Stability and Bifurcation Analysis of a Nonlinear DDE
Model for Drilling

Stone, Emily; Campbell, Sue Ann

doi:10.1007/s00332-003-0553-1

Cited by 49 publications

(38 citation statements)

References 27 publications

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“…As observed in [10], the system exhibits interesting behaviour for small values of . In particular, as is increased from the bifurcation value, there is a rapid transition from small amplitude to large amplitude limit cycles.…”

Section: Chatter In Machining Modelsmentioning

confidence: 57%

“…It was noted in [10] that the Hopf bifurcation in this model is degenerate, since the cubic coefficient of the normal form is zero. In fact, more can be said.…”

Section: The High Speed/small Delay Limit (S!0)mentioning

confidence: 90%

“…In [10] we compute the Hopf stability coefficient along this boundary via a centre manifold approximation [11][12][13]. In the case of the traditional vibration mode the Hopf bifurcation is supercritical for values of below 0.5.…”

Section: Chatter In Machining Modelsmentioning

confidence: 99%

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Delay induced canards in a model of high speed machining

2009

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To cite this Article Campbell, Sue Ann, Stone, Emily and Erneux, Thomas(2009) 'Delay induced canards in a model of high speed machining ',Dynamical Systems,24:3,[373][374][375][376][377][378][379][380][381][382][383][384][385][386][387][388][389][390][391][392] To link to this Article: DOI: 10.1080/14689360902852547 URL: http://dx.doi.org/10.1080/14689360902852547Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden.The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Vol. 24, No. 3, September 2009, 373-392 Delay induced canards in a model of high speed machining (2002), pp. 65-85] for regenerative chatter in a drilling process. The model is a nonlinear delay differential equation where the delay arises from the fact that the cutting tool passes over the metal surface repeatedly. For any fixed value of the delay, a large enough increase in the width of the chip being cut results in a Hopf bifurcation from the steady state, which is the origin of the chatter vibration. We show that for zero delay the Hopf bifurcation is degenerate and that for a small delay this leads to a canard explosion. That is, as the chip width is increased beyond the Hopf bifurcation value, there is a rapid transition from a small amplitude limit cycle to a large relaxation cycle. Our analysis relies on perturbation techniques and a small delay approximation of the DDE model due to Chicone [Inertial and slow manifolds for delay differential equations, J. Diff. Eqs 190 (2003), pp. 364-406]. We use numerical simulations and numerical continuation to support and verify our analysis. Dynamical Systems

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Section: Chatter In Machining Modelsmentioning

confidence: 57%

“…It was noted in [10] that the Hopf bifurcation in this model is degenerate, since the cubic coefficient of the normal form is zero. In fact, more can be said.…”

Section: The High Speed/small Delay Limit (S!0)mentioning

confidence: 90%

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Delay induced canards in a model of high speed machining

2009

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“…These coefficients satisfy a system of affine ODEs, in which the affine term depends on the orthogonal basis of the nullspace, denoted ψ(θ) (which is straightforward to calculate) arising from the projection of the original equation on N . These are now classical methods, introduced and used in [7,45,18]. In our case, the solution to the linear ordinary differential equation of h i jk yield relatively complex expressions, in terms of six constants C 1 · · · C 6 , that are then solved in order to match boundary values (and solve the original DDE on the center manifold).…”

Section: A Normal Form Reduction At Hopf Bifurcationmentioning

confidence: 99%

Liouville Correspondence Between the Modified KdV Hierarchy and Its Dual Integrable Hierarchy

Kang¹,

Liu²,

Olver

et al. 2015

J Nonlinear Sci

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Motivated by the dynamics of neuronal responses, we analyze the dynamics of the FitzHugh-Nagumo slow-fast system with delayed self-coupling. This system provides a canonical example of a canard explosion for sufficiently small delays. Beyond this regime, delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a delay-induced subcritical Bogdanov-Takens instability arising at the fold points of the S-shaped critical manifold. Underlying the transition from canardinduced to delay-induced dynamics is an abrupt switch in the nature of the Hopf bifurcation.

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“…Delay-differential equations are used as models in many areas of science, engineering, economics and beyond [2,9,12,14,15,16,17,18,21,22]. It is now well understood that retarded functional differential equations (RFDEs), a class which contains delay-differential equations, behave for the most part like ordinary differential equations on appropriate infinite-dimensional function spaces.…”

Section: Introductionmentioning

confidence: 99%

Zero-Hopf bifurcation in the Van der Pol oscillator with delayed position and velocity feedback

2014

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In this paper, we consider the traditional Van der Pol Oscillator with a forcing dependent on a delay in feedback. The delay is taken to be a nonlinear function of both position and velocity which gives rise to many different types of bifurcations. In particular, we study the Zero-Hopf bifurcation that takes place at certain parameter values using methods of centre manifold reduction of DDEs and normal form theory. We present numerical simulations that have been accurately predicted by the phase portraits in the Zero-Hopf bifurcation to confirm our numerical results and provide a physical understanding of the oscillator with the delay in feedback.

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Stability and Bifurcation Analysis of a Nonlinear DDE Model for Drilling

Cited by 49 publications

References 27 publications

Delay induced canards in a model of high speed machining

Delay induced canards in a model of high speed machining

Liouville Correspondence Between the Modified KdV Hierarchy and Its Dual Integrable Hierarchy

Zero-Hopf bifurcation in the Van der Pol oscillator with delayed position and velocity feedback

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