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The first charge-coupled device photometric light curves in B, V, R${_c}$, and I${_c}$ bands of the short-period contact binary star OQ Cam are presented. Through analyzing the light curves with the Wilson-Devinney code, it is found that OQ Cam is a W-type shallow contact binary star with a contact degree of f =20.2%. Based on the O-C analyzing, the orbit period displays a long term increasing with a rate of $dP/dt=4.40\times10{^{-7}} day \cdot year{^{-1}}$. The increasing in orbit period can be interpreted by mass transfer from the less massive component to the more massive one. As the orbit period increasing, OQ Cam would evolve from the present shallow contact configuration to a none contact stage. So it may be a potential candidate to confirm the thermal relaxation oscillation theory., 11 pages, 6 figures, and 5 tables, accepted by RAA

In this article, we study a class of non-linear neutral delay differential equations of third order. We first prove criteria for non-existence of non-Kneser solutions, and criteria for non-existence of Kneser solutions. We then use these results to provide criteria for the under study differential equations to ensure that all its solutions are oscillatory. An example is given that illustrates our theory.

In this chapter we present the oscillation theory of symplectic difference system ( SDS). In particular, we show that it is possible to develop this theory in a similar way to the oscillation theory of linear Hamiltonian systems or their special cases, the Sturm-Liouville differential and difference equations.

Among biomedical signals, repetitive or quasi-periodic signals are particularly widespread. While the periodic component is still presented these signals are characterized by period variations (fundamental frequency, amplitude, etc.). The lack of synchronization or phase shifts results in variations in similar segments’ durations, nominally identical signals demonstrate a variation at peak retention times, etc. The inverse methods of oscillation theory were proposed recently as a tool to solve the problems of modelling of repetitive signals with phase shift. In the article, the inverse method of oscillation theory is considered as a tool to solve the problems of supervised and non-supervised classification, and filtering of repetitive signals with phase shift. Examples of application are presented.

We present the results of our study of the eclipsing binary systems CSS J112237.1+395219, LINEAR 1286561 and LINEAR 2602707 based on new CCD $B$, $V$, $R_c$ and $I_c$ complete light curves. The ultra-short period nature of the stars \citep{Drake2014} is confirmed and the system's periods are revised. The light curves were modelled using the 2005 version of the Wilson-Devinney code. When necessary, cool spots on the surface of the primary component were introduced to account for asymmetries in the light curves. As a result, we found that CSS J112237.1+395219 is a W UMa type contact binary system belonging to W subclass with a mass ratio of $q=1.61$ and a shallow degree of contact of 14.8\% where the primary component is hotter than the secondary one by $500K$. LINEAR 1286561 and LINEAR 2602707 are detached binary systems with mass ratios $q=3.467$ and $q=0.987$ respectively. These detached systems are low-mass M-type eclipsing binaries of similar temperatures. The marginal contact, the fill-out factor and the temperature difference between components of CSS J112237.1+395219 suggest that this system may be at a key evolutionary state predicted by the Thermal Relaxation Oscillation theory (TRO). From the estimated absolute parameters we conclude that our systems share common properties with others ultra-short period binaries., Comment: 11 pages, 7 figures, accepted for publications in Research in Astronomy & Astrophysics

This paper studies the phenomenon of resonant oscillation in Modular Multilevel Converters (MMCs) for variable frequency operation. Without proper control, it may easily induce huge MMC inner circulating currents and damage major power devices. A Laplace model is established to clarify the mechanism of resonant oscillation in MMCs, and Bode diagrams are used to study the frequency response over a wide frequency range. Meanwhile, a straightforward resonant suppression method based on circulating current control is used to damp out the resonant condition, and a cross comparison is conducted in terms of system losses, capacitor voltage ripples, and load voltage harmonics with and without the resonant oscillation. Both simulation and experimental results are provided to validate the MMC resonant oscillation theory and its suppression control method.

In this work, the spectral analysis of signals with the use of spin-transfer nano-oscillator in the external phase locking regime is investigated. This work is motivated by the spintronics, spin-transfer nanooscillators (STNO), neuromorphic computing. A mathematical model of the spectral analyzer based on the STNO is considered. Using methods of oscillation theory we analytically found its main performance characteristics, which are compared with similar, obtained numerically. The results found in this paper can be used in the experimental study of spectrum analyzers based on the STNO and various spintronic devices.

