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Introduction to papers on chaos in nonquadratic dynamics: rational functions devised from doubling formulas

In: Fractals and Chaos

Author

Listed:
  • Benoit B. Mandelbrot

    (Yale University, Mathematics Department
    IBM T.J. Watson Research Center)

Abstract

THE PATH OF SCIENTIFIC INVESTIGATION AND DISCOVERY is not necessarily logical, as history never tires of reminding both laymen and scientists. From many viewpoints, the complex quadratic map, reducible to either z→z2 + c, z → λ(z2-2), or z → λz(1-z), is the simplest of all nonlinear maps. Its global action was therefore the first to be studied carefully, in Fatou 1906. Later, the 1960s and 1970s brought many studies of the restriction of z2-μ to the real quadratic map x 2-μ, and everyone became well aware that the dependence of this real map’s orbits on the parameter μ involves exquisite complications. It has its Myrberg sequence of bifurcations, its May tree, and its Feigenbaum number. Shortly thereafter, my papers reproduced in Part I made the complex quadratic map widely popular.

Suggested Citation

  • Benoit B. Mandelbrot, 2004. "Introduction to papers on chaos in nonquadratic dynamics: rational functions devised from doubling formulas," Springer Books, in: Fractals and Chaos, chapter 0, pages 137-145, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4757-4017-2_12
    DOI: 10.1007/978-1-4757-4017-2_12
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