Oscillation of second-order half-linear difference equations

AbstractSome new oscillation criteria are obtained for the second-order half-linear difference equation (−1)m+1δm yi(n) + σj=1Nqijyi(n−Tji)=0, m ⩾1, i=1,…N where α 0 is a ratio of odd positive integers. The method uses techniques based on a Riccati type difference inequality. Examples are inserted to illustrate the results

Asymptotic solutions of forced nonlinear second order differential equations and their extensions

Using a modified version of Schauder's fixed point theorem, measures of non-compactness and classical techniques, we provide new general results on the asymptotic behavior and the non-oscillation of second order scalar nonlinear differential equations on a half-axis. In addition, we extend the methods and present new similar results for integral equations and Volterra-Stieltjes integral equations, a framework whose benefits include the unification of second order difference and differential equations. In so doing, we enlarge the class of nonlinearities and in some cases remove the distinction between superlinear, sublinear, and linear differential equations that is normally found in the literature. An update of papers, past and present, in the theory of Volterra-Stieltjes integral equations is also presented

Speed-up of neutrino transformations in a supernova environment

When the neutral current neutrino-neutrino interaction is treated completely, rather than as an interaction among angle-averaged distributions, or as a set of flavor-diagonal effective potentials, the result can be flavor mixing at a speed orders of magnitude faster than that one would anticipate from the measured neutrino oscillation parameters. It is possible that the energy spectra of the three active species of neutrinos emerging from a supernova are nearly identical.Comment: 8 pages, 4 figure

Nonlinear gas oscillations in pipes. Part 1. Theory

The problem of forced acoustic oscillations in a pipe is studied theoretically. The oscillations are produced by a moving piston in one end of the pipe, while a variety of boundary conditions ranging from a completely closed to a completely open mouth at the other end are considered. All these boundary conditions are modelled by two parameters: a length correction and a reflexion coefficient equivalent to the acoustic impedance. The linear theory predicts large amplitudes near resonance and nonlinear effects become crucially important. By expanding the equations of motion in a series in the Mach number, both the amplitude and wave form of the oscillation are predicted there. In both the open- and closed-end cases the need for shock waves in some range of parameters is found. The amplitude of the oscillation is different for the two cases, however, being proportional to the square root of the piston amplitude in the closed-end case and to the cube root for the open end

Self-induced decoherence in dense neutrino gases

Dense neutrino gases exhibit collective oscillations where "self-maintained coherence" is a characteristic feature, i.e., neutrinos of different energies oscillate with the same frequency. In a non-isotropic gas, however, the flux term of the neutrino-neutrino interaction has the opposite effect of causing kinematical decoherence of neutrinos propagating in different directions, an effect that is at the origin of the "multi-angle behavior" of neutrinos streaming off a supernova core. We cast the equations of motion in a form where the role of the flux term is manifest. We study in detail the symmetric case of equal neutrino and antineutrino densities where the evolution consists of collective pair conversions ("bipolar oscillations"). A gas of this sort is unstable in that an infinitesimal anisotropy is enough to trigger a run-away towards flavor equipartition. The "self-maintained coherence" of a perfectly isotropic gas gives way to "self-induced decoherence."Comment: Revtex, 16 pages, 12 figure

Stochastic Perturbations of Periodic Orbits with Sliding

Vector fields that are discontinuous on codimension-one surfaces are known as Filippov systems and can have attracting periodic orbits involving segments that are contained on a discontinuity surface of the vector field. In this paper we consider the addition of small noise to a general Filippov system and study the resulting stochastic dynamics near such a periodic orbit. Since a straight-forward asymptotic expansion in terms of the noise amplitude is not possible due to the presence of discontinuity surfaces, in order to quantitatively determine the basic statistical properties of the dynamics, we treat different parts of the periodic orbit separately. Dynamics distant from discontinuity surfaces is analyzed by the use of a series expansion of the transitional probability density function. Stochastically perturbed sliding motion is analyzed through stochastic averaging methods. The influence of noise on points at which the periodic orbit escapes a discontinuity surface is determined by zooming into the transition point. We combine the results to quantitatively determine the effect of noise on the oscillation time for a three-dimensional canonical model of relay control. For some parameter values of this model, small noise induces a significantly large reduction in the average oscillation time. By interpreting our results geometrically, we are able to identify four features of the relay control system that contribute to this phenomenon.Comment: 44 pages, 9 figures, submitted to: J Nonlin. Sc

The Frequency Dependence of Critical-velocity Behavior in Oscillatory Flow of Superfluid Helium-4 Through a 2-micrometer by 2-micrometer Aperture in a Thin Foil

The critical-velocity behavior of oscillatory superfluid Helium-4 flow through a 2-micrometer by 2-micrometer aperture in a 0.1-micrometer-thick foil has been studied from 0.36 K to 2.10 K at frequencies from less than 50 Hz up to above 1880 Hz. The pressure remained less than 0.5 bar. In early runs during which the frequency remained below 400 Hz, the critical velocity was a nearly-linearly decreasing function of increasing temperature throughout the region of temperature studied. In runs at the lowest frequencies, isolated 2 Pi phase slips could be observed at the onset of dissipation. In runs with frequencies higher than 400 Hz, downward curvature was observed in the decrease of critical velocity with increasing temperature. In addition, above 500 Hz an alteration in supercritical behavior was seen at the lower temperatures, involving the appearance of large energy-loss events. These irregular events typically lasted a few tens of half-cycles of oscillation and could involve hundreds of times more energy loss than would have occurred in a single complete 2 Pi phase slip at maximum flow. The temperatures at which this altered behavior was observed rose with frequency, from ~ 0.6 K and below, at 500 Hz, to ~ 1.0 K and below, at 1880 Hz.Comment: 35 pages, 13 figures, prequel to cond-mat/050203

One-dimensional nonlinear stability of pathological detonations

In this paper we perform high-resolution one-dimensional time-dependent numerical simulations of detonations for which the underlying steady planar waves are of the pathological type. Pathological detonations are possible when there are endothermic or dissipative effects in the system. We consider a system with two consecutive irreversible reactions A[rightward arrow]B[rightward arrow]C, with an Arrhenius form of the reaction rates and the second reaction endothermic. The self-sustaining steady planar detonation then travels at the minimum speed, which is faster than the Chapman–Jouguet speed, and has an internal frozen sonic point at which the thermicity vanishes. The flow downstream of this sonic point is supersonic if the detonation is unsupported or subsonic if the detonation is supported, the two cases having very different detonation wave structures. We compare and contrast the long-time nonlinear behaviour of the unsupported and supported pathological detonations. We show that the stability of the supported and unsupported steady waves can be quite different, even near the stability boundary. Indeed, the unsupported detonation can easily fail while the supported wave propagates as a pulsating detonation. We also consider overdriven detonations for the system. We show that, in agreement with a linear stability analysis, the stability of the steady wave is very sensitive to the degree of overdrive near the pathological detonation speed, and that increasing the overdrive can destabilize the wave, in contrast to systems where the self-sustaining wave is the Chapman–Jouguet detonation