Systems In Normal Form Research Articles

Normal forms and versal deformations of linear Hamiltonian systems

Equivalence classes of time independent, linear, real Hamiltonian systems can be identified, up to canonical transformations, with the orbits of the adjoint action of the real symplectic group on its Lie algebra. A new set of representatives, also called normal forms, for these orbits is given. Versal deformations of systems in normal form are constructed. Applications of versal deformations to the study of bifurcations of linear systems with small codimension are indicated.

Zur Theorie der Polynomringe

Let\(\mathfrak{B}\) be a variety of rings,R a ring of\(\mathfrak{B}\) andx an indeterminate. The free compositionR(x,\(\mathfrak{B}\)) ofR and the free algebra of\(\mathfrak{B}\) generated byx, is called the\(\mathfrak{B}\)-polynomial ring inx the variety of rings, rings with identity, commutative rings or commutative rings with identity resp. We prove some results about relations between the polynomial ringsR(x,\(\mathfrak{B}\)), whereR is fixed and\(\mathfrak{B}\) runs over these varieties. Moreover we construct normal form systems for certain polynomial ringsR(x,\(\mathfrak{B}\)).