This issuePrevious ArticleDynamics of a circular cylinder interacting with point vorticesNext ArticleNon-universal features of forced 2D turbulence in the energy and enstrophy ranges
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 eigenvalues
are isolated, and the equilibrium is contained in a local two-dimensional invariant
manifold containing symmetric periodic solutions encircling the equilibrium
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
surface containing a two-parameter family of symmetric periodic solutions.