Versal Deformations and Normal Forms for Reversible and Hamiltonian Linear Systems

Abstract

The problem of this article is the characterization of equivalence classes and their versal deformations for reversible and reversible Hamiltonian matrices. In both cases the admissible transformations form a subgroup G of Gl(m). Therefore the Gl(m)-orbits of a given matrix may split into several G-orbits. These orbits are characterized by signs. For each sign we have a normal form and a corresponding versal deformation. The main tool in the characterization is reduction to the semi simple case.1996 Academic Press, Inc.

Bifurcation of a reversible Hamiltonian system from a fixed point with fourfold eigenvalue zero

“…We emphasise again that although the Hamiltonians H ∓ have been unfolded completely it is an open question whether the same holds true for the corresponding vector fields. Nevertheless, we note that the number of parameters in our unfoldings agrees with results by Hoveijn [12] about the linear codimension of the problem.…”

Reversible Equivariant Linear Systems

“…The normal form and unfolding theories for the case when \ is not _-selfdual and is of type R reduce to the standard theories for purely reversible matrices in gl(m+n; R) and have been treated by Hoveijn [9] in the case that m=n. The theories for the analogous cases with k=C or H will be discussed in [10].…”

“…For comparison, we also include the previously studied``purely reversible'' case (NSD-R) with m=n (Hoveijn [9]). The labels p 1 , p 2 , p 3 , q 1 , q 2 , q 3 , q 4 refer to the bifurcations depicted in Fig.…”

Quasi-periodic stability of normally resonant tori

“…The same is true for our first example (non-zero normal frequencies). Further information on these cases can be found in [23].…”

“…Let Ω 0 ∈ gl − (2p; R) be given; the aim of this appendix is to summarize some results from [17,23,27,37] which allow to describe a miniversal unfolding of Ω 0 , and to work out the details for two particular cases. See also [26].…”