuousmodelsarediscontinuous;

• Atbifurcationpointthebehaviorofdynamicalsystemchangessuddenlyandqualitatively;

• Profitoptimizationleadstoacatastrophicvalue;• Thedecisionconcerningtheamountofexploitation(harvesting,taxation,obligatorypayments)shouldbedonenotonthebaseofprescriptiveguidelines,butonthebasisoffeedback.

LogisticMap

Iteration• 𝑥(𝑛+1)=𝑟𝑥𝑛(1−𝑥𝑛)• 𝑓(𝑥)=𝑟𝑥(1−𝑥)• 0 ≤ 𝑥 ≤ 1• 0 ≤ 𝑟 ≤ 4

• Fixedpoints:𝑥 ∗= 𝑓(𝑥 ∗)• 𝑥 ∗= 0• 𝑥 ∗= 1 − 1/𝑟

CobwebPlot

r=0,5

r=2,9

r=3,2

r=3,5

r=3,9

BifurcationDiagram

r

...6692016,4

,

12

1lim

=

=

+-

+

-+

¥®

d

d

nrnr

nrnr

n

Feldman,DavidP.ChaosandFractals:AnElementaryIntroduction.OxfordUniversityPress,2012.

Chaos

• islong-termbehaviorofnonlineardynamicalsystem;

• lookslikearandomfluctuation,butstilloccursincompletelydeterministic,simpledynamicalsystems;

• exhibitssensitivitytoinitialconditions;• occurswhennoperiodictrajectoriesarestable;• isaprevalentphenomenonthatcanbefoundeverywhereinnature,aswellasinsocialreality.

ImportantResultsofModelling

• 1.Smallchangesinparametercanshiftthedynamicalbehaviorofthesystemfromstabletochaotic.

• 2.Achaoticsystembehavesasifitisrandom,notgovernedbyadeterministicrule.Sodeterministicsystemscanproducerandom,unpredictablebehavior.

• 3.Theperiod-doublingroutetochaosisuniversalscenario.Universalitygivesussomereasontobelievethatwecanunderstandcomplexsystemswithsimplemodels.

• Not only in research, but in the world of politics and economics, we would all be better off if more people realized that simple non-linear systems do not necessarily possess simple dynamical properties.

• R. M. May et al., “Simple mathematical models with very complicated dynamics,”

Nature, vol. 261, no. 5560, pp. 459–467, 1976.

Readinglist

• R.M.May,“Simplemathematicalmodelswithverycomplicateddynamics,”Nature,vol.261,no.5560,pp.459–467,1976.(beforeclass)

• KaoruYamaguchi“Stock-FlowFundamentals,DeltaTime(DT)andFeedbackLoop-FromDynamicstoSystemDynamics”,JournalofBusinessAdministration,OsakaSangyoUniversity,Vol.1No.2,March2000(afterclass)

AdditionalReading

• Gleick,James.Chaos:Makinganewscience.RandomHouse,1997.

• Stewart,Ian.DoesGodplaydice?:Thenewmathematicsofchaos.PenguinUK,1997.

• Feldman,DavidP.ChaosandFractals:AnElementaryIntroduction.OxfordUniversityPress,2012.

• JosLeys,EtienneGhys,andAurelien Alvarez,Chaos:AMathematicalAdventurehttp://www.chaos-math.org/en

Thankyouforattention