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