https://medium.com/@101summaries/making-a-new-science-4f0f4405362c Chaos theory shows how simple systems can behave in very complex and unpredictable ways. Scientists like Edward Lorenz and Mitchell Feigenbaum helped reveal patterns and rules behind this chaos. Today, we see these chaotic systems everywhere in nature, from weather to biology. # Highlights ## Key Concepts in this book After identifying the unpredictability of weather, meteorologist Edward Lorenz became the conceptual father of chaos theory. Nonlinear systems with simple inputs can yield immensely complicated behaviour. Nonlinear systems were first studied seriously by physicists and mathematicians in the 1970s. The behaviour of animal populations is similar to that of nonlinear dynamical systems. The indefinitely intricate patterns of complex dynamical systems were unveiled by Mandelbrot’s fractal geometry. Strange attractors aided scientists in their understanding of turbulence’s complex dynamics. Mitchell Feigenbaum raised chaos theory to new heights of legitimacy by discovering the universal principles of nonlinear systems. To popularize chaos theory, a group of young mathematicians at UC Santa Cruz employed computer graphics and ordinary phenomena. Nonlinear dynamical systems can be found all across nature, and they’re especially vital to our biology. Who can benefit the most from this book Curious minds interested in unraveling the mysteries of the universe. Anyone looking to understand and appreciate the chaos of life. 1. After identifying the unpredictability of weather, meteorologist Edward Lorenz became the conceptual father of chaos theory. Scientists in the 1950s were very confident about their ability to forecast – and even manipulate – the weather. Here's the main point: After identifying the unpredictability of weather, meteorologist Edward Lorenz became the conceptual father of chaos theory. Edward Lorenz started a weather simulation on his spanking new computer in 1960. He'd typed in.506. for airstream, for example. However, the computer's calculations were accurate to the sixth decimal place:.506127. This seemingly insignificant adjustment was enough to knock the weather forecast entirely off course. Instead, his blunder demonstrated how unstable, unpredictable, and chaotic complex systems can be. The butterfly effect was coined by Lorenz. This means that weather systems are so sensitive to minor perturbations that a butterfly flapping its wings in Beijing today could cause a devastating storm in New York next month. This is referred to as "sensitive dependency on initial conditions" in science, and it constituted the foundation of the new area of chaos theory. 2. Nonlinear systems with simple inputs can yield immensely complicated behaviour. One of the reasons Lorenz was attracted by the weather was because of the butterfly effect. It means that even if we have strewn weather sensors a foot apart throughout the globe, we wouldn't be able to forecast the weather for a few weeks. This is the most important message: Nonlinear systems with simple inputs can yield immensely complicated behaviour. A simple waterwheel, turning as the flow of water fills its buckets, was one of the most renowned chaotic systems he discovered. Lorenz discovered that if the water flow is fast enough, the buckets cease filling fully and the wheel's motion slows or reverses. The motion gets chaotic at high speeds. When Lorenz plotted his three equations on a graph, he discovered a distinctive shape: a weird three-dimensional double spiral that resembles a pair of butterfly wings. Like the weather, the waterwheel, or a playground swing, its motions were virtually cyclical but never exactly repeat themselves. 3. Nonlinear systems were first studied seriously by physicists and mathematicians in the 1970s. What is the most important message? Nonlinear systems were first studied seriously by physicists and mathematicians in the 1970s. Pendulums become one of the most popular objects to study for scientists interested in chaos. Stephen Smale, a mathematician at UC Berkeley, was one of the first to take chaos seriously, even though he had never heard of Lorenz's work. Smale studied topology, a branch of mathematics that investigates which properties of geometric objects remain constant when they are distorted, twisted, or stretched. He was able to visualize chaotic systems thanks to his geometric method. Smale discovered, to his astonishment, that chaos and instability are not the same things. Nonlinear systems, he discovered, can have substantially more stable average behaviour than linear systems. Even when exposed to external noise and perturbations, a nonlinear system quickly reverts to its original chaotic structure. When individuals started integrating Lorenz’s and Smale’s work, it ushered in a new generation of chaos experts who were enthralled by the richness and complexity that simple, predictable systems may produce. 