Article

Reversible Equivariant Linear Systems

11/1999;
Source: CiteSeer

ABSTRACT In this paper we classify the structure of linear reversible systems (vector fields) on R that are equivariant with respect to a linear representation of a compact Lie group H . We assume the time-reversal symmetry R also acts linearly and is such that the group G that is generated by H and R is again a compact Lie group. The main tool for the classification is the representation theory of compact Lie groups. The results are applied to some generic eigenvalue movements of linear reversible equivariant systems.

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    Article: Reversible Hamiltonian Liapunov center theorem
    Claudio A Buzzi, Jeroen S W Lamb
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    ABSTRACT: We study the existence of periodic solutions in the neighbourhood of symmetric (partially) elliptic equilibria in purely reversible Hamiltonian vector fields. These are Hamiltonian vector fields with an involutory reversing symmetry R. We contrast the cases where R acts symplectically and anti-symplectically. In case R acts anti-symplectically, generically purely imaginary eigenval-ues are isolated, and the equilibrium is contained in a local two-dimensional invariant manifold containing symmetric periodic solutions encircling the equi-librium point. In case R acts symplectically, generically purely imaginary eigenvalues are doubly degenerate, and the equilibrium is contained in two two-dimensional invariant manifolds containing nonsymmetric periodic solutions encircling the equilibrium point. In addition, there exists a three-dimensional invariant sur-face containing a two-parameter family of symmetric periodic solutions.
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Keywords

compact Lie group
 
compact Lie group H
 
compact Lie groups
 
equivariant
 
generic eigenvalue movements
 
group G
 
linear representation
 
linear reversible equivariant systems
 
linear reversible systems
 
representation theory
 
time-reversal symmetry R
 
vector fields