Abstract
The scope of this chapter will be limited to systems of finite dimension; i.e., those which can be described by a limited number of ordinary differential equations of finite order. Any such set of equations can, of course, be brought into a canonical set of first-order equations by augmenting the number of variables, but for most of the analysis here it is more convenient to deal with the equations in the form in which they are generated.
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References and Bibliography
13.7.1 References
Goldstein, H., Classical Mechanics, Addison-Wesley, Reading, Mass., 1953.
Crandall, S. H., et al., Dynamics of Mechanical and Electromechanical Systems, McGraw-Hill, New York, 1968.
Poincaré, H., Les Methodes Nouvelles de la Mécanique Céleste, 3 vols., Gauthiers-Villars, Paris, 1892–99. Also in translation, New Methods of Celestial Mechanics, 3 vols., NASA TTF-450, 451, 452, CFSTI, Springfield, Virginia 22151.
Whittaker, E. I., and Watson, G. N., Modern Analysis, Chapter XIX, Cambridge University Press, Cambridge, 1915.
Krylov, N. and Bogoliubov, N., “Introduction to Nonlinear Mechanics,” Translated from Russian by S. Lefschetz, Annals of Mathematical Studies, No. 11, Princeton, 1947.
Cole, J. D., and Kevorkian, J., “Uniformly Valid Asymptotic Approximation for Certain Non-Linear Differential Equations,” Proceedings of the International Symposium on Non-Linear Differential Equations and Non-Linear Mechanics, Academic Press, New York, 1963. pp. 1120.
Morrison, J. A., “Comparison of the Modified Method of Averaging and the Two-Variable Expansion Procedure,” SIAM Review, 8(1), Jan. 1966.
Mahoney, J. J., “On the Validity of Averaging Methods for Systems with Periodic Solutions,” Report No. 1, 1970 Fluid Mechanics Research Institute, University of Essex, England.
Andronov, A. A.; Witt, A. A.; and Khaikin, S. E., Theory of Oscillators, (translated from Russian), Pergamon Press, New York, 1966.
Cartwright, M. L., “Forced Oscillations in Nearly Sinusoidal Systems,” Journal of the Institute of Electrical Engineers, (London), 96(3), 88–96, 1948.
Hayashi, C., Nonlinear Oscillations in Physical Systems, Chapters 12 and 13, McGraw-Hill, New York, 1964.
Kronauer, R. E., and Musa, S. A., “Necessary Conditions for Subharmonic and Superharmonic Synchronization in Weakly Nonlinear Systems,” Quarterly of Applied Mathematics, 24(2), July 1966.
Gambill, R., and Hale, J., “Subharmonic and Ultraharmonic Solutions for Weakly Nonlinear Systems,” Journal of Rational Mechanical Analysis, 5, 353–394, 1956.
Hénon, M., and Heiles, C., “The Applicability of the Third Integral of Motion: Some Numerical Experiments,” Astronomical Journal, 69(1), 73–79, 1964.
Barbanis, B., “On the Isolating Character of the Third Integral in a Resonance Case,” Astronomical Journal, 71(6), 415–424, 1966.
Lefschetz, S., Differential Equations: Geometric Theory, Pure and Applied Mathematics, vol. VI, Wiley-Interscience, New York, 1957.
13.7.2 Bibliography
Bogoliubov, N. N., and Mitropolsky, Y. A., Asymptotic Methods in the Theory of Nonlinear Oscillations, (translated from Russian), Gordon and Breach, New York, 1961.
Hale, J. K., Ordinary Differential Equations, Pure and Applied Mathematics, vol. XXI, Wiley-Interscience, New York, 1969.
Minorsky, N., Nonlinear Oscillations, Van Nostrand Reinhold, New York, 1962.
Stoker, J. J., Nonlinear Vibrations, Wiley-Interscience, New York, 1950.
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© 1990 Van Nostrand Reinhold
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Kronauer, R.E. (1990). Oscillations. In: Pearson, C.E. (eds) Handbook of Applied Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-1423-3_13
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