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Making sense of a complex world Chris Budd

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Many natural (and human!) systems appear complex and hard to understand National Electricity Grid

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Atmosphere and climate El Nino Clouds

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Flocking Turbulence Geology

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Aircraft undercarriage Complex designs

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Human behaviour Crowds

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What makes a system complex ? Many components with individual behavior Coupling between components Many different scales in space and time

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Can scientists, mathematicians and engineers make any sense of complexity? And can we use this knowledge to our advantage?

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Traditional view Things are complicated because there are lots of independent things all going on at once

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A complicated example: The tides h(t) t Bombay tides 1872

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Kelvin decomposed h(t) into 37 independent components He found these out using past data and added them up using an analogue computer

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US Tidal predictorKelvins Tidal predictor

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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

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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

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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

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Aircraft undercarriage can be very similar

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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

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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

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Coherent Patterns

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Emergent Patterns often arise because of the way that things interact and communicate with each other Slime mould All described using mathematical equations Flocking

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Patterns in rocks

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Crowds at a scramble crossing

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Scaling laws

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Microstructure of a ceramic. Al 2 O 3 -TiO 2 R TiO 2 C Al 2 O 3

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Frequency Conductivity

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PERCOLATION DETERMINED DC CONDUCTIVITY POWER LAW EMERGENT PROPERTY 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 Conductivity

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We see examples of scaling laws in many other complex systems: The Internet Epidemics Mechanical systems Rocks and water Homogeneous system

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A very complex example.. The H Bomb r: Radius of fireball E: Energy of the bomb t: Time after the explosion G I Taylor Scaling law

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Bringing this all together … forecasting the weather The atmosphere/ocean is a very complex system with many length and time scales

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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!

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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!!

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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!!!

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Intestinal wall: Villi and Microvilli Stomach Intestine Eg. The digestive system

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What have mathematicians done for us? Chris Budd.

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