Abstract
This paper explores the application of the method of variable-coefficient harmonic balance to nonautonomous nonlinear equations of the form XsF(X, t:λ), and in particular, a one-degree-of-freedom nonlinear oscillator equation describing escape from a cubic potential well. Each component of the solution, X(t), is expressed as a truncated Fourier series of superharmonics, subharmonics and ultrasubharmonics. Use is then made of symbolic manipulation in order to arrange the oscillator equation as a Fourier series and its coefficient are evaluated in the traditional way. The time-dependent coefficients permit the construction of a set of amplitude evolution equations with corresponding stability criteria. The technique enables detection of local bifurcations, such as saddle-node folds, period doubling flips, and parts of the Feigenbaum cascade. This representation of the periodic solution leads to local bifurcations being associated with a term in the Fourier series and, in particular, the onset of a period doubled solution can be detected by a series of superharmonics only. Its validity is such that control space bifurcation diagrams can be obtained with reasonable accuracy and large reductions in computational expense.
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References
Summers, J. L. and Savage, M. D., ‘Two timescale harmonic balance. I. Application to autonomous onedimensional nonlinear oscillators’, Phil. Trans. R. Soc. Lond. A 340, 1992, 473–501.
Nayfeh, A. H., Perturbation Methods, Wiley-Interscience, New York, 1973.
Nayfeh, A. H., Introduction to Perturbation Techniques, John Wiley, New York, 1981.
Nayfeh, A. H. and Mook, D. T., Non-Linear Oscillations, John Wiley, New York, 1979.
Kuzmak, G. E., ‘Asymptotic solutions of nonlinear second order differential equations with variable coefficients’, P.M.M., 23, 1959, 515–526.
Bogoliubov, N. N. and Krylov, N. M., Introduction to Nonlinear Mechanics, Pub. of the Acad. Sci. Ukr. SSR, 1937 (English translation Princeton University Press, Princeton, 1947).
Bejarano, J. D. and Yuste, S. B., ‘Construction of approximate analytical solutions to a new class of nonlinear oscillator equations’, J. Sound Vibration 110, 1986, 347–350.
Garcia-Margallo, J. and Bejarano, J. D., ‘A generalization of the method of harmonic balance’, J. Sound Vibration 116, 1987, 591–595.
Savage, M. D., Slowly varying amplitude and phase, Internal Report, Department of Applied Mathematical Studies, University of Leeds, Leeds, England, 1986.
Summers, J. L., Brindley, J., and Savage, M. D., ‘Two timescale harmonic balance. II. Application to nonautonomous one-dimensional nonlinear oscillator equations’, Phil. Trans. R. Soc. Lond. A, 1992, submitted for publication.
Kaas-Petersen, C., PATH: User's guide, Internal Report, Leeds University, Leeds, England, Centre for Nonlinear Studies, 1987.
Codling, D., Gaskell, P. H., Savage, M. D., and Summers, J. L., ‘Application of 2THB to quasi-periodically forced nonlinear oscillations’, 1994, to be submitted.
Thompson, J. M. T., Bishop, S. R., and Leung, L. M., ‘Fractal basins and chaotic bifurcations prior to escape from a potential well’, Phys. Lett. A 121, 1987, 116–120.
Virgin, L. N., ‘On the harmonic response of an oscillator with unsymmetric restoring force’, J. Sound Vibration 126, 1988, 157–165.
Thompson, J. M. T., ‘Chaotic phenomena triggering escape from a potential well’, Proc. R. Soc. Lond. A 421, 1989, 195–225.
Thompson, J. M. T. and Soliman, M. S., ‘Fractal control boundaries of driven oscillators and their relevance to safe engineering design’, Proc. R. Soc. Lond. A 428, 1990, 1–13.
Holmes, C. and Holmes, P., ‘Second order averaging and bifurcations to subharmonics in Duffing's equation’, J. Sound Vibration 78, 1981, 161–174.
Leung, A. Y. T. and Fung, T. C., ‘Construction of chaotic regions’, J. Sound Vibration 131, 1989, 445–455.
Nayfeh, A. H. and Sanchez, N. E., ‘Bifurcations in a forced softening Duffing oscillator’, Int. J. Non-Linear Mechanics 24, 1989, 483–497.
Szemplińska-Stupnicka, W., ‘Bifurcations of harmonic solution leading to chaotic motion in the softening type Duffing's oscillator’, Int. J. Non-Linear Mechanics 23, 1988, 257–277.
van Dooren, R. and Janssen, H., ‘Period doubling solutions in the Duffing oscillator: A Galerkin's approach’, J. Comp. Physics 82, 1989, 161–171.
Greenspan, B. D. and Holmes, P. J., ‘Repeated resonance and homoclinic bifurcation in a periodically forced family of oscillators’, SIAM J. Math. Anal. 15, 1984, 69–97.
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Summers, J.L. Variable-coefficient harmonic balance for periodically forced nonlinear oscillators. Nonlinear Dyn 7, 11–35 (1995). https://doi.org/10.1007/BF00045123
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DOI: https://doi.org/10.1007/BF00045123