Flashcards in Week 10: Chaos Deck (11)

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1

## What was the worldview like in the 19th C?

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In the 19th century, a Newtonian worldview was commonplace

• World is deterministic, and its trajectory is perfectly predictable given initial conditions and the dynamic equations which constrain it

• Akin to the laws of planetary motion

• Many intellectuals, such as Laplace, subscribed to this belief

2

## Where did the holes in the Newtonian worldview appear?

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1. Henri Poincaré

Examined the three-body problem in 1889

Newtonian planetary motion already held for two bodies, but the equations became unsolvable when applied to a system with three planetary bodies

Concluded that such a system was nonlinear and unstable

Expectation: If initial position is slightly off, the predicted future location will be too

Reality: A slight perturbation in initial conditions leads to wild variations in future

Challenged the predicting capabilities of Newtonian systems

Even a small error in the initial conditions would destroy the predictions, making them impractical for use

These weren’t unreasonable far-fetched predictions – it was the motion of planetary bodies, which should be bread and butter for Newtonian mechanics

Was a controversial finding, and did not see wide adoption

2. Aleksandr Lyapunov

Examined turbulence in fluid mechanics

Wanted to uncover the point where fluid goes from steady to turbulent

Created the idea of the “Lyapunov Exponent”

Suppose we have a system which we examine under two conditions at time 0:

X0, the reference initial conditions

X0 + 𝛿; a condition slightly deviating from X0

Consider the difference between the two system’s status at some time as Δ(t)

We know Δ(t0) = 𝛿

Would hope that Δ(t) = 𝛿 as well, but not always the case

Lyapunov modelled this deviation as follows:

Δ(t)≅Δ(t_0 ) e^λt

Note the “Lyapunov Exponent”, λ

> 0 : system is unstable and deviations increase rapidly with t

< 0: system is stable, and system converges on a common result

λ = 0 is the “tipping point” – moving it slightly to either side will have dramatic impacts on the system’s behaviour

3

## What was Lorenz studying? What did he observe?

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In 1961, Edward Lorenz (M.I.T.) published his findings on “Deterministic Nonperiodic Flow”

• Examined the behaviour of weather

• Tried to model the flow of fluid (like gasses)

o Used three variables, (x,y,z)

o Variables governed by three differential functions of time

• System required solving non-linear differential equations

o No closed form solution existed – used numerical methods

• Performed repeated simulations, but found strange issue

o Even repeating with the same output, output could be wildly different

o Unexpected since input was identical, and the program was the same

• Found the issue was due to the computation itself

o Eventually, system would need to round data off

Was leading to slightly different numbers each time

o Usually, an error of 10-10 was acceptable and unnoticeable

But in Lorenz’s simulations, that little difference would lead to completely different results

4

## What was the Butterfly effect? Who, when?

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Lorenz, 1961, studying weather.

• Believed they weren’t errors – but the nature of the system was just that unpredictable

• Coined as the “Butterfly Effect” (the term “Chaos” wouldn’t come until ~1975)

o A minor change like a butterfly flapping its wings can cause a storm a few days later

5

## What were the symptoms of Chaos?

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The Smale Horseshoe

"Self similarity" in the logistic map

"Scale invariance" > Fractal Geometry

Strange Attractor

6

## What is the Universality of Chaos?

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While these were all breakthroughs, they were just symptoms of chaos and still not enough.

Mitchell Feigenbaum was working at the Argonne Labs, and tried to extract a common point; some meaningful link between these chaotic systems.

• Evaluated a limit relating to the location of doubling locations (recall May)

• Showed that it approached a limit

o argued that it was invariant various chaotic systems

7

## What was the first demonstration of the production fo chaos?

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In 1980, Albert Lichbacher experimented with the flow of water

• When does laminar flow end and the vortices begin?

• Saw at low speed, flow was uniform

• But as speed increased, boundary layers formed

o Observed convection from the outer side of the pipe to the centre

o First just one convection cycle – but then it doubles as flow increases further

• Was one of the first demonstrations of chaos, and was experimental validation of period doubling

8

## Smale Horseshoe?

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At UC Berkeley, Stephen Smale examined a type of electronic oscillator, the “Van der Pol” Oscillator

• Key difference compared to other oscillators was that it had non-linear feedback loop

Smale examined the time evolution of the phase space

• Found that the Van der Pol Oscillator transformed the phase space strangely

o Would bend the phase space into a horseshoe, and compress areas near the tips

o Areas which previously were disconnected could now be very close, due to the bending of the horseshoe bringing them side by side

• Repeated iterations would then cause stripes, doubling with each iteration

Smale’s findings were a topological study of chaos. While his was a more macroscopic perspective compared to Lorenz, it demonstrated a similar result.

9

## Who studied teh Logistic maps? What was seen?

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In 1976 at Princeton, Robert May studied logistic maps in Ecology, and showed how a system might go towards chaotic behaviour:

Consider the population in an area over time

Modelled as the following second order equation

x_(n+1)=r×(x_n )×(1-x_n)

Xn is a fraction of the current population over the area’s capacity, [0,1]

‘r’ is the Malthusian parameter

Found that it was dependent on “r”

Very small r: population plummets

Low r: Population converges

As r increases though, population becomes an increasingly complex oscillator

May plotted a logistic map, showing the values the population would converge to under varying ‘r’ values

< 1: Population converges at 0

As r increases, the converging point increases

Eventually the converging point splits in two: system is now bistable

Continuing further, each of the points will split, and those will split, etc

Like a fractal, the map kept performing the same splits again and again

Known as “self-similarity”

10

## What was Fractal geometry? Who?

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Benoit Mandelbrot took this further in the 1950’s and 1960’s while working at IBM Watson

• Studied fluctuating time-series

o Considered the error in data transmissions

• Amplifying waveform yielded repeating structures

o Same “self-similarity” that May saw

• Takes this further and explores fractals

o In fact, they area central characterization of chaotic systems

o Can be thought of as a system; plotting a chaotic system will show this “scale invariance”, where examining it in any region demonstrates the same structures

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