Delay-periodic solutions and their stability using averaging in delay-differential equations, with applications

Carr, Thomas W.; Haberman, Richard; Erneux, Thomas

doi:10.1016/j.physd.2012.06.001

Cited by 4 publications

(4 citation statements)

References 21 publications

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“…Further details on the stability analysis of state-dependent delayed differential equations and their applications can be found, among others, in [16,18,[25][26][27][28][29] and the references therein.…”

Section: State-dependent Delaymentioning

confidence: 99%

Instability regimes and self-excited vibrations in deep drilling systems

Depouhon

Detournay

2014

Journal of Sound and Vibration

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This paper analyzes the stability of the discrete model proposed by Richard et al. [1,2] to study the self-excited axial and torsional vibrations of deep drilling systems. This model, which relies on a rate-independent bit/rock interaction law, reduces to a coupled system of state-dependent delay differential equations governing the axial and angular perturbations to the stationary motion of the bit. A linear stability analysis indicates that, although the steady-state motion of the bit is always unstable, the nature of the instability depends on the nominal angular velocity Ω 0 of the drillstring imposed at the rig. On the one hand, if Ω 0 is larger than a critical velocity Ω c , the angular dynamics is responsible for the instability. However, on the timescale of the resonance period of the drillstring viewed as a torsional pendulum, the system behaves like a marginally stable one, provided that exogenous perturbations are of limited magnitude. The instability then only appears on a much larger timescale, in the form of slowly growing oscillations that ultimately lead to an undesired drilling regime such as bit-bouncing or stick-slip vibrations. On the other hand, if Ω 0 is smaller than Ω c , the instability manifests itself on the timescale of the bit motion due to a dominating unstable axial dynamics; perturbations to the steady-state motion then rapidly degenerate into stick-slip limit cycles or bit-bouncing. For typical deep drilling field conditions, the critical angular velocity Ω c is virtually independent of the axial force acting on the bit and of the bit bluntness. It can be approximated by a power law monomial, a function of known parameters of the drilling system and of the intrinsic specific energy (a quantity characterizing the energy required to drill a particular rock). This approximation holds on account that the dissipation in the drilling structure is negligible with respect to that taking place through the bit/rock interaction, as is typically the case. These findings are further illustrated on an example of deep drilling and shown to match the trends observed in the field.

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Section: State-dependent Delaymentioning

confidence: 99%

Instability regimes and self-excited vibrations in deep drilling systems

Depouhon

Detournay

2014

Journal of Sound and Vibration

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“…Using asymptotic formulae (2.6) and the ansatz λ = iω for the eigenvalues of the linearisation, we obtain the following asymptotic formulae for the frequency and the bifurcation value of the parameter at each Hopf bifurcation point 5 :…”

Section: Bifurcations At the Positive Equilibriummentioning

confidence: 99%

“…All the asymptotic formulae are compared with numerical simulations. We also note an alternative perturbation technique of the fixed-point analysis based on averaging, which was proposed in [5].…”

Section: Introductionmentioning

confidence: 99%

Periodic pulsating dynamics of slow–fast delayed systems with a period close to the delay

2017

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We consider slow–fast delayed systems and discuss pulsating periodic solutions, which are characterised by specific properties that (a) the period of the periodic solution is close to the delay, and (b) these solutions are formed close to a bifurcation threshold. Such solutions were previously found in models of mode-locked lasers. Through a case study of population models, this work demonstrates the existence of similar solutions for a rather wide class of delayed systems. The periodic dynamics originates from the Hopf bifurcation on the positive equilibrium. We show that the continuous transformation of the periodic orbit to the pulsating regime is simultaneous with multiple secondary almost resonant Hopf bifurcations, which the equilibrium undergoes over a short interval of parameter values. We derive asymptotic approximations for the pulsating periodic solution and consider scaling of the solution and its period with the small parameter that measures the ratio of the time scales. The role of competition for the realisation of the bifurcation scenario is highlighted.

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“…In order to make the models more consistent, delay differential equations (DDEs) are used to describe these phenomena. Many real life applications in the areas as diverse as engineering, medicines, economics and other physical sciences can be modeled using DDEs, see for example [1][2][3]. Since the role of DDEs in modeling real life phenomena is increasing, the need to develop numerical solutions for DDEs is becoming more important than ever.…”

Section: Introductionmentioning

confidence: 99%

Block method with Hermite interpolation for numerical solution of delay differential equations

Ishak¹,

Majid

Suleiman

2013

AIP Conference Proceedings

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Phase-amplitude method for numerically exact solution of the differential equations of the two-center Coulomb problem Abstract. In this paper, we consider a two-point implicit block method for solving delay differential equations. For greater efficiency, the block method is implemented in variable stepsize technique. The most optimal stepsize is taken while achieving the desired accuracy. The implicit method is solved using predictor-corrector scheme where the corrector is iterated until convergence. Grid point formulae are derived using a predictor and a corrector of order five. The formulae produce two new values in a single integration step. Delay solutions are approximated using Hermite interpolation of order five. The advantage of using Hermite interpolation is that it requires less support points than the existing interpolation technique in order to achieve the overall accuracy. Numerical results indicate that the two-point block method with Hermite interpolation technique is efficient and reliable in solving a wide range of delay differential equations.

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Delay-periodic solutions and their stability using averaging in delay-differential equations, with applications

Cited by 4 publications

References 21 publications

Instability regimes and self-excited vibrations in deep drilling systems

Instability regimes and self-excited vibrations in deep drilling systems

Periodic pulsating dynamics of slow–fast delayed systems with a period close to the delay

Block method with Hermite interpolation for numerical solution of delay differential equations

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