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Kindle Notes & Highlights
by
Iwona Abrams
Read between
February 10 - February 15, 2022
The ancient Greeks seem to have accepted that chaos precedes order, in other words, that order comes from disorder.
Chaos is exciting for all these reasons ... It connects our everyday experiences to the laws of nature by revealing the subtle relationships between simplicity and complexity and between orderliness and randomness.
It shows that predictability is a rare phenomenon operating only within the constraints that science has filtered out from the rich diversity of our complex world.
It shows that there are inherent limits to our understanding and predicting the future at all levels of complexity.
tributaries
A deterministic system is one that is predictable, stable and completely knowable.
In linear systems, variables are simply and directly related. Mathematically, a linear relationship can be expressed as a simple equation where the variables involved appear only to the power of one: X = 2y + Z
Nonlinear relationships involve powers other than one.
A period is an interval of time characterized by the occurrence of a certain condition or event.
Aperiodic behaviour occurs when no variable affecting the state of the system undergoes a completely regular repetition of values
Unstable aperiodic behaviour is highly complex. It never repeats itself and continues to show the effects of any small perturbation to the system. This makes exact predictions impossible and produces a series of measurements that appear random.
What is Unstable Aperiodic Behaviour?
broad patterns in the rise and fall of civilizations. We can see that these patterns are periodic. But we know that events never actually repeat themselves exactly. In this realistic sense, history is aperiodic.
chaos is the occurrence of aperiodic, apparently random events in a deterministic system. In chaos there is order, and in order there lies chaos.
Most forces in real life are nonlinear. So why have we not discovered this before? The reason that chaotic behaviour has not been studied until now is because scientists reduced difficult nonlinear problems to simpler linear ones in order to analyze them.
it is much easier to study population as a simple linear system than one involving feedback and complexity.
Chaotic behaviour results when nonlinear forces are turned back on themselves. This is called nonlinear feedback – and is an essential prerequisite for chaos.
The Three Body Problem
The reason why the three-body problem cannot be solved is that gravity is a nonlinear force (specifically, it is “inverse square”), and in a three-body system each body exerts its force on the other two. This produces nonlinear feedback and results in chaotic motion of the moons’ orbits. But we have now “solved” the three-body problem by demonstrating that the orbits are inherently unpredictable. Such a solution would have been considered nothing short of sacrilege a few years ago.
An amateur Biblical scholar, Immanuel Velikovsky (1895–1979), was dismissed by astronomers as a complete crank when he argued in his Worlds in Collision (1948) that the orbits of Mars and Venus had changed drastically around 1000 B.C. His theory did help to resolve some difficulties with the chronology of the ancient world.
If we take the simple equation x2 + c = result, where x is a complex number that changes and c is a fixed complex number, and continuously feedback the result into the changing number (x) slot – that is, we iterate the equation – chaotic patterns like these are produced
Chaos theory works by asking questions about the long-term behaviour of a system.
Conventional astronomy, for example, is interested in knowing when a system of three planets will line up.
"In contrast, chaos theory will ask, what circumstances sbould lead to elliptical orbits."
A distinguishing feature of systems studied by chaos theory is that unstable aperiodic behaviour can be found in mathematically simple systems. Very simple, rigorously defined mathematical models can display behaviour that is awesomely complex.
Another distinguishing characteristic of chaotic systems is their sensitive dependence on initial conditions – infinitesimally small changes at the start lead to bigger changes later. This behaviour is described as the signature of chaos.
To explain sensitive dependence, mathematical physicist David Ruelle tells this story. “The little devil, presumably having nothing else to do, decides one day to upset your life. The devil does this by altering the motion of a single electron in the atmosphere. But you don’t notice. Not yet. After a minute, the structure of turbulence in the air has changed. You still don’t notice that anything is amiss.
Benoit Mandelbrot (b. 1924), a Polish-born French mathematical physicist who worked for IBM, developed the field of fractal geometry
Chaos on the Telephone Lines
Engineers were puzzled by the problem of noise in the lines. The current carries the information in “discrete packets” in the lines. But some spontaneous noise could not be eliminated. Sometimes it would wipe out the signal, creating an error. The interference was random, yet it occurred in clusters. I started to investigate by working with time, dividing the day into two, then dividing that period into two, and so on. He found an hour with no errors. But when he divided the hour with errors into two, he found again, a period with no errors and a period with errors. Again, when he divided the
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the Cantor Set – a pattern that is created by removing segments of a line, then removing segments of the segments through to infinity, leaving a dust of points arranged in clusters.
Mandelbrot asked “How Long is the Coast of Britain?”. Suppose we measure the British coastline with a metre stick. The answer will be approximate,
if we measure the coastline with smaller and smaller scale, the answer will become larger and larger. As we approach towards a very small scale of measurement, the coastline becomes longer and longer without a limit.
Fractal Dimensionality Mandelbrot suggested that what we observe depends on where we are positioned and how we measure it.
Mandelbrot described systems with fractional dimensionality with the term fractals. The coastline of Britain is an example of fractals. And, he argued, the only way to solve this problem is to move from ordinary three dimensions into what he called “fractal dimensions”.
Fractal geometry is the geometry of special types of irregular shapes. Fractals are a way of measuring qualities that otherwise have no clear definition: the degree of roughness or brokenness or irregularity in an object.
"A fractal is a way of seeing infinity." Mandelbrot
Fractals are geometrical shapes that, contrary to those of Euclid, are not regular at all. First, they are irregular all over. Secondly, they have the same degree of irregularity on all scales. A fractal object looks the same when examined from far away or nearby – it is self-similar.”
Self-similarity implies that any subsystem of a fractal system is equivalent to the whole system.
Julia set. It explores imaginary numbers in a complex plane.
Fractals help us understand turbulence, not just how it arises, but the motion of the turbulence itself.
Blood vessels can also be considered as fractals, as they can be divided down into smaller and smaller sections. They perform what has been described as “dimensional magic”, squeezing a large surface area into a limited volume.
The fractal dimensions of a metal’s surface also tell us a lot about its strength.
the Mandelbrot set.
fractals are now an important part of special effects in films.
Edward Lorenz (b. 1917),
became the director of a project in statistical weather forecasting, a field that was pioneered by his department.
Lorenz studied the interrelationship between three nonlinear meteorological factors: temperature, pressure and wind speed.
Lorenz’s discovery of the phenomena of chaos is often related in an interesting story. One day in 1961, the story goes, Lorenz decided to take a short cut with his weather machine. He wanted to examine one sequence at greater length. So instead of starting the whole computer run from the beginning, he started half-way through. He tapped in the numbers straight from an earlier printout and went away to get a coffee. When he came back he could hardly believe his eyes. The newly generated weather was nowhere near the original. They were two completely different systems! Then he realized what had
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two states differing by imperceptible amounts may eventually evolve into two considerably different states.
