Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations

We study small perturbations of three linear Delay Differential Equations (DDEs) close to Hopf bifurcation points. In analytical treatments of such equations, many authors recommend a center manifold reduction as a first step. We demonstrate that the method of multiple scales, on simply discarding the infinitely many exponentially decaying components of the complementary solutions obtained at each stage of the approximation, can bypass the explicit center manifold calculation. Analytical approximations obtained for the DDEs studied closely match numerical solutions.

Self-interrupted regenerative metal cutting in turning

A new approach is used to study the global dynamics of regenerative metal cutting in turning. The cut surface is modeled using a partial differential equation (PDE) coupled, via boundary conditions, to an ordinary differential equation (ODE) modeling the dynamics of the cutting tool. This approach automatically incorporates the multiple-regenerative effects accompanying self-interrupted cutting. Taylor's 3/4 power law model for the cutting force is adopted. Lower dimensional ODE approximations are obtained for the combined tool–workpiece model using Galerkin projections, and a bifurcation diagram computed. The unstable solution branch off the subcritical Hopf bifurcation meets the stable branch involving self-interrupted dynamics in a turning point bifurcation. The tool displacement at that turning point is estimated, which helps identify cutting parameter ranges where loss of stability leads to much larger self-interrupted motions than in some other ranges. Numerical bounds are also obtained on the parameter values which guarantee global stability of steady-state cutting, i.e., parameter values for which there exist neither unstable periodic motions nor self-interrupted motions about the stable equilibrium.

Self-interrupted regenerative turning

A preliminary study of self-interrupted regenerative turning is performed in this paper. To facilitate the analysis, a new approach is proposed to model the regenerative effect in metal cutting. This model automatically incorporates the multiple-regenerative effects accompanying self-interrupted cutting. Some lower dimensional ODE approximations are obtained for this model using Galerkin projections. Using these ODE approximations, a bifurcation diagram of the regenerative turning process is obtained. It is found that the unstable branch resulting from the subcritical Hopf bifurcation meets the stable branch resulting from the self-interrupted dynamics in a turning point bifurcation. Using a rough analytical estimate of the turning point tool displacement, we can identify regions in the cutting parameter space where loss of stability leads to much greater amplitude self-interrupted motions than in some other regions.

Infinite dimensional slow modulations in a well known delayed model for cutting tool vibrations

We apply the method of multiple scales (MMS) to a well-known model of regenerative cutting vibrations in the large delay regime. By ``large'' we mean the delay is much larger than the timescale of typical cutting tool oscillations. The MMS up to second order, recently developed for such systems, is applied here to study tool dynamics in the large delay regime. The second order analysis is found to be much more accurate than the first order analysis. Numerical integration of the MMS slow flow is much faster than for the original equation, yet shows excellent accuracy in that plotted solutions of moderate amplitudes are visually near-indistinguishable. The advantages of the present analysis are that infinite dimensional dynamics is retained in the slow flow, while the more usual center manifold reduction gives a planar phase space; lower-dimensional dynamical features, such as Hopf bifurcations and families of periodic solutions, are also captured by the MMS; the strong sensitivity of the slow modulation dynamics to small changes in parameter values, peculiar to such systems with large delays, is seen clearly; and though certain parameters are treated as small (or, reciprocally, large), the analysis is not restricted to infinitesimal distances from the Hopf bifurcation.

A simplified approach to a three-part Wiener-Hopf problem arising in diffraction theory

A direct and simple approach, utilizing Watson's lemma, is presented for obtaining an approximate solution of a three-part Wiener-Hopf problem associated with the problem of diffraction of a plane wave by a soft strip.

