References and Notes

{ } [10]
[1]
There are several excellent reviews of the quadratic map. Some of the oldest are still the best, and all of the following are quite accessible to undergraduates: E. N. Lorenz, The problem of deducing the climate from the governing equations, Tellus 16, 1-11 (1964); E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos.\ Sci. 20, 130-141 (1963); R. M. May, Simple mathematical models with very complicated dynamics, Nature 261, 459-467 (1976); M. J. Feigenbaum, Universal behavior in nonlinear systems, Los Alamos Science 1, 4-27 (1980). These last two articles are reprinted in the book, P. Cvitanovic, ed., Universality in Chaos (Adam Hidgler Ltd: Bristol, 1984). All four are models of good expository writing. A simple circuit providing an analog simulation of the quadratic map suitable for an undergraduate lab is described by T. Mishina, T. Kohmoto, and T. Hashi, Simple electronic circuit for the demonstration of chaotic phenomena, Am. J. Phys. 53 (4), 332-334 (1985).

[2]
A good review of the dynamics of the quadratic map from a dynamical systems perspective is given by R. L. Devaney , Dynamics of simple maps, in Chaos and fractals: The mathematics behind the computer graphics, Proc. Symp. Applied Math. 39, edited by R. L. Devaney and L. Keen (AMS: Rhode Island,1989).

[3]
For an elementary proof, see E. A.\ Jackson, Perspectives in nonlinear dynamics, Vol. 1 (Cambridge University Press: New York, 1989), pp. 152-153.

[4]
Counting the number of periodic orbits of period n is a very pretty combinatorial problem. See Hao B.-L., Elementary symbolic dynamics and chaos in dissipative systems (World Scientific: New Jersey, 1989), pp. 196-201. An updated version for some results in this book can be found in Zheng W.-M. and Hao B.-L., Applied symbolic dynamics, in Experimental study and characterization of chaos, edited by Hao B.-L. (World Scientific: New Jersey, 1990). Also see the original analysis published in the article by M. Metropolis, M. L. Stein, and P. R. Stein, On the finite limit sets for transformations of the unit interval, J. Comb. Theory 15, 25-44 (1973). Also reprinted in Cvitanovic, reference [1].

[5]
For a discussion concerning the existence of a single stable attractor in the quadratic map and its relation to the Schwarzian derivative, see Jackson, reference [3], pp. 148-149, and Appendix D, pp. 396-399.

[6]
A more complete mathematical account of local bifurcation theory is presented by S. N. Rasband, Chaotic dynamics of nonlinear systems (John Wiley & Sons: New York, 1990), pp. 25-31 and pp. 108-109. Chapter 3 deals with universality theory from the quadratic map.

[7]
G. Iooss and D. D. Joseph, Elementary stability and bifurcation theory (Springer-Verlag: New York, 1981).

[8]
For more about this tale see the chapter ``Universality'' of J. Gleick, Chaos: Making a new science, (Viking: New York, 1987).

[9]
See the introduction of P. Cvitanovic, ed., Universality in Chaos (Adam Hidgler Ltd: Bristol, 1984).

[10]
R. L. Devaney, An introduction to chaotic dynamical systems, second edition (Addison-Wesley: New York, 1989). Section 1.10 covers Sarkovskii's Theorem and section 1.18 covers kneading theory. Another elementary proof of Sarkovskii's Theorem can be found in H. Kaplan, A cartoon-assisted proof of Sarkovskii's Theorem, Am. J.\ Phys. 55, 1023-1032 (1987).

[11]
K. Falconer, Fractal geometry (John Wiley & Sons: New York, 1990).

[12]
S. Smale, The mathematics of time. Essays on dynamical systems, economic processes, and related topics (Springer-Verlag: New York, 1980).

[13]
P. Cvitanovic, G. H. Gunaratne, and I. Procaccia, Topological and metric properties of Hénon-type strange attractors, Phys. Rev.\ A 38 (3), 1503-1520 (1988).

[14]
Some papers providing a hands-on approach to kneading theory include: P. Grassberger, On symbolic dynamics of one-humped maps of the interval, Z. Naturforsch. 43a, 671-680 (1988); J.-P. Allouche and M. Cosnard, Itérations de fonctions unimodales et suites engendrées par automates, C. R. Acad. Sc. Paris Série I 296, 159-162 (1983); and reference [13]. Also see section 1.18 of Devaney's book , reference [10], for a nice mathematical account of kneading theory. classic reference in kneading theory is J. Milnor and W. Thurston, On iterated maps of the interval, Lect. Notes in Math. 1342, in Dynamical Systems Proceedings, University of Maryland 1986-87, edited by J. C. Alexander (Springer-Verlag: Berlin, 1988), pp. 465-563. The early mathematical contributions of Fatou, Julia, and Myrberg to the dynamics of maps, symbolic dynamics, and kneading theory are emphasized by C. Mira, Chaotic dynamics: From the one-dimensional endomorphism to the two-dimensional diffeomorphism (World Scientific: New Jersey, 1987).

[15]
H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Vol. 1-3 (Gauthier-Villars: Paris, 1899); reprinted by Dover, 1957. English translation: New methods of celestial mechanics (NASA Technical Translations, 1967). See Vol. 1, section 36 for the quote.

[16]
See sections 2.1.1 and 3.6.4 of Hao B.-L., reference [4], for some results on periodic windows in the quadratic map.