References and Notes

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[1]
J. R. Tredicce and N. B. Abraham, Experimental measurements to identify and/or characterize chaotic signals, in Lasers and quantum optics, edited by L. M. Narducci, E. J. Quel, and J. R. Tredicce, CIF Series Vol. 13 (World Scientific: New Jersey, 1988); N. Gershenfeld, An experimentalist's introduction to the observation of dynamical systems, in Directions in Chaos, Vol. 2, edited by Hao B.-L. (World Scientific, New Jersey, 1988), pp. 310-382; N. B. Abraham, A. M. Albano, and N. B. Tufillaro, Introduction to measures of complexity and chaos, in NATO ASI Series B:\ Physics. Proceedings of the international workshop on Quantitative Measures of Dynamical Complexity in Nonlinear Systems, Bryn Mawr, PA, USA, June 22-24, 1989, edited by N. B. Abraham, A. M. Albano, T. Passamante, and P. Rapp, (Plenum Press: New York, 1990); J. Crutchfield, D. Farmer, N. Packard, R. Shaw, G. Jones, and R. Donnelly, Power spectral analysis of a dynamical system, Phys. Lett. 76A (1), 1-4 (1980).

[2]
C. Gough, The nonlinear free vibration of a damped elastic string, J. Acoust. Soc. Am. 75, 1770-1776 (1984).

[3]
The original version of this apparatus was constructed and examined for possible chaotic motions by T. C. A. Molteno at the University of Otago, Dunedin, New Zealand. For more details, see T. C. A. Molteno and N.\ B. Tufillaro, Torus doubling and chaotic string vibrations: Experimental results, J. Sound Vib. 137 (2), 327-330 (1990).

[4]
R. C. Cross, A simple measurement of string motion, Am. J.\ Phys. 56 (11), 1047-1048 (1988).

[5]
D. Martin, Decay rates of piano tones, J. Acoust. Soc. Am.\ 19 (4), 535-541 (1974).

[6]
R. Hanson, Optoelectronic detection of string vibrations, The Physics Teacher (March 1987), 165-166.

[7]
N. B. Tufillaro, Nonlinear and chaotic string vibrations, Am. J. Phys. 57 (5), 404-414 (1989); N. B. Tufillaro, Torsional parametric oscillations in wires, Eur. J. Phys. 11, 122-124 (1990).

[8]
For theoretical and experimental details on chaos in an elastic beam, see F. C. Moon, Chaotic vibrations: An introduction for applied scientists and engineers (John Wiley & Sons: New York, 1987). In particular, Appendix C contains a detailed description of the construction of an inexpensive beam experiment showing chaos. Some discussion of the Duffing equation as it relates to the motion of a beam can also be found in J. M. Thompson and H. B. Steward, Nonlinear dynamics and chaos (John Wiley & Sons: New York, 1986).

[9]
J. Elliot, Intrinsic nonlinear effects in vibrating strings, Am. J. Phys. 48 (6), 478-480 (1980); J. Elliot, Nonlinear resonance in vibrating strings, Am. J. Phys. 50 (12), 1148-1150 (1982).

[10]
See Chapter 4 of A. Nayfeh and D. Mook, Nonlinear oscillators (Wiley-Interscience: New York, 1979). A more elementary account of perturbative solutions to the Duffing equation is presented by A. Nayfeh, Introduction to perturbation techniques (John Wiley & Sons: New York, 1981).

[11]
A classic in nonlinear studies is C. Hayashi's Nonlinear oscillations in physical systems (Princeton University Press:\ Princeton, NJ, 1985). The method of harmonic balance is used extensively by Hayashi; see pages 28-30, 114-120, and 238-251 for a discussion and examples of this technique applied to the Duffing oscillator .

[12]
A real gem of mathematical exposition is V. I. Arnold's Ordinary differential equations (MIT Press: Cambridge, MA, 1973). The basic theorems of ordinary differential equations are proved in Chapter 4. Arnold's text is the best introductory exposition to the geometric study of differential equations available. A must read for all students of dynamical systems.

[13]
S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos (Springer-Verlag: New York, 1990), pp. 64-82.

[14]
N. B. Tufillaro and G. A. Ross, Ode--A program for the numerical solution of ordinary differential equations, Bell Laboratories Technical Memorandum 83-52321-39 (1983); program and documentation are in the public domain. Also see Appendix C.

[15]
U. Parlitz and W. Lauterborn, Superstructure in the bifurcation set of the Duffing equation, Phys. Lett. 107A (8), 351-355 (1985); U. Parlitz and W. Lauterborn, Resonance and torsion numbers of driven dissipative nonlinear oscillators, Z. Naturforsch. 41a, 605-614 (1986).

[16]
The method of slowly varying amplitude is also known as the averaging method of Krylov-Bogoliubov-Mitropolsky; see E. A.\ Jackson, Perspectives in nonlinear dynamics, Vol. 1 (Cambridge University Press: New York, 1989), pp. 264, 308-322. In the optics community, it is also known as the rotating-wave approximation ; see P.\ W. Milonni, M.-L. Shih, and J. R. Ackerhalt, Chaos in laser-matter interactions (World Scientific:\ Singapore, 1987), pp. 58-70.