Degenerate rhombohedral and orthorhombic states in Ca-substituted Na0.5Bi0.5TiO3

Neutron powder diffraction and temperature dependent dielectric studies were carried out on Ca-substituted Na0.5Bi0.5TiO3, i.e., (Na0.5Bi0.5)(1-x)CaxTiO3. Stabilization of an orthorhombic phase even at a low Ca concentration (0.05 < x < 0.10) suggests that Na0.5Bi0.5TiO3 (NBT) is susceptible to orthorhombic distortion. The orthorhombic and rhombohedral phases coexist for x=0.10, suggesting these phases to be nearly degenerate. The orthorhombic distortion favoring tendency of Ca assists in promoting the inherent instability with regard to this structure in pure NBT, which was reported recently.

The dynamic stability of degenerate systems under parametric excitation

The parametric resonance in a system having two modes of the same frequency is studied. The simultaneous occurence of the instabilities of the first and second kind is examined, by using a generalized perturbation procedure. The region of instability in the first approximation is obtained by using the Sturm's theorem for the roots of a polynomial equation.

Degenerate surface modes in a magnetized plasma

Electromagnetic surface waves propagating along the plasma-vacuum interface parallel to an applied magnetic field are studied. New modes for which the field components are degenerate, not reported in the earlier investigation of Kotsarenko and Fedorchenko (1969), are found and discussed. These modes, which propagate up to the plasma frequency for all values of the magnetic field, start as forward waves at low frequency but smoothly change into the backward mode as the frequency increases.

Low-frequency degenerate surface modes of an electron-hole plasma

The computations of Flahive and Quinn1 of the dispersion curves of low frequency degenerate surface (DS) modes propagating along the magnetic field in an electron-hole plasma are extended to higher values of the wavenumber. We find that beyond a certain value of the wavenumber the DS mode re-enters the allowed region of surface wave propagation and tends to an asymptotic frequency ωR (<ωLH). These low frequency resonances of an electron-hole plasma are discussed with reference to the experimental observations.

Undamped Cooper pairs in a non-degenerate electron system

Following the reaction matrix technique and the Kanamori approximation. a condition is obtained for the occurence of undamped Cooper pairs in a non-degenerate electron system. Its relevance to induced superconductivity in systems with artificially populated (optically pumped) bands is pointed out.

Type I seesaw mechanism for quasi-degenerate neutrinos

We discuss symmetries and scenarios leading to quasi-degenerate neutrinos in type I seesaw models. The existence of degeneracy in the present approach is not linked to any specific structure for the Dirac neutrino Yukawa coupling matrix y(D) and holds in general. Basic input is the application of the minimal flavour violation principle to the leptonic sector. Generalizing this principle, we assume that the structure of the right-handed neutrino mass matrix is determined by y(D) and the charged lepton Yukawa coupling matrix y(l) in an effective theory invariant under specific groups G(F) contained in the full symmetry group of the kinetic energy terms. G(F) invariance also leads to specific structure for the departure from degeneracy. The neutrino mass matrix (with degenerate mass m(0)) resulting after seesaw mechanism has a simple form Mv approximate to m(0)(I - py(l)y(l)(T)) in one particular scenario based on supersymmetry. This form is shown tolead to correct description of neutrino masses and mixing angles. The thermal leptogenesis after inclusion of flavour effects can account for the observed baryon asymmetry of the universe within the present scenario. Rates for lepton flavour violating processes can occur at observable levels in the supersymmetric version of the scenario. (c) 2010 Elsevier B.V. All rights reserved.

A generalised cole-hopf transformation for nonlinear parabolic and hyperbolic equations

The Cole-Hopf transformation has been generalized to generate a large class of nonlinear parabolic and hyperbolic equations which are exactly linearizable. These include model equations of exchange processes and turbulence. The methods to solve the corresponding linear equations have also been indicated.La transformation de Cole et de Hopf a été généralisée en vue d'engendrer une classe d'équations nonlinéaires paraboliques et hyperboliques qui peuvent être rendues linéaires de façon exacte. Elles comprennent des équations modèles de procédés d'échange et de turbulence. Les méthodes pour résoudre les équations linéaires correspondantes ont également été indiquées.