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Making sense of a complex world
Chris Budd
Many natural (and human!) systems appear complex and hard to understand
National Electricity Grid
Atmosphere and climate
El Nino
Clouds
Flocking Turbulence
Geology
Aircraft undercarriage
Complex designs
Human behaviour
Crowds
What makes a system complex ?
Many components with individual behavior
Coupling between components
Many different scales in space and time
Can scientists, mathematicians and engineers make any sense of complexity?
And can we use this knowledge to our advantage?
Traditional view
Things are complicated because there are lots of independent things all going on at once
A complicated example: The tides
h(t)
t
Bombay tides 1872
Kelvin decomposed h(t) into 37 independent components
He found these out using past data and added them up using an analogue computer
US Tidal predictorKelvin’s Tidal predictor
But many examples of complexity in nature are not like this!
In the tides we see complicated behaviour due to a large number of independent uncoupled systems combining their effects
The tides are a resultant property of this combination
The Double Pendulum .. An example of complex behaviour in a simple coupled system
Motion can be
• Periodic in phase : predictable
• Periodic out of phase : predictable
• Chaotic : unpredictable
Each part of the system is relatively simple, with easy to understand behavior
It is the coupling which leads to new complex emergent behavior which we understand by using maths
0)sin()sin()cos( 112
2
2122
22
21
2
dt
dm
dt
dm
dt
d
0)sin()sin()cos( 212
2
1122
12
22
2
dt
d
dt
d
dt
d
Aircraft undercarriage can be very similar
Emergence .. A property of a complex system which is more than the sum of its parts
Emergence arises from the way that the components interact with each other and not just from their individual properties
Emergent properties of complex systems can allow us to make predictions and even to new designs. They include …
• Coherent Patterns ..
Much of science and maths involves the search for, and study of, these patterns
•Scaling laws
Coherent Patterns
Emergent Patterns often arise because of the way that things interact and communicate with each other
Slime mould
All described using mathematical equations
Flocking
Patterns in rocks
Crowds at a scramble crossing
Scaling laws
Microstructure of a ceramic.
Al2O3-TiO2
RTiO2
CAl2O3
Frequency
Con
du
ctiv
ity
PERCOLATION DETERMINED DCCONDUCTIVITY
POWER LAW EMERGENT PROPERTY
25 frequencytyconductivi
The ac conductivity of 255 2D squae networks randomly filled with 512 components 60% 1 k resistors
& 40% 1 nF capacitors
Emergent scaling law
Frequency
Random
Con
du
ctiv
ity
We see examples of scaling laws in many other complex systems:
• The Internet
• Epidemics
• Mechanical systems
•Rocks and waterHomogeneous system
A very complex example .. The H Bomb
r: Radius of fireball
E: Energy of the bomb
t: Time after the explosion
5/25/1 tCEr G I Taylor
Scaling law
Bringing this all together … forecasting the weather
The atmosphere/ocean is a very complex system with many length and time scales
Need to make predictions but …
• System has far more degrees of freedom than data
• Small scale behavior is very can be chaotic
• Small and large scales interact
• Lots of random events
Turbulence
• Computations are hard!
Make use of all of the previous ideas to improve predictability
Scaling laws show how energy is transferred from small to large scales and from small heights to large heights and greatly speed up computations
Fit coherent patterns of weather eg. depressions to the sparse data to start and monitor computer weather forecasts (data assimilation)
1987!!
Complexity .. May apply to many many other problems
Where many things interact with each other
• Spread of disease
• Customer behavior
• Transport networks
• The national grid
• Chemical reactions
Much still to be discovered!!!
Intestinal wall:
Villi and Microvilli
Stomach
Intestine
Eg. The digestive system