Abstract
The existence of normal modes in the resonant 1:N cases for time-reversible, equivariant vectorfields is shown. The spatial symmetries studied here includes the usual compact Lie Groups. An application of the main theorem to a family of time-reversibleD n x0(2) vectorfields which includes the Stokeslet Model from sedimentation theory is given.
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References
Arnold, V. I. (1988).Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd Ed., Springer-Verlag, New York.
Devaney, R. L. (1976). Reversible diffeomorphisms and flows.Trans. Amer. Math. Soc. 218, 89–113.
Golubitsky, M., Krupa, M., and Lim, C. (1990). Time-reversibility and particle sedimentation.SIAM J. Appl. Math. 51(1), 49–72.
Golubitsky, M., and Schaeffer, D. G. (1985). Singularities and Groups in Bifurcation Theory: Vol. I.Appl. Math. Sci., Vol. 51, Springer-Verlag, New York.
Golubitsky, M., Stewart, I. N., and Schaeffer, D. G. (1988). Singularities and Groups in Bifurcation Theory: Vol. II.Appl. Math. Sci., Vol. 69, Springer-Verlag, New York.
McComb, I., and Lim, C. (to appear). Stability of equilibria for a class of time-reversible,D n ×O(2)-symmetric homogeneous vector fields.SIAM J. Math. Anal.
Moser, J. (1976). Periodic orbits near an equilibrium and a theorem by Alan Weinstein.Comm. Pure Appl. Math. XXIX, 727–747.
Roels, J. (1971a). An extension to resonant cases of Liapunov's theorem concerning the periodic solutions near a Hamiltonian equilibrium.J. Diff. Eqns. 9, 300–324.
Roels, J. (1971b). Families of periodic solutions near a Hamiltonian equilibrium when the ratio of two eigenvalues is 3.J. Diff. Eqns. 10, 431–447.
Sattinger, D. H. (1983).Branching in the Presence of Symmetry. SIAM, Philadelphia.
Schmidt, D. S. (1974). Periodic solutions near a resonant equilibrium of a Hamiltonian system.Cel. Mech. 9, 81–103.
Schmidt, D. S., and Sweet, D. (1978). A unifying theory in determining periodic families for Hamiltonian systems at resonance.J. Diff. Eqns. 14, 597–609.
Sevryuk, M. (1986).Reversible Systems, Lecture Notes in Mathematics, Vol. 1211, Springer-Verlag, Berlin.
Siegel, C., and Moser, J. (1971).Lectures on Celestial Mechanics. Springer-Verlag, New York.
Vanderbauwhede, A. (1982).Local Bifurcaion and Symmetry. Pitman, Boston.
van der Meer, J. C. (1985).The Hamiltonian Hopf Bifurcaion, Lecture Notes in Mathematics, Vol. 1160, Springer-Verlag, Berlin.
Weinstein, (1973).
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McComb, IH., Lim, C. Resonant normal modes for time-reversible, equivariant vectorfields. J Dyn Diff Equat 7, 287–339 (1995). https://doi.org/10.1007/BF02219359
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DOI: https://doi.org/10.1007/BF02219359