In this paper we study a class of hybrid systems defined by Pfaffian maps.
It is a sub-class of o-minimal hybrid systems which capture rich continuous
dynamics and yet can be studied using finite bisimulations. The existence of
finite bisimulations for o-minimal dynamical and hybrid systems has been shown
by several authors.
The next natural question to investigate is how the sizes of such
bisimulations can be bounded. The first step in this direction was done by
authors where a double exponential upper bound was shown for Pfaffian dynamical
and hybrid systems. In the present paper we improve this bound to a single
exponential upper bound. Moreover we show that this bound is tight in general,
by exhibiting a parameterized class of systems on which the exponential bound
The bounds provide a basis for designing efficient algorithms for
computing bisimulations, solving reachability and motion planning problems.