We study a neutral delay
differential equation that arises naturally in the Cosserat description
of torsional waves on a driven drillstring. The equation under study has
the following form: 
(1) 
where F is a nonlinear function.
Parameters are fixed as =2,
J=1, as well as parameters of F. 
We analyze bifurcations in Eq.
(1) as two control parameters
and A are varied. As a result,
a map of regimes is obtained, splitting the
plane of the above parameters
into regions where the system has qualitatively
different behavior. We also
characterize the observed regimes by quantities
useful from the viewpoint of
engineering practice, namely, amplitude and
average frequency of oscillations,
thickness of Poincare map and time of
relaxation to the given attractor
from physicaly reasonable initial
conditions. 
Attractors found in system (1) for
practically meaningful values of control parameters are: a periodone and
a periodtwo limit cycles, a torus and a doubled torus, and chaotic attractors.
A domain in the parameter plane is found where multistability occurs, that
is, coexistence in the phase space of several attractors for the same set
of control parameters. On the basis of this analysis, practical recommendations
are elaborated regarding the optimal choice of control parameters in a
real drilling device described by Eq. (1).
