Systems Theory Tiago Garcia de Senna Carneiro Pedro Ribeiro de Andrade Gilberto Câmara Münster, 2013
Jan 10, 2016
Systems Theory
Tiago Garcia de Senna CarneiroPedro Ribeiro de AndradeGilberto Câmara
Münster, 2013
Geoinformatics enables crucial links between nature and society
Nature: Physical equations Describe processes
Society: Decisions on how to Use Earth´s resources
How to model Natural-Society systems?
If (... ? ) then ...
Desforestation?
Connect expertise from different fieldsMake the different conceptions explicit
“A hypothesis or theory [model] is clear, decisive, and positive, but it is believed by no one but the man who created it. Experimental findings [observations], on the other hand, are messy, inexact things, which are believed by everyone except the man who did that work”Harlow Shapley (1885-1972), American astronomer
“[The] advantage of a mathematical statement is that it is so definite that it might be definitely wrong…..Some verbal statements have not this merit; they are so vague that they could hardly be wrong, and are correspondingly useless.”Lewis Fry Richardson (1881-1953) – first to apply mathematical methods to numerical weather prediction
Models
How reality is conceived Any measurable part of reality can be modelled as a system
Systems are represented as stocks and flows Stocks represent storages of energy, matter, or information Flows connect and transport stocks
Real systems are opened only theoretical ones are closed
Environment System 2
System 3
System 1
System 4
What is a System?
Definition: A system is a group of components with different functions, which interact with each other
Example: The climate system includes the atmosphere, oceans, polar caps, clouds, vegetation…and lots of other things
How do we study systems?
• Identify the components
• Determine the nature of the interactions between components
Atmospheric Physics/Dynamics
Tropospheric Chemistry
Global Moisture
Ocean Dynamics
MarineBiogeochemistry
Terrestrial Ecosystems
Terrestrial Energy/Moisture
Climate Change
Pollutants
CO2
CO2
Soil
Land Use
Physical Climate System
Biogeochemical Cycles
Human Activities
(from Earth System Science: An Overview, NASA, 1988)
Earth as a system
Systems Theory
Provides a unified classification for scientific knowledge. Enunciated by biologist Ludwig Von Bertalanffy:
1920s: earliest developments 1937: Charles Morris Philosophy Seminar, University of Chicago 1950: “An Outline of General Systems Theory”, Journal for the Philosophy
of Science
Scientists that introduced Systems Theory in their fields: Parsons, sociologist (1951) J.G Miller, psychiatrist & psychologist (1955) Boulding, economist (1956) Rapoport, mathematician (1956) Ashby, bacteriologist (1958)
A system
Can you identify parts? and Do the parts affect each other? and Do the parts together produce an effect that is different from
the effect of each part on its own? and perhaps Does the effect, the behavior over time, persist in a variety of
circumstances?
Source: (Meadows, 2008)
A system
Can you identify parts? and Do the parts affect each other? and Do the parts together produce an effect that is different from
the effect of each part on its own? and perhaps Does the effect, the behavior over time, persist in a variety of
circumstances?
Source: (Meadows, 2008)
Systems can grow in different ways...População
600
450
300
150
0
0 10 20 30 40 50 60 70 80 90 100Time (Month)
População : C:\Users\Alexandre\Desktop\linear
População
6,000
4,500
3,000
1,500
0
0 10 20 30 40 50 60 70 80 90 100Time (Month)
População : C:\Users\Alexandre\Desktop\linear
população
600
450
300
150
0
0 10 20 30 40 50 60 70 80 90 100Time (Month)
população : Current
forever...
explode...
stabilize...
Run code #1 – Linear Growth
Feedbacks
Feedback is how the system affect itself
Essential to systems be able to reach their goal
Inflow OutflowSystem
Feedback
Population growth
Births Deaths
Fertility
Mortality
Population
Positive Coupling
AtmosphericCO2
Greenhouseeffect
• An increase in atmospheric CO2 causes a corresponding increase in the greenhouse effect, and thus in Earth’s surface temperature• Conversely, a decrease in atmospheric CO2
causes a decrease in the greenhouse effect
Negative Coupling
Earth’s albedo(reflectivity)
Earth’ssurface
temperature
• An increase in Earth’s albedo causes a corresponding decrease in the Earth’s surface temperature by reflecting more sunlight back to space• Or, a decrease in albedo causes an increase in surface temperature
The interesting thing to do is to putcouplings together in feedback loops…
person A’sbodytemperature
person A’sblankettemperature
Negative Feedback Loops:Electric Blankets
person B’sblankettemperature
person B’sbodytemperature
person A’sbodytemperature
person A’sblankettemperature
A Positive Feedback Loop:Mixed-up Electric Blankets
person B’sblankettemperature
person B’sbodytemperature
A Positive Feedback Loop:Mixed-up Electric Blankets
Any perturbation will cause both people to adjust their blanket controls, but with undesired consequences.