[17]
This is a bit sneaky. here because there is no damping term causing a frequency shift in the approximation to first order. In general , and then the slowly varying amplitude approximation results in both a detuning and a phase shift. Both data are encoded by .

[18]
A marvelous pictorial introduction to dynamical systems theory is the unique series of books by R. Abraham and C. Shaw, Dynamics--The geometry of behavior, Vol. 1-4 (Aerial Press: Santa Cruz, CA, 1988). Volume 1, pp. 143-151, contains a good description of the geometry of the phase space associated with hysteresis in the Duffing oscillator, and Volume 3, pp. 83-87, shows how homoclinic tangles are formed.

[19]
J. M. Johnson and A. K. Bajaj, Amplitude modulated and chaotic dynamics in resonant motion of strings, J. Sound Vib. 128 (1), 87-107 (1989).

[20]
J. Miles, Resonant, nonplanar motion of a stretched string, J. Acoust. Soc. Am. 75 (5), 1505-1510 (1984).

[21]
O. M. O'Reilly, The chaotic vibration of a string, Ph.D. thesis, Cornell University, 1990.

[22]
P. Bak, The devil's staircase, Physics Today 39 (12), 38-45 (1986); P. Bak, T. Bohr, and M. H.\ Jensen, Mode-locking and the transition to chaos in dissipative systems, Physica Scripta, Vol. T9, 50-58 (1985); P. Cvitanovic, B. Shraiman, and B. Söderberg, Scaling laws for mode lockings in circle maps, Physica Scripta 32, 263-270 (1985) .

[23]
For a fuller account of torus attractors and quasiperiodic motion in a nonlinear system, see Chapter 7 of P. Bergé, Y.\ Pomeau, and C. Vidal, Order within chaos (John Wiley & Sons: New York, 1984). An elementary discussion of fractal dimensions and the correlation integral is found on pages 146-154.

[24]
A. Arnédo, P. Coullet, and E. Spiegel, Cascade of period doublings of tori, Phys. Lett. 94A (1), 1-6 (1983); K. Kaneko, Doubling of torus, Prog. Theor. Phys. 69 (6), 1806-1810 (1983); F. Argoul and A. Arnédo, From quasiperiodicity to chaos: an unstable scenario via period doubling bifurcation or tori, J. de Mécanique Théorique et Appliquée, Numeréro spécial, 241-288 (1984).

[25]
W. Ditto, M. Spano, H. Savage, S. Rauseo, J. Heagy, and E. Ott, Experimental observation of a strange nonchaotic attractor, Phys. Rev. Lett. 65 (5), 533-536 (1990).

[26]
N. Packard, J. Crutchfield, J. Farmer, and R. Shaw, Geometry from a time series, Phys. Rev. Lett. 45, 712 (1980). Also see Chapter 6 of D. Ruelle, Chaotic evolution and strange attractors (Cambridge University Press: New York, 1989). For a geometric approach to the embedding problem see Th. Buzug, T. Reimers, and G. Pfister, Optimal reconstruction of strange attractors from purely geometrical arguments, Europhys. Lett. 13 (7), 605-610 (1990).

[27]
R. Hamming, An introduction to applied numerical analysis (McGraw-Hill: New York, 1971).

[28]
W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical recipes in C (Cambridge University Press: New York, 1988).

[29]
K. Wiesenfeld, Virtual Hopf phenomenon: A new precursor of period doubling bifurcations, Phys. Rev. A 32, 1744 (1985); K. Wiesenfeld, Noisy precursors of nonlinear instabilities, J. Stat.\ Phys. 38, 1701 (1985); P. Bryant and K. Wiesenfeld, Suppression of period doubling and nonlinear parametric effects in periodically perturbed systems, Phys. Rev. A 33, 2525-2543 (1986); B. McNamara and K. Wiesenfeld, Theory of stochastic resonance, Phys. Rev. A 39 (9), 4854 (1989); K. Wiesenfeld and N. B. Tufillaro, Suppression of period doubling in the dynamics of the bouncing ball, Physica 26D, 321-335 (1987).

[30]
P. Grassberger and I. Procaccia, Characterization of strange attractors, Phys. Rev. Lett. 50, 346-349 (1983); P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors, Physica 9D, 189 (1983).

[31]
The BINGO  program runs on the IBM-PC. For a copy contact A. Albano, Department of Physics, Bryn Mawr College, Bryn Mawr, PA 19010-2899.

[32]
P. Grassberger, An optimized box-assisted algorithm for fractal dimensions, Phys. Lett. A 148, 63-68 (1990). This brief paper provides a Fortran program for calculating the correlation integral. For another new approach to calculating the correlation integral see Xin-Jun Hou, Robert Gilmore, Gabriel B. Mindlin, and Hernán Solari, An efficient algorithm for fast box counting, Phys. Lett. 151 (1,2), 43-46 (1990).