>[!citation] >Gleick, J. (1998). _Chaos: Making a new science_. Vintage. > [!note] Reading notes > ## Prologue > - 5 “Fifteen years ago, science was headed for a crisis of Increasing specialization […] Dramatically, that specialization has reversed because of chaos”. > - 6 “Relativity eliminated the Newtonian illusion of absolute space and time; quantum theory eliminated the Newtonian dream of a controllable measurement process; and chaos eliminates the Laplacian fantasy of deterministic probability”. > - 8 “Only a new kind of science could begin to cross the great gulf between knowledge of what one thing does —one water molecule, one cell of heart tissue, one neuron— and what millions of them do”. > - 8 “Traditionally, when physicists saw complex results, they looked for complex clauses […] The modern study of chaos began with the creeping realization in the 1960s that quite simple mathematical equations could model systems every bit as violent as a waterfall”. > ## Chapter 1 — The Butterfly Effect > - 13 “Not only did meteorologists scorn forecasting, but in the 1960s, virtually all serious scientists mistrusted computers […] so numerical weather modeling was something of a bastard problem”. > - 14 The advent of powerful computers led to believe that [[Laplace's demon]] would become possible; “Van Neumann [in the 1950s] recognized that weather modeling could be an ideal task for a computer”. > - 15 It was already understood that the measurements fed into the computers would be less than perfect, but it was assumed under Newtonian tradition that this would be fine: “Given an *approximate* knowledge of a system’s initial conditions and an understanding of natural law, one can calculate the *approximate* behavior of the system […] Very small influences can be neglected”. Reminds me of [[All models are wrong]]. > - 16 But one day in 1961, Edward Lorenz ran a computer forecast twice, except that he manually entered the initial conditions from the first run’s printout into the second. The resulting forecast diverged catastrophically due to the fact that the printout had just three decimals of precision, whereas the computer memory had six. > - 17 “That first day, [Lorenz] decided that long-range weather forecasting must be doomed”. > - 18 In fact, predicting other phenomena, such as eclipses and tides, is just as complex; but because they are periodic, they are not as sensitive to initial conditions, which makes small imprecisions at the start of the run inconsequential even for long-term simulations. Weather, on the other hand, is aperiodic. > - 22 “Lorenz saw that there must be a link between the unwillingness of the weather to repeat itself and the inability of forecasters to predict it — a link between aperiodicity and unpredictability”. > - 23 “The [[Butterfly Effect]] acquired a technical name: sensitive dependence on initial conditions”. Gleick illustrates with the parable of [[For want of a nail]. > - 24 Lorenz then looked for other [[Nonlinear system|nonlinear systems]] and discovered the [[Malkus waterwheel]]. He examined the [[Navier–Stokes equations]], and, simplifying the convection modeling with just three equations (the [[Lorenz system]] introduced in his paper [[Deterministic nonperiodic flows]]), he found that the outputs of those equations plotted in the 3D plane form a Lorenz attractor. > ## Chapter 2 — Revolution > - 36 The historian of science Thomas S. Kuhn developed the view that science progresses not by “mopping up operations” (carrying out previous experiments slightly differently, or adding a theoretical brick to the wall), so early chaos theorists met with hostility and resistance. > - 39 The pendulum was to Galileo what the bathtub was to Archimedes or the apple was to Newton. But the pendulum is not as predictable as it looks; it does have a small amount of non-linearity which makes it behave seemingly randomly. > - 48 In the 1960s, mathematician Stephen Smale found out about chaotic systems whose irregularity came back after being disturbed. They were locally unpredictable but globally stable. Chaos and instability were unrelated concepts. > - 51 Smale studied oscillators and came up with the idea of a [[Smale’s horseshoe|horseshoe]], borrowed from topology, which consists of transforming a system repeatedly through stretching, squeezing, and folding. > > ## Chapter 3 — Life’s Ups and Downs > - 60 Biologists use simplified time-step models (e.g., from one species’ generation to the next) rather than differential equations which describe how processes change smoothly