Blinkist Summary Book

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order to visualize the onset of turbulence in dynamical systems he plotted their motions in phase space. A phase space is an abstract space that tracks all possible states of a system at any point in time and helps scientists visualize how the system evolves. Some systems have “attractors” in phase space – like a fixed state at which they reach an equilibrium or dynamic states that they cyclically exhibit.
What Ruelle discovered is that many nonlinear dynamical systems have what he called “strange attractors.” These systems orbit around certain points in phase space but never quite in the exact same cycle. After Ruelle published this paper other scientists began constructing their own strange attractors and finding them in the chaos of nature. Astronomer Michel Hénon for instance found that the orbit of stars around the center of “globular clusters” corresponds to a strange attractor.
Once again Edward Lorenz had been there first. Remember the endlessly looping pair of butterfly wing Open / Comment

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dom. But when a flow turns turbulent particles gain more and more degrees of freedom creating ever-more turbulence.
In 1973 Landau’s US colleagues Harry Swinney and Jerry Gollub teamed up to prove that turbulence builds up in this linear fashion. They chose a simple kind of fluid motion to study in action building a system of two cylinders one rotating inside the other with a liquid flowing in the space between them. At first the liquid flows smoothly. But as the rotation of the cylinders speeds up the liquid begins flowing in wavy bands. At even greater speeds the motion becomes chaotic and turbulence emerges. The process didn’t look gradual at all. Most importantly even in turbulence the flow of the liquid wasn’t uniformly chaotic – regions of smooth flow jostled alongside regions of turbulence.
Belgian physicist David Ruelle came to the rescue. In the early 1970s he attended a talk about chaos by Steve Smale and was working on an alternative to Landau’s theory on fluid motion. In Open / Comment

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uid suddenly turns messy breaking up into whorls and eddies. Turbulence is everywhere and poses a big problem for engineers. When the smoke ring of a cigarette rises steadily before breaking up into little curls we might watch in fascination. But when the same thing happens beneath the wings of an airplane we’re traveling in we’re more likely to panic.
For a long time studying fluid dynamics seemed such a hopeless task that physicists left it to the engineers grappling with its practical implications. But chaos theory promised to shed new light on fluid dynamics.
Here’s the key message: Strange attractors helped physicists understand the complicated motions of turbulence.
Before chaos theory the most important theory of turbulence in physics came from Russian scientist Lev D. Landau who proposed that any liquid or gas is formed from a multitude of individual particles whose motion depends on the motion of its neighboring particles. In a smooth flow particles have few degrees of free Open / Comment

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t makes up the coastline? As the units of measurements become smaller you’ll find the coastline grows longer. As your units of measurements become smaller – down to an atom say – Britain's coastline length approaches infinity.
Mandelbrot’s new fractal geometry accounts for this infinity – the rugged scattered and fragmented nature of our world. Because fractal geometry was so elegant and beautiful Mandelbrot became somewhat of a superstar in the academic community. His infinitely intricate geometrical structures became the visual representation of chaos theory.
Strange attractors helped physicists understand the complicated motions of turbulence.
On his deathbed quantum physicist Werner Heisenberg swore to ask God two questions about physics: Why relativity? And why turbulence? “I really think He may have an answer to the first question ” Heisenberg joked.
Of all the longstanding problems in physics turbulence is among the thorniest. It arises when a smooth flow of a gas or liq Open / Comment

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ctuations for daily prices matched the fluctuations for monthly prices with small trends nested inside bigger trends and so on.
Mandelbrot was fascinated by this symmetry of scale and he soon discovered it in other structures – both abstract mathematical ones and real-world phenomena. For instance he began to notice how many structures in nature like mountains and clouds can be broken up into smaller and smaller versions of themselves. Mandelbrot called these types of structures self-similar or fractal.
To illustrate his discovery Mandelbrot liked to ask a simple question: How long is the coast of Britain?
If you look at a map measure the coastline with a ruler and then bring the measurement up to scale you’ll easily have an answer to this question. But did you really measure all the nooks and crannies of the coastline? Probably not. For that you’d have to walk along the entire coast and measure each of its twists and turns. But wait – can you the outline of each rock and pebble tha Open / Comment