4. The behaviour of animal populations is similar to that of nonlinear dynamical systems. For example, animal populations change in a nonlinear, dynamical manner. Ecology is a branch of biology that analyzes how organisms behave over time, and it was one of the first fields to link its findings to chaos theory. The math behind population growth is straightforward. The greater the number of animals you have, the more offspring they can have. However, animal populations do not continue to increase indefinitely for a variety of reasons. When you include limited food resources, for example, the calculation becomes considerably more difficult. A tiny population may increase swiftly and exponentially at first. However, the larger it becomes, the slower it grows. It occasionally, and inexplicably, collapses totally. This is referred to as a "boom-and-bust cycle" in ecology and economics. What is the main point here? The behaviour of animal populations is similar to that of nonlinear dynamical systems. 5. The indefinitely intricate patterns of complex dynamical systems were unveiled by Mandelbrot's fractal geometry. Benoit Mandelbrot, a mathematician and polymath, has spent most of his life in settings where he was not wanted. When he was studying cotton price swings in the nineteenth century, he got a glimpse of the finding that would make him famous: our universe's densely nested nature. Here's the main point: The indefinitely intricate patterns of complex dynamical systems were unveiled by Mandelbrot's fractal geometry. And he discovered something intriguing: daily price variations matched monthly price changes, with little trends nested inside larger trends, and so on. This symmetry of scale captivated Mandelbrot, and he soon recognized it in numerous structures, both abstract mathematical structures and real-world phenomena. Mandelbrot coined the term "fractal" to describe structures that are self-similar. The length of Britain's coastline approaches infinity as your units of measurement shrink - say, to an atom. This infinite — the rugged, scattered, and fragmented nature of our reality – is accounted for by Mandelbrot’s novel fractal geometry. Mandelbrot became something of a celebrity in academia because fractal geometry was so elegant and beautiful. His inexhaustibly complex geometrical constructions became the visual embodiment of chaos theory. 6. Strange attractors aided scientists in their understanding of turbulence's complex dynamics. Werner Heisenberg, a quantum scientist, pledged on his deathbed to ask God two questions regarding physics: What is the point of relativity? What is the significance of turbulence? Heisenberg said, "I honestly think He may have an answer to the first question." Turbulence is one of the most difficult long-standing problems in physics. It occurs when a smooth flow of a gas or liquid becomes clogged, forming whorls and eddies. Turbulence is all around us, and it's a major issue for engineers. For a long time, researching fluid dynamics appeared hopeless, so physicists delegated it to engineers who had to deal with its practical ramifications. Chaos theory, on the other hand, was expected to throw new light on fluid dynamics. Here's the main point: Strange attractors aided scientists in their understanding of turbulence's complex dynamics. Harry Swinney and Jerry Gollub, two of Landau's American colleagues, worked up in 1973 to prove that turbulence grows up in a linear method. They built a system of two cylinders, one spinning inside the other, with a liquid flowing in the gap between them, to explore fluid motion in action. At higher speeds, the motion becomes chaotic, resulting in turbulence. The progression did not appear to be progressive at all. Most notably, even in turbulence, the liquid flow was not evenly disordered – smooth flow zones jostled with turbulence regions. David Ruelle, a Belgian physicist, came to the rescue. He plotted the motions of dynamical systems in phase space to visualize the onset of turbulence. A phase space is an abstract space that tracks all potential states of a system at any given time and aids scientists in visualizing how it evolves. In phase space, certain systems have "attractors," such as a stable state where they find equilibrium or dynamic states where they cycle. Many nonlinear dynamical systems have what Ruelle calls "strange attractors," as he discovered. These systems revolve around specific places in phase space, but never in the same cycle. Edward Lorenz had arrived first once more. When he was calculating his first nonlinear system, he noticed an eternally looping set of butterfly wings. The Lorenz attractor was the world’s first odd attractor. 