Abstract
Normal forms for versal unfoldings of linear Hamiltonian systems anti-commute with an anti-symplectic involution are given in this paper. They can be derived from suitable chosen versal unfoldings of linear Hamiltonians without an involution. The results are expressed in an alternative basis and in a symplectic basis compatible with this involution. Descriptions of unfoldings of codimension ≤ 2 are given for an illustration.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arnol’d, V., On matrices depending on parameters, Russian Math. Surveys 26 (1971), 29–43.
Burgoyne, N. and Cushman, R., Normal forms for real linear Hamiltonian systems, The 1976 NASA Conference on Geometric Control Theory, pp. 483–529, Math. Sci. Press, Brookline, MA, 1977.
Galin, D. M., Versal deformations of linear Hamiltonian systems, Amer. Math. Soc. Transl. (2) vol. 118, 1982.
Kocak, H., Normal forms and versal deformations of linear Hamiltonian systems, Journal of Differential Equations 51 (1984), 359–407.
Laub, A. J., & Meyer, K., Canonical forms for symplectic and Hamiltonian matrices, Cels. Mech. 9 (1974), 213–238.
Mackay, R., Stability of equilibria of Hamiltonian systems, in Nonlinear Phenomena and Chaos (ed. S. Sarkar), Adam Hilges, Bristol, (1986), 254–270.
Meyer, K., Hamiltonian systems with discrete symmetry, Journal of Differential Equations 41 (1981), 228–238.
Nickerson, H. K., spencer, D. C., and Steenrod, N.E., Advanced Calculus.
Sevryuk, M. B., Reversible systems; Lecture Notes in Mathematics 1211; Springer-Verlag (1986).
Wan, Y. H., Normal forms for infinitesimally symplectic transformations with involutions. Preprint, State University of New York at Buffalo, 1989.
Williamson, J., On the algebraic problem concerning the normal forms of linear dynamical systems, Am. J. Math. 58 (1936), 141–163.
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag
About this paper
Cite this paper
Wan, YH. (1991). Versal deformations of infinitesimally symplectic transformations with antisymplectic involutions. In: Roberts, M., Stewart, I. (eds) Singularity Theory and its Applications. Lecture Notes in Mathematics, vol 1463. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085438
Download citation
DOI: https://doi.org/10.1007/BFb0085438
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-53736-6
Online ISBN: 978-3-540-47047-2
eBook Packages: Springer Book Archive