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ound work at the IBM research center in New York.
In his research too Mandelbrot often chose unfriendly territory. One of his first interests at IBM was studying economic patterns like income distribution and price changes. When he studied fluctuations in cotton prices in the nineteenth century he got a first glimpse of the discovery that would make him famous: the intricately nested nature of our universe.
Here’s the key message: Mandelbrot's fractal geometry revealed the infinitely intricate patterns of complex dynamical systems.
Economists at the time believed that prices tended to fluctuate randomly over the short term but responded to real forces in the economy over the long term – like economic policy and new technology. Even more they thought that most prices should converge around an average. But the cotton prices of the last century clearly hadn’t done that. Using the latest computers that IBM had to offer Mandelbrot investigated. And he found something interesting: the flu Open / Comment

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s ” Yorke demonstrated that when a system starts breaking up into period-doubling bifurcations it's only a matter of degree before chaos emerges.
He thought that scientists tended to overlook such bifurcations because they didn’t want to see the chaos lurking in the systems they studied. May who later applied his interest in chaos theory to the study of epidemics was one of the first scientists to take them seriously.
Mandelbrot's fractal geometry revealed the infinitely intricate patterns of complex dynamical systems
Throughout his life the mathematician and polymath Benoit Mandelbrot ended up in places where he wasn’t welcome. First as a child he fled with his family from Poland to France in the 1930s. After the war he felt stifled by the intellectual climate at the École Polytechnique where he studied mathematics. At the time pure math was all the rage and his colleagues didn’t appreciate his visual approach to mathematical problems. So Mandelbrot escaped to the US where he f Open / Comment

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ions which measure the change in little jumps – year by year for instance.
A realistic difference equation that describes how the gypsy moth population changes year by year needs to restrain growth after a certain point. The simplest equation that fulfills this criterion is a logistic differential equation. For a long time biologists believed that this type of equation would always reach an equilibrium – just as the animal population would.
Ecologist Robert May experimented with a logistic differential equation when he made a startling discovery. May found that if he ramped up the level of “boom-and-bustiness” of his fictional animal population it would start behaving strangely. First the periodic cycles of the population would double in time then double again – looping into so-called period-doubling bifurcations . Eventually the whole system would turn chaotic.
May turned to his mathematician friend James Yorke to find an explanation. In his seminal paper “Period Three Implies Chao Open / Comment

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theory.
The basic math of population growth is simple. The more animals you have the more offspring they can produce. But for several reasons animal populations don’t just grow and grow. Factor in limited food resources for example and the math gets much more complicated. At the beginning a small population might grow rapidly and exponentially. But the bigger it gets the slower it grows. Sometimes unpredictably it collapses completely. In ecology and economy this is known as a “boom-and-bust cycle.”
The key message here? Animal populations behave like nonlinear dynamical systems.
Let’s consider how an ecologist might study changes to the population of gypsy moths over time. In the real world this happens smoothly one moth at a time. The mathematical equations that could describe such a smooth nonlinear change are called differential equations . But differential equations are complicated to calculate with and most biologists don’t like math to begin with. So they use difference equat Open / Comment

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e stable in their average behavior than linear systems. Even in the face of outside noise and disturbances a nonlinear system soon returns to its same old chaotic pattern.
Smale only found out about Lorenz’s work later and he was surprised that a meteorologist had anticipated the mathematics of chaos ten years before him. When people began connecting Lorenz’s and Smale’s work it paved the way for a new generation of chaos specialists who were fascinated by the richness and complexity that simple deterministic systems can create.
Animal populations behave like nonlinear dynamical systems.
Nonlinear dynamical systems aren’t just a pet project of mathematicians and physicists. As Lorenz showed when he studied the weather nonlinear systems are fundamental to nature.
Animal populations for example change in a nonlinear dynamical way. The subfield of biology that studies how they behave over time is called ecology – and it was one of the first fields to connect its findings to chaos Open / Comment