References and Notes

{ } [10]
[1]
Good general references for the material in this chapter are listed here. For ordinary differential equations, see V. I. Arnold, Ordinary differential equations (MIT Press:\ Cambridge, MA, 1973). Suspensions are discussed in Z. Nitecki, Differentiable Dynamics (MIT Press: Cambridge, MA, 1971), and S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos (Springer-Verlag: New York, 1990). The standard reference for applied dynamical systems theory is J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, second printing (Springer-Verlag: New York, 1986).

[2]
IHJM stands for Ikeda, Hammel, Jones, and Moloney. See S. K. Hammel, C. K. R. T. Jones, and J. V. Moloney, Global dynamical behavior of the optical field in a ring cavity, J. Opt. Soc. Am. B 2, 552-564 (1985).

[3]
The quotes are from Paul Blanchard. Some of the introductory material is based on notes from Paul Blanchard's 1984 Dynamical Systems Course, MA771. We are indebted to Paul and to Dick Hall for explaining to us the ``chain recurrent point of view.''

[4]
C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Conference Series 38 (1978); J. M. Franks, Homology and Dynamical Systems, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics Number 49 (American Mathematical Society, Providence, 1980); R. Easton, Isolating blocks and epsilon chains for maps, Physica 39D, 95-110 (1989).

[5]
J. Marsden and A. Tromba, Vector Calculus, third ed. (W. H. Freeman: New York, 1988).

[6]
S. Rasband, Chaotic dynamics of nonlinear systems (John Wiley: New York, 1990). Chapter 7 discusses fixed point theory for maps.

[7]
J. Thompson and H. Stewart, Nonlinear dynamics and chaos (John Wiley: New York, 1986).

[8]
T. Parker, and L. Chua, Practical numerical algorithms for chaotic systems (Springer-Verlag: New York, 1989).

[9]
A nice description of the homoclinic periodic orbit theorem of Birkhoff and Smith is presented by E. V. Eschenazi, Multistability and Basins of Attraction in Driven Dynamical Systems, Ph.D. Thesis, Drexel University (1988); also see section 6.1 of R. Abraham and C. Shaw, Dynamics--The geometry of behavior. Part three: Global behavior (Aerial Press: Santa Cruz, CA, 1984).

[10]
H. G. Solari and R. Gilmore, Relative rotation rates for driven dynamical systems, Phys. Rev. A 37 (8), 3096-3109 (1988); L. M. Narducci and N. B. Abraham, Laser physics and laser instabilities (World Scientific: New Jersey, 1988).

[11]
S. Smale, The mathematics of time: Essays on dynamical systems, economic processes, and related topics (Springer-Verlag: New York, 1980); J. Yorke and K.\ Alligood, Period doubling cascades for attractors: A prerequisite for horseshoes, Comm. Math. Phys. 101, 303 (1985).

[12]
P. Holmes, Knotted periodic orbits in the suspensions of Smale's horseshoe: Extended families and bifurcation sequences, Physica D 40, 42-64 (1989).

[13]
R. L. Devaney, An introduction to chaotic dynamical systems, second edition (Addison-Wesley: New York, 1989).

[14]
S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos (Springer-Verlag: New York, 1990).

[15]
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).

[16]
An elementary introduction to Lyapunov exponents is provided by S. Souza-Machado, R. Rollins, D. Jacobs, and J. Hartman, Studying chaotic systems using microcomputers and Lyapunov exponents, Am. J. Phys. 58 (4), 321-329 (1990). For the state of the art in computing Lyapunov exponents, see P. Bryant, R. Brown, and H. Abarbanel, Lyapunov exponents from observed time series, Phys. Rev. Lett. 65 (13), 1523-1526 (1990).