Exponential stability of linear stochastic differential equations with bounded delay and the W-transform

We demonstrate how the method of auxiliary ('reference') equations, also known as N. V. Azbelev's W-transform method, can be used to derive efficient conditions for the exponential Lyapunov stability of linear delay equations driven by a vector-valued Wiener process. For the sake of convenience the W-method is briefly outlined in the paper, its justification is however omitted. The paper contains a general stability result, which is specified in the last section in the form of seven corollaries providing sufficient stability conditions for some important classes of It\^{o} equations with delay

Hybrid dynamical systems vs. ordinary differential equations: Examples of a "pathological" behavior

We investigate the controlled harmonic oscillator \begin{equation*}\label{eq3.1}\tag{1} \ddot{\xi}+\xi=u, \end{equation*} where an external force (the control function) $u$ depends on the coordinate $\xi$, only. It can be shown that no ordinary (even nonlinear) feedback controls of the form $u=f(\xi(t))$ can asymptotically stabilize the solutions of the system \eqref{eq3.1}. However, one is able to make the system \eqref{eq3.1} asymptotically stable if one designs a special feedback control $u$ depending on $\xi(\cdot)$ which is called a hybrid feedback control. We demonstrate in the paper that the dynamics of a typical linear system of ordinary differential equations equipped with a linear hybrid feedback control possesses some irregular properties that dynamical systems without delay do not have. For example, solutions with different initial conditions may cross or even partly coincide. This proves that the hybrid dynamics cannot, in general, be described by a system of ordinary differential equations, neither linear, nor nonlinear, so that time-delays have to be incorporated into the system

Spatially localized solutions of the Hammerstein equation with sigmoid type of nonlinearity

We study the existence of fixed points to a parameterized Hammerstein operator Hβ, β∈(0,∞], with sigmoid type of nonlinearity. The parameter β<∞ indicates the steepness of the slope of a nonlinear smooth sigmoid function and the limit case β=∞ corresponds to a discontinuous unit step function. We prove that spatially localized solutions to the fixed point problem for large β exist and can be approximated by the fixed points of H∞. These results are of a high importance in biological applications where one often approximates the smooth sigmoid by discontinuous unit step function. Moreover, in order to achieve even better approximation than a solution of the limit problem, we employ the iterative method that has several advantages compared to other existing methods. For example, this method can be used to construct non-isolated homoclinic orbit of a Hamiltonian system of equations. We illustrate the results and advantages of the numerical method for stationary versions of the FitzHugh–Nagumo reaction–diffusion equation and a neural field model

Collective and fractal properties of pion jets in the four-velocity space at intermediate energies

Experimental results are presented for study of collective and fractal properties of soft pion jets in the space of relative four-dimensional velocities. Significant decreasing is obtained for mean square of second particle distances from jet axis for pion-proton interactions at initial energies $\sim 3$ GeV in comparison with hadron-nuclear collisions at close energies. The decreasing results in power dependence of distance variable on collision energy for range $\sim 2 - 4$ GeV. The observation allows us to estimate the low boundary of manifestation of color degree of freedom in pion jet production. Cluster dimension values were deduced for pion jets in various reactions. Fractional values of this dimension indicate on the manifestation of fractal-like properties by pion jets. Changing of mean kinetic energy of jet particles and fractal dimension with initial energy increasing is consistent with suggestion for presence of color degrees of freedom in pion jet production at intermediate energies.Comment: The conference "Physics of fundamental interactions". ITEP, Moscow, Russia. November 23 - 27, 200

Application of Speckle Dynamics and Raman Light Scattering to Study the Fracture Features of Pipe Steel at High-cycle Fatigue

Despite a long history of research and a large number of publications, currently there are no methods for assessing and calculating the residual life of structural elements with their multi-cycle fatigue that would meet the requirements of engineering practice. In this regard, the role of physical methods to record the features of accumulation of local fatigue damage without stopping the operation or testing of various objects for fatigue increases. In the article two laser methods are used to study the origin of fatigue crack. Earlier, after testing for high-cycle fatigue of polished steel specimen with a Charpy notch, two zones of different sizes with different roughness were found near the notch. The first zone of 50x100 μm was located directly on the top of the notch. It consisted of inhomogeneities up to 10 μm in diameter and about 100 nm in height. In the center of the zone a macro-crack was discovered. The second zone with a diameter of 500-700 microns had a form of a hole (tie) with a depth of about 1 micron. Its center was located at a distance of 250-300 microns from the top of the notch. The aim of the work was to determine the formulation of inhomogeneities in a small zone and the sequence of the two zones’ occurrence. By using Raman microscopy, it is shown that the inhomogeneities are pieces of iron carbide. By the peculiarities of speckle image changes it is shown that the formation of two zones begins almost simultaneously. After the origination of a macro crack with a length of about 100 microns, a new plasticity zone at its top begins to form. Possible formation mechanisms of two zones are discussed. The disadvantages of the speckle method and the direction of further research are considered. © PNRPU.The authors thank I. S. Kamentsev, N.. Drukarenko, I. Tikhonova for help with sample preparation and fatigue testing. The work was carried out on the equipment of centers for collective use &quot; Plastometry&quot; of Institute of Engineering Science of Ural Branch of RAS and &quot; Nanomaterials and Nanotechnology&quot; of Ural Federal University with partial funding RFBR grant № 16-08-01077_a and act 211 of Government of the Russian Federation, agreement No. 02.A03.21.0006

