Making sense of a complex world
Chris Budd
Much of natural (and human!) behavior
appears complex and hard to understand
Rocks underground
Atmosphere and climate
El Nino
Flocking Turbulence
Geology
Complex designs
Aircraft undercarriage
Photonic crystals
Human behavior
Stock markets Crowds
What do we mean by a complex system?
Many components with individual behavior
Nonlinear Coupling between components
Many different scales in space and time
• The weather.. Air, oceans, sun, CO2
• The earth ..
• Disease spread .. People, viruses, pollutants
Human body
Stomach
Small intestine:
7m x 1.25cm
Intestinal wall:
Villi and Microvilli
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
Example: The tides
a complicated system which isn’t complex
Bombay tides 1872
h(t)
t
Kelvin decomposed h(t) into 37
independent periodic functions
37
h(t ) a j sin( j t j )
j 1
Kelvin calculated the coefficients using past data
and added them up using an analogue computer
Kelvin’s Tidal predictor US Tidal predictor
In the tides we see complicated
behavior due to a large number of
independent uncoupled systems
combining their effects
The tides are a resultant property of
this combination
But many examples of complexity in
nature are not like this!
The Double Pendulum .. An example of
complex behavior in a simple coupled system
Motion can be
• Periodic in phase : predictable
• Periodic out of phase : predictable
• Chaotic : unpredictable
Newton’s laws apply to the double pendulum!
1 Angle of top part
2 Angle of bottom part
d 1 d 2 d 2
2 2 2
m 2 cos( 2 1 ) m sin( 2 1 ) sin(1 ) 0
dt
2
dt dt
d 2 d 1 d
2 2 2
2 cos( 2 1 ) 1 sin( 2 1 ) sin( 2 ) 0
dt 2 dt dt
Each part of the system is relatively simple,
with easy to understand behavior
It is the coupling which leads to new complex
emergent behavior
In this case chaotic motion
Aircraft undercarriage can be very similar
Motion of the asteroids is chaotic:
will the human race survive?
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
Emergent Properties Include
• Coherent Patterns .. Exotic macroscopic
behavior
• Scaling laws
• Understandable behavior ‘in the large’
Coherent Patterns
Emergent Patterns often arise because of the way that
things interact and communicate with each other
Flocking
BZ reaction
Slime mould
Can often describe using differential equations
Patterns in rocks
Singularity
Crowds
Scaling laws
Microstructure of a real technical ceramic.
Al2O3-TiO2
RTiO2
CAl2O3
Frequency
Conductivity
Frequency
PERCOLATION POWER LAW
DETERMINED DC Random EMERGENT
CONDUCTIVITY percolation PROPERTY
Conductivity
Emergent
scaling law
conductivity frequency 2 / 5
An emergent scaling law
If
a is something we can measure
b is something that changes
They are related by an equation of the form
a Cb
A very complex example .. The H Bomb
r: Radius of fireball
Scaling law E: Energy of the bomb
t: Time after the explosion
r CE t 1/ 5 2 / 5
G I Taylor
We see examples of scaling laws in many
other complex systems:
• The Internet
• Networks of friends
• Disease
Homogeneous system
• Mechanical systems
• Protein and gene interactions
• Porous media
This is VERY useful for environmental predictions
Scaling law allows us to make calculations at a finer scale
than any computational mesh
These computations are important in understanding the
transport of pollutants underground over long times
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 indicate how energy is transferred from
small to large scales and from small heights to large
heights which allows us to greatly speed up computations
Can fit expected patterns of weather such as
depressions and fronts to the sparse data to start
and monitor computer weather forecasts allowing
for uncertainty
Data assimilation
Homogenisation
Stochastic
Complexity .. May apply to many many other problems
Where many things interact with each other
• Spread of disease
• Customer behavior
• Transport networks
• Chemical reactions
Much still to be discovered!!!
The BICS team:
Darryl Almond, Chris Bowen, Nick Britton, Chris
Budd, Guler Ergun, Ivan Graham, Giles Hunt,
Merilee Hurn, Ilia Kamotski,Vladimir Kamotski, Jan
van Lent, Ann Linfield, Nick McCullen, Cathryn
Mitchell, Ruth Salway, Rob Scheichl, Hartmut
Schwetlick, Valery Smyshlyaev, Chris Williams,
Johannes Zimmer