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Chaos, Drawn

SourcesWhere this comes from

24 sources — the papers that built the field, and the books to read next. Every claim on this site traces to one of these; every figure is computed by the code, not taken from them.

  1. Henri Poincaré, “Sur le problème des trois corps et les équations de la dynamique”, Acta Mathematica 13 (1890) 1–270 — the first sight of a system whose future depends without limit on where it began. ↗
  2. Henri Poincaré, Science et méthode (1908), book I, chapter 4: “a very small cause, which escapes us, determines a considerable effect which we cannot help seeing.” ↗
  3. Jacques Hadamard, “Les surfaces à courbures opposées et leurs lignes géodésiques”, Journal de mathématiques pures et appliquées 4 (1898) 27 — nearby paths on a curved surface pulling apart exponentially. ↗
  4. Aleksandr Lyapunov, The General Problem of the Stability of Motion (1892, doctoral thesis, Kharkov); English translation, Taylor & Francis 1992 — the exponent that measures how fast neighbours separate. ↗
  5. Edward N. Lorenz, “Deterministic Nonperiodic Flow”, Journal of the Atmospheric Sciences 20 (1963) 130–141 — three equations for convection, and the discovery that they never repeat. ↗
  6. Edward N. Lorenz, “Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?”, address to the AAAS, 29 December 1972. ↗
  7. Robert M. May, “Simple mathematical models with very complicated dynamics”, Nature 261 (1976) 459–467 — the logistic map, and a call to teach it in schools. ↗
  8. Tien-Yien Li & James A. Yorke, “Period Three Implies Chaos”, American Mathematical Monthly 82 (1975) 985–992 — the paper that put the word “chaos” into mathematics. ↗
  9. Mitchell J. Feigenbaum, “Quantitative universality for a class of nonlinear transformations”, Journal of Statistical Physics 19 (1978) 25–52 — the constant 4.669, the same for a whole class of maps. ↗
  10. Oleksandr Sharkovsky, “Co-existence of cycles of a continuous map of the line into itself”, Ukrainian Mathematical Journal 16 (1964) 61 — the ordering of periods that has period three at its head. ↗
  11. David Ruelle & Floris Takens, “On the nature of turbulence”, Communications in Mathematical Physics 20 (1971) 167 — the phrase “strange attractor”. ↗
  12. Benoit Mandelbrot, “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension”, Science 156 (1967) 636. ↗
  13. Stephen Smale, “Differentiable dynamical systems”, Bulletin of the AMS 73 (1967) 747 — the horseshoe: stretch and fold, the engine under every chaotic map. ↗
  14. Giancarlo Benettin, Luigi Galgani, Antonio Giorgilli & Jean-Marie Strelcyn, “Lyapunov Characteristic Exponents for smooth dynamical systems”, Meccanica 15 (1980) 9 — the method used here to measure the exponent. ↗
  15. Stanisław Ulam & John von Neumann, “On combination of stochastic and deterministic processes”, Bulletin of the AMS 53 (1947) 1120 — the map x → 4x(1−x) as a source of pseudo-random numbers, and its arcsine distribution. ↗
  16. Jacob Bernoulli, Ars Conjectandi (Basel, 1713), part IV — the first proof that the observed frequency closes on the true chance as the trials pile up. His “golden theorem”. ↗
  17. Andrey Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung (1933); English, Foundations of the Theory of Probability, Chelsea 1956 — the strong law and the axioms under it. ↗
  18. Harry Markowitz, “Portfolio Selection”, Journal of Finance 7 (1952) 77–91 — spreading a stake across uncorrelated bets shrinks the swing without shrinking the return. ↗
  19. C. S. Holling, “Resilience and Stability of Ecological Systems”, Annual Review of Ecology and Systematics 4 (1973) 1–23 — resilience as the size of the shock a system can take and still recover, as distinct from how fast it returns. ↗
  20. James Clerk Maxwell, “On Governors”, Proceedings of the Royal Society 16 (1868) 270 — the mathematics of a machine that corrects its own error, the ancestor of control theory. ↗
  21. Norbert Wiener, Cybernetics: or Control and Communication in the Animal and the Machine (MIT Press, 1948) — feedback as the common thread of stable systems, living and built. ↗
  22. Steven H. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed. (Westview/CRC, 2015) — the standard undergraduate text, and a kind one. ↗
  23. James Gleick, Chaos: Making a New Science (Viking, 1987) — how the field came together, told for everyone. ↗
  24. Nassim Nicholas Taleb, Antifragile: Things That Gain from Disorder (Random House, 2012) — a working vocabulary for things that get stronger under stress, not merely survive it. ↗