  A steady state (or equilibrium point) of a dynamical system is hyperbolic if
the Jacobian at the steady state has no eigenvalues with zero real parts. In
this case, the linearized system does qualitatively capture the dynamics in a
small neighborhood of the hyperbolic steady state. However, one is often forced
to consider non-hyperbolic steady states, for example in the context of
bifurcation theory. A geometric technique to desingularize non-hyperbolic
points is the blow-up method. The classical case of the method is motivated by
desingularization techniques arising in algebraic geometry. The idea is to blow
up the steady state to a sphere or a cylinder. In the blown-up space, one is
then often able to gain additional hyperbolicity at steady states. In this
paper, we discuss an explicit example where we replace the sphere in the
blow-up by hyperbolic space. It is shown that the calculations work in the
hyperbolic space case as for the spherical case. This approach may be even
slightly more convenient if one wants to work with directional charts. Hence,
it is demonstrated that the sphere should be viewed as an auxiliary object in
the blow-up construction. Other smooth manifolds are also natural candidates to
be inserted at steady states.
