×

zbMATH — the first resource for mathematics

Bifurcation analysis of a neutral delay differential equation modelling the torsional motion of a driven drill-string. (English) Zbl 1037.34075
This 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 Poincaré map, and the relaxation time toward the attractor for the various regimes.

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Tucker, R.W.; Wang, C., On the effective control of torsional vibrations in drilling systems, J. sound vib., 224, 101-122, (1999)
[2] Tucker, R.W.; Wang, C., An integrated model for drill-string dynamics, J. sound vib., 224, 123-165, (1999)
[3] Antman, S., Non-linear problems in elasticity, (1991), Springer-Verlag Berlin
[4] Brayton, R.K., Bifurcation of periodic solutions in a non-linear difference – differential equation of neutral type, Appl. math., XXIV, 3, 215-224, (1966) · Zbl 0143.30701
[5] Bauer, A., Utilisation of chaotic signals for radar and sonar purposes, Norsig, 96, 33-36, (1996)
[6] Ottesen, J.T., Modelling of the baroflex-feedback mechanism with time-delay, J. math. biol., 36, 41-63, (1997) · Zbl 0887.92015
[7] Seidel, H.; Herzel, H., Bifurcations in a non-linear model of the baroreceptor-cardiac reflex, Physica D, 115, 145-160, (1998) · Zbl 0932.92024
[8] Jamaleddine, R.; Vinet, A., Role of gap junction resistance in rate-induced delay in conduction in a cable model of the atrioventricular node, J. biol. sys., 7, 475-499, (1999)
[9] Iserles, A., On the generalised pantograph functional – differential-equation, Euro. J. appl. math., 4, 1-38, (1993) · Zbl 0767.34054
[10] Hale, J., Functional differential equations, () · Zbl 0222.34003
[11] Driver, R.D., Ordinary and delay differential equations, () · Zbl 0374.34001
[12] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, () · Zbl 0752.34039
[13] Farmer, J., Chaotic attractors of a finite-dimensional dynamical system, Physica D, 4, 366-393, (1982) · Zbl 1194.37052
[14] Ikeda, K.; Matsumoto, K., High-dimensional chaotic behavior in systems with time-delayed feedback, Physica D, 29, 223-235, (1987) · Zbl 0626.58014
[15] Simmendbeiger, C.; Wunderlin, A.; Pelster, A., Analytical approach for the Floquet theory of delay differential equations, Phys. rev. E, 59, 5, 5344-5353, (1999)
[16] Le Berre, M.; Ressayre, E.; Tallet, A.; Gibbs, H., High-dimension chaotic attractors of a non-linear ring cavity, Phys. rev. lett., 56, 4, 274-277, (1986)
[17] Ueda, Y.; Ohta, H., Bifurcations in a system described by a non-linear differential equation with delay, Chaos, 4, 1, 75-83, (1994) · Zbl 1055.34509
[18] Luzyanina, T.; Engelborghs, K.; Lust, K.; Roose, D., Computation, continuation and bifurcation analysis of periodic solutions of delay differential equations, Int. J. bif. chaos, 7, 11, 2547-2560, (1997) · Zbl 0910.34057
[19] Luzyanina, T.; Engelborghs, K.; Roose, D., Numerical bifurcation analysis of differential equations with state-dependent delay, Int. J. bif. chaos, 11, 3, 737-753, (2001) · Zbl 1090.65551
[20] Bunner, M.; Popp, M.; Meyer, Th.; Kittel, A.; Parisi, J., Tool to recover scalar time-delay systems from experimental time series, Phys. rev. E, 54, 4, R3082-R3085, (1996)
[21] Bunner, M.; Meyer, Th.; Kittel, A.; Parisi, J., Recovery of the time-evolution equation of time-delay systems from time series, Phys. rev. E, 56, 3, 5083-5089, (1997)
[22] Voss, H.; Kurths, J., Reconstruction of non-linear time delay models from data by the use of optimal transformations, Phys. lett. A, 234, 336-344, (1997) · Zbl 1044.34510
[23] Engelborghs, K.; Roose, D.; Luzyanina, T., Bifurcation analysis of periodic solutions of neutral functional differential equations: a case study, Int. J. bif. chaos, 8, 10, 1889-1905, (1998) · Zbl 0941.34070
[24] Engelborghs K, Luzyanina T, Samaey G. DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations, Technical Report TW-330, Department of Computer Science, KU Leuven, Leuven, Belgium, 2001
[25] Engelborghs, K.; Luzyanina, T.; Roose, D., Numerical bifurcation analysis of delay differential equations, J. comput. appl. math, 25, 1-2, 265-275, (2000) · Zbl 0965.65099
[26] Campbell, S., Resonant codimension two bifurcation in a neutral functional differential equation, Nonlinear anal. theor. meth. appl., 30, 7, 4577-4584, (1997) · Zbl 0892.34071
[27] Ladyzhenskaya, O.A., On the dynamical system generated by navier – stocks equations, J. sov. math., 3, 458-479, (1972) · Zbl 0336.35081
[28] Ladyzhenskaya, O.A., Dynamical system generated by navier – stocks equations, Sov. phys. (doklady), 17, 647-649, (1973) · Zbl 0301.35077
[29] Mallet-Paret, J., Negatively invariant sets of compact maps and an extension of a theorem of carturiaht, J. differ. equations, 22, 331-348, (1976) · Zbl 0354.34072
[30] Ruelle, D., Characteristic exponents and invariant manifolds in Hilbert space, Ann. math., 115, 243-290, (1982) · Zbl 0493.58015
[31] Moon, F., Chaotic vibrations: an introduction for applied scientists and engineers, (1987), Wiley · Zbl 0745.58003
[32] Sauer, T., Reconstruction of dynamical systems from interspike intervals, Phys. rev. lett., 72, 3811-3814, (1994)
[33] Hegger, R.; Kantz, H., Embedding of sequences of time intervals, Europhys. lett., 38, 4, 267-272, (1997)
[34] Janson, N.B.; Pavlov, A.N.; Neiman, A.B.; Anishchenko, V.S., Reconstruction of dynamical and geometrical properties of chaotic attractors from threshold-crossing interspike intervals, Phys. rev. E, 58, R4-R7, (1998)
[35] Corwin, S.P.; Sarafyan, D.; Thompson, S., DKLAG6: a code based on continuously imbedded sixth-order runge – kutta methods for the solution of state-dependent functional differential equations, Appl. numer. math., 24, 2-3, 319-330, (1997) · Zbl 0899.65046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.