Ultimately, one person will freeze (become infinitely cold) and the other person to swelter (become infinitely hot).
Equilibrium State:
Conditions under which the system will remain indefinitely
--If left unperturbed
Reinforcing feedbacks
Also named: positive, self-reinforcing, discrepancy-enhancing, degenerative
Self-enhancing behavior that leads to growth or even collapses
População
6,000
4,500
3,000
1,500
0
0 10 20 30 40 50 60 70 80 90 100Time (Month)
População : C:\Users\Alexandre\Desktop\linear
Run code #2 – Exponential Growth
Balancing feedback
Also named: negative, self-correcting, discrepancy-reducing, regenerative
Equilibrating or goal-seeking structures
população
600
450
300
150
0
0 10 20 30 40 50 60 70 80 90 100Time (Month)
população : Current
Homeostasis
It is a tendency that all systems have to maintain their equilibrium state through negative feedbacks
Initial condition = 3.2
Initial condition = 8
Run code #3 – Homeostasis
Equilibrium state (some times steady-state)
Equilibrium means a state of balance.
There are no net flows of matter or of energy
Input flow == Output flow
Inflow OutflowSystem
Equilibrium state (some times steady-state)
Equilibrium means a state of balance.
There are no net flows of matter or of energy
Input flow == Output flow
Inflow OutflowSystem
An Unstable Equilibrium State
low resilience
An Unstable Equilibrium State
Perturbation
When pushed by a perturbation, an unstable equilibrium state shifts to a new, stable state.
A Stable Equilibrium State
higher resilience
A Stable Equilibrium State
Perturbation
When pushed by a perturbation, a stable equilibrium state, returns to (or near) the original state.
Run code #4 – Logistic Growth
Verify and analyse models with visualizations TerraME provides you different types of Observers However, it can only observes TerraME types: Cell, Agent, CellularSpace, Timer,
Environment, etc.
Ant agents eat sugar on a cellular space
Run codes #5, #6 – Logistic Growth
Discrete & Continuous Systems
Discrete systems jump from one state to other without intermadiate valuas, like the traffic light.
Continuous system change from a state to other going through all intermadiate states, like the speed of a car.
Depending on your point of view you can model a system as discrete or continuos, like a lift.
ht+1 =ht ± 1 = ± 0.1 hdt
dh
There are different types of equlibrium
Discrete systems: Fixed point - System converges to a one-dimension fixed
value.
N-dimensional attractors – System converges to attractors composed by several N fixed points
Deterministic CAOS – System is locked in a high dimensional attractor composed theorically by a infinite number of fixed points and will never repeat itself (this is the caos).
Run codes #7 – Discrete Logistic Growth
As the system is discrete we should use a difference equation istead of a differential equation:
)/1(1 KNrNN ttt
Logistic Map
From smooth behavior to deterministic caos through duplication of periods. Feigenbaum, M. (1983) – in Physics. May, R. (1976) – in Ecology.
Discrete Growth – It is no error propagation!
(a) r = 1,2, (b) r = 3,0, (c) r = 3,5 e (d) r = 4,0.
There are different types of equlibrium
Cotinuous systems: One single system
Static equilibrium - System converges to a one-dimension fixed value.
Coupled sytems (like prey-predator) Static equilibrium - System converges to a one-dimension fixed value.
Dynamic equilibrium – System converges to cyclical behavior and keep repeating itself
Erratic outcomes of deterministics rules should be treated as error propagation in the integration method
Run code #8 – Contiuous System
How CONTINUOUS systems grow?
