Abstract
Models of impact oscillators using an instantaneous impact law are by their very nature discontinuous. These discontinuities geve rise to bifurcations which cannot be classified using the usual tools of bifurcation analysis. However, we present numerical evidence which suggests that these discontinuous bifurcations are just the limits (in some sense) of standard bifurcations of smooth dynamical systems as the impact is hardened. Finally we show how one dimensional maps of the interval with essentially similar characteristics can exhibit the same kinds of bifurcational behaviour, and how these bifurcations are related to standard bifurcations.
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References
Thompson, J. M. T., Bokaian, A. R., and Ghaffari, R., ‘Subharmonic and chaotic motions of compliant offshore structure and articulated mooring towers’, Journal of Energy Resources Technology 106, 1984, 191–198.
Shaw, S. W. and Holmes, P. J., ‘A periodically forced piecewise linear oscillator’, Journal of Sound and Vibration 90, 1983, 129–155.
Shaw, S. W., ‘Dynamics of harmonically excited systems having rigid amplitude constraints, Part I — Subharmonic motions and local bifurcations’, Journal of Applied Mechanics 52, 1985, 453–458.
Shaw, S. W., ‘Dynamics of harmonically excited systems having rigid amplitude constraints, Part II-Chaotic motions and global bifurcations’, Journal of Applied Mechanics 52, 1985, 459–464.
Nordmark, A. B., ‘Non-periodic motion caused by grazing incidence in an impact oscillator’, Journal of Sound and Vibration 145, 1991, 279–297.
Foale, S. and Bishop, S. R., ‘Dynamical complexities of forced impacting systems’, Philosophical Transactions of the Royal Society of London 338A, 1992, 547–556.
Whiston, G. S., ‘Singularities in vibro-impact dynamics’, Journal of Sound and Vibration 152, 1992, 427–460.
Nordmark, A. B., Grazing Conditions and Chaos in Impacting Systems, Doctoral Thesis, Royal Institute of Technology, Stockholm, Sweden, 1992.
Goldsmith, W., Impact, Edward Arnold, London, 1960.
Foale, S. and Thompson, J. M. T., ‘Geometrical concepts and computational techniques of nonlinear dynamics’, Computer Methods in Applied Mechanics and Engineering 89, 1991, 381–394.
Parker, T. S. and Chua, L. O., Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, New York, 1989.
Block, L. S. and Coppel, W. A., Dynamics in one dimension, Lecture Notes in Mathematics 1513, Springer-Verlag, Berlin, 1991.
Nusse, H. E. and Yorke, J. A. ‘Border-collision bifurcations including “period two to period three” for piecewise smooth systems’, Physica D 57, 1992, 39–57.
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Foale, S., Bishop, S.R. Bifurcations in impact oscillations. Nonlinear Dyn 6, 285–299 (1994). https://doi.org/10.1007/BF00053387
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DOI: https://doi.org/10.1007/BF00053387