  We provide a systematic study of equilibria of contact vector fields and the
bifurcations that occur generically in 1-parameter families, and express the
conclusions in terms of the Hamiltonian functions that generate the vector
fields.
  Equilibria occur at points where the zero-level set of the Hamiltonian
function is either singular or is tangent to the contact structure. The
eigenvalues at an equilibrium have an interesting structure: there is always
one particular real eigenvalue of any equilibrium, related to the contact
structure, that we call the principal coefficient, while the other eigenvalues
arise in quadruplets, similar to the symplectic case except they are translated
by a real number equal to half the principal coefficient.
  There are two types of codimension 1 equilibria, named Type I, arising where
the zero-set of the Hamiltonian is singular, and Type II where it is not, but
there is a degeneracy related again to the principal coefficient and the
contact of the zero level-set of the Hamiltonian with the contact structure.
Both give rise generically to saddle-node bifurcations.
  Some special features include: (i) for Type II singularities, Hopf
bifurcations cannot occur in dimension 3, but they may in dimension 5 or more;
(ii) for Type I singularities, a fold-Hopf bifurcation can occur with
codimension 1 in any dimension, and (iii) again for Type I, and in dimension at
least 5, a fold-multi-Hopf bifurcation (where several pairs of eigenvalues pass
through the imaginary axis simultaneously together with one through the origin)
may also occur with codimension 1.