over time. > - 63 The logistic equation x(next) = rx(1-x) where r is the rate of population growth gives the population at step next. It’s an evolution of the linear Malthusian function x(next) = rx. It grows small populations and shrinks large ones to some equilibrium. > - 66 The mathematician James Yorke is the one who “discovered” the work of meteorologist Edward Lorenz’ 1963 paper “Deterministic Nonperiodic Flow” in 1972 and bringing it to the attention of Stephen Smale who was studying nonlinear systems from a pure math standpoint. > - 68 The mathematician Stanislaw Ulam remarked that to call the study of chaos “nonlinear science” was like calling zoology “the study of nonelephant animals”. Linear systems are actually the exception. > - 68 Yorke: “The first message is that there is disorder. Physicists and mathematicians want to discover regularities. People say, what use is disorder. But people have to know about disorder if they are going to deal with it. The auto mechanic who doesn’t know about sludge in valves is not a good mechanic”. > - 69 Yorke coined the word “chaos” in chaos theory in his paper “Period three implies chaos” which was meant to broadcast the messages of Lorenz and Smale. > - 70 The logistics map bifurcates when the parameter r exceeds 3 or so. > - 76 Chaos is ubiquitious; it is stable; it is structured. > > **Chapter 4 — A Geometry of Nature** > - 83 Anecdote on serendipity: Houthakker, a Harvard economics professor, had invited Mandelbrot to give a talk, and when the young mathematician arrived […], he was startled to see his findings already charted on the older man’s blackboard. Mandelbrot made a querulous joke — *how should my diagram have materialized ahead of my lecture?* — but Houthakker didn’t know what Mandelbrot was talking about. The diagram had nothing to do with income distribution; it represented eight years of cotton prices. > - 92 The [[Noah effect]] means discontinuity instead of smoothness: it refutes the idea that a stock must sell for $50 in its way down from $60 to $10. > - 93 [[Cantor dust]]: begin with a line; remove the middle third; then remove the middle third of the remaining segments; and so on. The Cantor set is the dust of points that remains. They are infinitely many, but their total length is 0. It is indispensable in modeling intermittency. > - 93 The [[Joseph effect]] means persistence (similar to [[hysteresis]]?): despite the underlying randomness, the longer a place has suffered drought, the likelier it is to suffer more. > - 96 The coastline measurement problem. The finer (shorter) your measuring instrument, the longer the coastline. Intuitively, as you keep using a shorter ruler, you should converge to some finite value for the length of the coastline. But a coastline is a fractal shape, not a Euclidean one. So the measured length rises without limit, at least until we reach an atomic scale. > - 98 Mandelbrot originated the word “fractal” in 1975 from his son’s latin dictionary in which he referenced the adjective “fractus” (broken). It seemed appropriate not only because of the infinite breaking up of shapes but also because of fraction (he was exploring fractional dimensions, not just 2D or 3D). > - 98 “In the mind’s eye, a fractal is a way of seeing infinity”. > - 99 The Koch snowflake (or Koch curve): being with a triangle with sides of length 1; at the middle of each side, add a new triangle one-third the size; and so on. The length of the boundary is 3 x 4/3 x 4/3 … which is infinity. Yet the area remains less than the area of a circle drawn around the original triangle. Thus, an infinitely long line surrounds a finite area (which is not much bigger than the original triangle, in fact). If you drew a circle around the original triangle, the Koch curve would never extend beyond it, so the area is clearly bounded, even though the curve is infinitely long. > - 107 Scale matters. There are objects that scale and others don’t. Human bones would collapse under their weight if humans were twice the size. But a large earthquake is just a scaled-up version of a small earthquake. > - 126 Discontinuity in physics: Harry Swinney was studying the thermal conductivity of helium at the phase transition point between vapor to liquid. A small change in thermal energy completely changes the property of the material, in this case by a factor of 1,000. The same applies for other phase transitions: fluid to superfluid, conductor to superconductor, nonmagnet to magnet, etc. These are bifurcations, in similar fashion to the bifurcation diagram.