7. Mitchell Feigenbaum raised chaos theory to new heights of legitimacy by discovering the universal principles of nonlinear systems. Feigenbaum, like May, was enthralled by the notion that basic systems can exhibit highly complicated behaviour, and that some systems never achieve equilibrium. Here's the main point: Mitchell Feigenbaum raised chaos theory to new heights of legitimacy by discovering the universal principles of nonlinear systems. Feigenbaum was particularly interested in systems that were almost intransitive. These are nonlinear systems that oscillate for a long period around one average state before kicking into a whole new average state at random. Feigenbaum was fascinated by the point where order and chaos collide and a new average state emerges. The Feigenbaum constants indicated a significant new characteristic of chaos theory's expanding field: universality. Feigenbaum demonstrated that certain characteristics of nonlinear systems stay constant and can even be predicted. 8. To popularize chaos theory, a group of young mathematicians at UC Santa Cruz employed computer graphics and ordinary phenomena. A group of young mathematicians at the University of California's new Santa Cruz campus took matters into their own hands. Robert Stetson Shaw, a modest young graduate student, was the catalyst. He'd heard of the Lorenz attractor and began experimenting with it by graphing its equations on the campus's large analogue computer. The main message is that a group of young mathematicians from UC Santa Cruz popularized chaos theory by using computer graphics and daily happenings. Other plotters, converters, and filters were added to the Dynamical Systems Collective's computer lab. They could see the random motion of nonlinear systems and uncover patterns in the chaos using all of their computers and equipment. They looked into what the shape of a weird attractor indicated about the system is depicted, for example. The concept of entropy, which argues that our world and all physical systems progress toward higher levels of disorder, is central to information theory. Strange attractors, according to the Dynamical Systems Collective, are information engines. They increase a system's entropy, resulting in chaotic, novel, and unpredictable behaviour. The group suggested that this chaotic information production could be at the root of our mental processes and biological evolution's trajectory. They didn't only bring chaos theory to the computer age, though. They also demonstrated how it may be applied to ordinary events. They used to sit in a café together and wonder aloud, "Where is the nearest unusual attractor?" Robert Shaw demonstrated that even a dripping faucet can produce a random, endlessly creative pattern in one experiment. The Dynamical Systems Collective brought chaos theory to the pinnacle of its popularity using common examples and cutting-edge computer visualizations. Scientists from the fields of economics, ecology, and meteorology took notice, and the field exploded. 9. Nonlinear dynamical systems can be found all across nature, and they're especially vital to our biology. The main point is that nonlinear dynamical systems may be found everywhere in nature, and they are especially crucial to our biology. Doctors tend to think of the body as a collection of different organs, each with its own microstructure and function. The universal laws of motion apply in the human body just as they do everywhere else, according to Huberman's address. Random motion patterns, nonlinear oscillation patterns, and bifurcating rhythms can all be observed in biological structures. Einstein famously stated, "God does not play dice with the universe." Physicist Joseph Ford was prompted by the discovery of chaos theory to oppose him: God does play dice, but they’re loaded dice. "To find out by what rules they were loaded," Ford contended, was the goal of modern physics. The important message in this summary is that scientists have been discovering the chaos hidden in simple physical systems since Edward Lorenz's weather simulations in the 1960s. Simple laws, they discovered, can yield complicated, unpredictable, and chaotic behaviour, much to their surprise. The weather, animal populations, and even human heartbeat are all examples of nonlinear dynamical systems. The chaos, on the other hand, is never aimless. Mathematicians like Benoit Mandelbrot and physicists like Mitchell Feigenbaum have demonstrated that the chaos of our planet has a weird, beautiful order. Try out the chaos game. All you need is a penny, some paper, and a pen. Begin writing anywhere on the sheet of paper. Then make a rule for head or tail, such as "move 25% closer to the centre for the head" or "move two inches south for the tail." Begin flipping the coin and making a fresh mark on the paper for each new point. The chaotic game does not yield a random scattering of dots; instead, it begins to reveal a particular shape. Because there is order in the chaos, even with random processes.