Week 10: Chaos Flashcards
What was the worldview like in the 19th C?
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
Where did the holes in the Newtonian worldview appear?
- 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 - 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
What was Lorenz studying? What did he observe?
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
What was the Butterfly effect? Who, when?
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
What were the symptoms of Chaos?
The Smale Horseshoe
“Self similarity” in the logistic map
“Scale invariance” > Fractal Geometry
Strange Attractor
What is the Universality of Chaos?
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
What was the first demonstration of the production fo chaos?
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
Smale Horseshoe?
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.
Who studied teh Logistic maps? What was seen?
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”
What was Fractal geometry? Who?
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
What was teh strange attractor? Who?
Traditional systems: Point & Periodic attractor
David Ruelle and Floris Takens studied the dynamic behaviour of chaotic systems – and discovered they often demonstrated a different class of behaviour. 3. Chaotic system: Strange Attractor o Chaotic systems don’t tend to converge on one point o Instead, they seem to orbit around two points, dangling between the two o Within each area, there are infinite trajectories passing through that area Known as “dense coverage” property o As May and Mandelbrot saw, it displays self-similarity
