Abstract
In this paper, a perturbation method based on multiple scales is used forcomputing the normal forms of nonlinear dynamical systems. The approach,without the application of center manifold theory, can be used to systematicallyfind the explicit normal form of a system described by a general n-dimensionaldifferential equation. The attention is focused on the dynamic behaviour ofa system near a critical point characterized by two pairs of purely imaginaryeigenvalues without resonance. The method can be easily formulated and implemented using a computer algebra system. Maple programs have been developedwhich can be `automatically' executed by a user without knowing computeralgebra. Examples chosen from mathematics, electrical circuits, mechanics andchemistry are presented to show the applicability of the technique and theconvenience of using computer algebra.
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References
Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
Elphick, C., Tirapegui, E., Brachet, M. E., Coullet, P., and Iooss, G., ‘A simple global characterization for normal forms of singular vector fields’, Physica D 29, 1987, 95–127.
Nayfeh, A. H., Method of Normal Forms, Wiley, New York, 1993.
Chow, S.-N., Li, C.-Z., and Wang, D., Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994.
Chow, S.-N., Drachman, B., and Wang, D., ‘Computation of normal forms’, Journal of Computational and Applied Mathematics 29, 1990, 129–143.
Yu, P., ‘Computation of normal forms via a perturbation technique’, Journal of Sound and Vibration 211, 1998, 19–38.
Bi, Q. and Yu, P., ‘Symbolic computation of normal forms for semi-simple cases’, Journal of Computational and Applied Mathematics 102, 1999, 195–220.
Bi, Q. and Yu, P., ‘Double Hopf bifurcations and chaos of a nonlinear vibration system’, Nonlinear Dynamics 19, 1999, 313–332.
Yu, P., Zhang, W., and Bi, Q., ‘Vibration analysis on a thin plate with the aid of computation of normal forms’, International Journal of Non-Linear Mechanics 36, 2001, 597–627.
Zhang, W. and Yu, P., ‘A study of the limit cycles associated with a generalized codimension-3 Lienard oscillator’, Journal of Sound and Vibration 231, 2000, 145–173.
Nayfeh, A. and Chin, C. M., Perturbation Methods with Maple, Dynamics Press, Blacksburg, VA, 1999.
Nayfeh, A. and Balachandran, B., Applied Nonlinear Dynamics, Wiley, New York, 1995.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
Huseyin, K. and Yu, P., ‘On bifurcations into non-resonant quasi-periodic motions’, Applied Mathematics Modelling 12, 1988, 189–201.
Yu, P. and Bi. Q., ‘Analysis of non-linear dynamics and bifurcations of a double pendulum’, Journal of Sound and Vibration 217, 1998, 691–736.
Kuznetsov, Y. A., Elements of Applied Bifurcation Theory, Applied Mathematical Science, Vol. 112, Springer-Verlag, New York, 1995.
Takens, F., ‘Singularities of vector fields’, Publications Mathématiques IHES 43, 1974, 47–100.
Lefever, R. and Nicolis, G., ‘Chemical instabilities and sustained oscillations’, Journal of Theoretical Biology 30, 1971, 267–284.
Tyson, J., ‘Some further studies of non-linear oscillations in chemical systems’, Journal of Chemical Physics 58, 1972, 3919–3930.
Huseyin, K., Multiple-Parameter Stability Theory and Its Applications, Oxford University Press, Oxford, 1986.
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Yu, P. Analysis on Double Hopf Bifurcation Using Computer Algebra with the Aid of Multiple Scales. Nonlinear Dynamics 27, 19–53 (2002). https://doi.org/10.1023/A:1017993026651
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DOI: https://doi.org/10.1023/A:1017993026651