Bifurcation analysis of a neutral delay differential equation modelling the torsional motion of a driven drill-string.

*(English)*Zbl 1037.34075This paper is devoted to a neutral delay differential equation arising in drill-string modelling. A particular feature of this neutral equation is that its finite-difference part \(\dot x(t)-\dot x(t-\tau)\) is not stable.

The authors study the dynamics of the system by linearization in the unique equilibrium, and, subsequently, by systematic numerical long-time simulations varying two parameters, the target force \(A\) on the drill and the rotary speed \(\Omega\) of the drill.

The result is a map of regimes in the \((A,\Omega)\)-plane for \(A\leq 1\) and \(\Omega\in[0,1]\). The equilibrium is stable for \(A<0\) and unstable for \(A>0\). Most of the region of instability is covered by single-round periodic orbits. However, there are also periodic orbits of double period, invariant tori, chaotic attractors, and multistability between the various regimes in smaller parts of the plane. Furthermore, the authors provide practically useful information about the average oscillation amplitude and frequency, the thickness of the PoincarĂ© map, and the relaxation time toward the attractor for the various regimes.

The authors study the dynamics of the system by linearization in the unique equilibrium, and, subsequently, by systematic numerical long-time simulations varying two parameters, the target force \(A\) on the drill and the rotary speed \(\Omega\) of the drill.

The result is a map of regimes in the \((A,\Omega)\)-plane for \(A\leq 1\) and \(\Omega\in[0,1]\). The equilibrium is stable for \(A<0\) and unstable for \(A>0\). Most of the region of instability is covered by single-round periodic orbits. However, there are also periodic orbits of double period, invariant tori, chaotic attractors, and multistability between the various regimes in smaller parts of the plane. Furthermore, the authors provide practically useful information about the average oscillation amplitude and frequency, the thickness of the PoincarĂ© map, and the relaxation time toward the attractor for the various regimes.

Reviewer: Jan Sieber (Bristol)

##### MSC:

34K40 | Neutral functional-differential equations |

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

34K18 | Bifurcation theory of functional-differential equations |

34D45 | Attractors of solutions to ordinary differential equations |

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\textit{A. G. Balanov} et al., Chaos Solitons Fractals 15, No. 2, 381--394 (2003; Zbl 1037.34075)

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