Atomic and electronic structure of a copper/graphene interface as prepared and 1.5 years after

We report the results of X-ray spectroscopy and Raman measurements of as-prepared graphene on a high quality copper surface and the same materials after 1.5 years under different conditions (ambient and low humidity). The obtained results were compared with density functional theory calculations of the formation energies and electronic structures of various structural defects in graphene/Cu interfaces. For evaluation of the stability of the carbon cover, we propose a two-step model. The first step is oxidation of the graphene, and the second is perforation of graphene with the removal of carbon atoms as part of the carbon dioxide molecule. Results of the modeling and experimental measurements provide evidence that graphene grown on high-quality copper substrate becomes robust and stable in time (1.5 years). However, the stability of this interface depends on the quality of the graphene and the number of native defects in the graphene and substrate. The effect of the presence of a metallic substrate with defects on the stability and electronic structure of graphene is also discussed.Comment: 18 pages, 6 figures, accepted to Appl. Surf. Sc

Raman spectroscopy of the low dimensional antiferromagnet with large Neel temperature SrRu2O6

We report results of the Raman measurements for SrRu$_2$O$_6$ having extraordinary high N$\acute{\textrm{e}}$el temperature for a layered material. No additional phonon modes were detected at the temperature of magnetic transition thus excluding lowering of the symmetry in the magnetically ordered phase. An unusual increase in softening and damping of some phonons as the temperature approaches $T_N$ indicate the appearance of a continuum of interacting electronic excitations at $T\geq300K$. We also observe an intensive Raman response at 2050 cm$^{-1}$. Analysis of the polarization dependence and comparison with available theoretical data shows that this peak likely originates from the transitions between molecular orbitals previously proposed to explain the magnetic properties of SrRu$_2$O$_6$

Raman study of coupled electronic and phononic excitations in LuB12

Electronic Raman scattering and optical phonon self-energies are studied on single crystals of LuB12with different isotopic composition in the temperature region 10–650 K and at pressures up to 10 GPa. The shape and energy position of the spectral peaks depend on the magnitude of the probed wave vector, temperature, and symmetry of excitations. We simulated experimental spectra using electronic structure obtained in the density functional theory and taking into account the electron-phonon scattering. The emergence of a broad continuum in the spectra is identified with the inelastic scattering of light from the electronic intraband excitations. Their coupling to non-fully symmetric phonon modes is the source of both the Fano interference and temperature-dependent phonon self-energies. In addition, long wavelength vibrations of the boron atoms are in nonadiabatic regime, so the electronic contribution to their self-energies provides a temperature dependence that is similar to the anharmonic contribution. Comparison of calculation and experiment allowed us to determine the coupling constant λ = 0.32, which gives correct critical temperature of the transition to the superconducting state. © 2017 Elsevier B.V

Raman study of coupled electronic and phononic excitations in LuB12

Electronic Raman scattering and optical phonon self-energies are studied on single crystals of LuB12with different isotopic composition in the temperature region 10–650 K and at pressures up to 10 GPa. The shape and energy position of the spectral peaks depend on the magnitude of the probed wave vector, temperature, and symmetry of excitations. We simulated experimental spectra using electronic structure obtained in the density functional theory and taking into account the electron-phonon scattering. The emergence of a broad continuum in the spectra is identified with the inelastic scattering of light from the electronic intraband excitations. Their coupling to non-fully symmetric phonon modes is the source of both the Fano interference and temperature-dependent phonon self-energies. In addition, long wavelength vibrations of the boron atoms are in nonadiabatic regime, so the electronic contribution to their self-energies provides a temperature dependence that is similar to the anharmonic contribution. Comparison of calculation and experiment allowed us to determine the coupling constant λ = 0.32, which gives correct critical temperature of the transition to the superconducting state. © 2017 Elsevier B.V