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

Aircraft undercarriage

Complex designs

Photonic crystals

Human behavior

CrowdsStock markets

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

h(t)

t

Bombay tides 1872

Kelvin decomposed h(t) into 37 independent periodic functions

)sin()(37

1j

jjj tath

Kelvin calculated the coefficients 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 behavior 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 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!

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

21 Angle of top part

Angle of bottom part

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

Slime mouldBZ reaction

Can often describe using differential equations

Flocking

Singularity

Patterns in rocks

Crowds

Scaling laws

Microstructure of a real technical ceramic.

Al2O3-TiO2

RTiO2

CAl2O3

Frequency

Con

du

ctiv

ity

PERCOLATION DETERMINED DCCONDUCTIVITY

POWER LAW EMERGENT PROPERTY

5/2frequencytyconductivi

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 percolation

Con

du

ctiv

ity

An emergent scaling law

Cba

a is something we can measure

b is something that changes

They are related by an equation of the form

If

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

We see examples of scaling laws in many other complex systems:

• The Internet

• Networks of friends

• Disease

• Mechanical systems

• Protein and gene interactions

• Porous media

Homogeneous system

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