Linear growth
Exponential growth
Logistic growth
rdt
dN
rNdt
dN
kNkrNdt
dN/)(
População
600
450
300
150
0
0 10 20 30 40 50 60 70 80 90 100Time (Month)
População : C:\Users\Alexandre\Desktop\linearPopulação
6,000
4,500
3,000
1,500
0
0 10 20 30 40 50 60 70 80 90 100Time (Month)
População : C:\Users\Alexandre\Desktop\linear
população
600
450
300
150
0
0 10 20 30 40 50 60 70 80 90 100Time (Month)
população : Current
N
rk
)/1( kNrNdt
dN
How to implement stochastic models?
Create a random object that is able to generate numbers in a uniform distributionrandom = Random()
probability density function
Call function number(a, b) to generate real numbers within the interval [a, b]:
n = random:number(0,1)
Call function integer(a,b) to generate integers within the interval [a,b]: n =
random:integer(10,20)
How to implement stochastic models?
random = Random()
counter = 0for i = 1, 1000 do
local n = random:number(0, 1)if ( n < 0.3) then -- try 0.0 0.5 0.95 1.0
counter = counter + 1end
end
print( (counter/1000) * 100 ) -- 30%
Run codes #9 and #10 – Stochastic process
Coupled systems – Dynamic Equilibrium
Run code #11 – Prey-predator model
Short History of System Dynamics
The System Dynamics approach was developed in the 1960s at M.I.T. by Jay Forrester.
A system in Modelica
Conception of Reality
Any measurable part of reality can be modeled Systems are represented as stocks and flows
Stocks represent energy, matter, or information Flows connect and transport stocks
Systems are opened or closed
A system
Can you identify parts? and Do the parts affect each other? and Do the parts together produce an effect that is different from
the effect of each part on its own? and perhaps Does the effect, the behavior over time, persist in a variety of
circumstances?
Source: (Meadows, 2008)
slide 59
Systems Building Blocks
Stocks Flows Information Links Decision Points Converters Auxiliary Variables
slide 60
Stocks
“Things” that accumulate in a system Physical or non-physical things Value is a quantity or level Persistent (remain even if all flows stop) Conservation (stock units enter from environment
and return to environment)
slide 61
Flows
Movement of “things” in and out of stocks Not persistent (can be stopped and started) Value is a rate of change (will always have a time
dimension) Flow unit = stock unit / time The unit of measurement for a flow will always be
the unit of measurement of a stock divided by an element of time
slide 62
Stock and Flow Diagram
Stocks in boxes Flows as straight double arrows Information Links as thin curved arrows Decision Points as closed in X
Control Material Flaw
to Stock
Add New information
Send informationfrom the Stock
Control Material Flaw
from Stock
Stock
System Dynamics Modelling
Shrimp farming
Simple model for shrimp farm
Results?
Figure 7
An Unstable Equilibrium State
An Unstable Equilibrium State
Perturbation
When pushed by a perturbation, an unstable equilibrium state shifts to a new, stable state.
A Stable Equilibrium State
A Stable Equilibrium State
Perturbation
When pushed by a perturbation, a stable equilibrium state, returns to (or near) the original state.
Tools for system dynamics
Dinamo Vensim Simile STELLA
Water in the tub
Initial stock: water in tub = 40 gallons water in tub(t) = water in tub(t – dt) – outflow x dt t = minutes dt = 1 minute Runtime = 8 minutes Outflow = 5 gal/min
Cell
Not yet
(description extracted from “TerraME types and functions”)
Event
Not yet
Not yet
Temporal model
Source: (Carneiro et al., 2013)
1:32:10 ag1:execute( )
1:38:07 ag2:execute( )
1:42:00 cs:save()
. . .
(4) ACTIONreturn value
true
(1) Get first EVENT
1:32:00 cs:load( ) (2) Update current time
(3) Execute the ACTION
false
(5) Schedule EVENT again
Observer
Not yet
Water in the tub
Initial stock: water in tub = 40 gallons water in tub(t) = water in tub(t – dt) – outflow x dt t = minutes dt = 1 minute Runtime = 8 minutes Outflow = 5 gal/min
Water in the tub 2
Initial stock: water in tub = 40 gallons water in tub(t) = water in tub(t – dt) – outflow x dt t = minutes dt = 1 minute Runtime = 8 minutes Outflow = 5 gal/min Inflow = 40 gal every 10 min
Conclusions
Two ways to increase stocks Stocks act as delays or buffers Stocks allow inflows and outflows to be decoupled