Bifurcations in a nonlinear neutral delay differential equation

A. Balanov, N. Janson*, P.V.E. McClintock, R.W. Tucker, C. Wang


We study a neutral delay differential equation that arises naturally in the Cosserat description of torsional waves on a driven drill-string. The equation under study has the following form:
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 period-one and a period-two 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).