In [Bull. Acad. Polon. Sci. S\'{e}r. Sci. Math. 29 (1981), no.~7-8, 367--370], Philos proved the following result: Let $f:[t_{0},\infty)_{\mathbb{R}}\to\mathbb{R}$ be an $n$-times differentiable function such that $f^{(n)}(t)\leq0$ ($\not\equiv0$) and $f(t)>0$ for all $t\geq{}t_{0}$. If $f$ is unbounded, then $f(t)\geq\frac{\lambda{}t^{n-1}}{(n-1)!}f^{(n-1)}(t)$ for all sufficiently large $t$, where $\lambda\in(0,1)_{\mathbb{R}}$. In this work, we first present time scales unification of this result. Then, by using it, we provide sufficient conditions for oscillation and asymptotic behaviour of solutions to higher-order neutral dynamic equations.

WOS: 000423158800026 The oscillation and nonoscillation theories for nonlinear systems have recently received a lot of attention. We consider a two-dimensional time-scale system and find the oscillation criteria for solutions of the system by using some improper integrals and inequalities. We also give a few examples in order to highlight our main results.

The study of oscillation theory for fractional differential equations has been initiated by Grace et.al. 14. In this paper we establish some new criteria for the oscillation of fractional differential equations with the Caputo derivative of the form c D a α x ( t ) = e ( t ) + f ( t , x ( t ) ) , a 1 , α ? ( 1 , 2 ) .We also present the conditions under which all solutions of this equation are asymptotic to at + b as t ? ∞ for some real numbers a, b. We shall employ a different technique rather than that in 14.

We investigate oscillatory properties of second order Euler type half-linear differential equations whose coefficients are given by periodic functions and functions having mean values. We prove the conditional oscillation of these equations. In addition, we prove that the known oscillation constants for the corresponding equations with only periodic coefficients do not change in the studied more general case. The presented results are new for linear equations as well.

The oscillation of rigid rods is a simple and classical problem. The oscillation theory of rigid rods is widely applicable in various fields. However, the deformation effect on the oscillation periods of slender rods has not yet been studied. Herein we employ the energy method to derive the expression for the oscillation periods of deformable and slender rods. The derivation of the expression is straightforward. We present the deformation effect on the oscillation periods of slender iron wires. As the diameter of wires decreases and the length increases, the deformation effect becomes more significant and the percentage error of the oscillation period without including deformation effect increases. The expression for the oscillation periods of deformable and slender rods is verified by oscillation experiments.

Aim. The elaboration of the theory of the impact of the vibrational digging-out working tool on the body of the beetroot, attached to the soil, and the substantiation of the rational kinematic and constructive parameters of the advanced vibrational digging-out working tool on the basis of the results obtained. Methods. The main provisions of advanced mathematics and theoretical mechanics were used, in particular, the general theory of mechanical impact and the oscillation theory. The equation systems obtained were solved using PC and the elaborated programme. Results. The main provisions of the theory of the impact of the digging-out working tool on the beetroot while extracting the latter from the soil were elaborated. The mathematical model of the impact interaction of the digging-out working tool and the body of the beetroot, attached to the soil, during the vibrational digging-out of sugar beets was created. The equations, describing the mentioned process, were used as a basis for the impact impulse and the impact force; the conditions of not damaging the beetroot during the impact were established. The analytic expression for the determination of the allowed frequency of vibra- tions of the working vibrational digging-out tool with the consideration of its constructive parameters and the translational motion was obtained. The estimates, obtained using PC calculations, allow designing advanced working vibrational digging-out tools to harvest sugar beets, taking the condition of avoiding damage to the root into account. Conclusions. The analytic expression for the determination of the allowed frequency of vibrations of the vibrational digging-out working tool, taking into account its constructive parameters and the translational motion, was obtained taking the condition of avoiding damage to the root into consideration. The algorithm and the PC analysis programme for the allowed frequency of oscillations of the vibrational digging- out working tool with the consideration of no damage to the beetroot during the impact interaction were elabo- rated. Numerous PC estimates demonstrated that on condition of the mentioned weight of the working tool of m = 1.0 kg the allowed frequency, at which there is no damage to the beetroot, should not exceed v = 10.0 hertz (with the oscillation amplitude of a = 0.015 m and the maximum depth of the digging-out working tool in the soil of h = 0.1 m). Here the translational motion should be in the range of 1.3 ... 2.2 m/s. The obtained results of the theoretical research allow designing advanced vibrational digging-out working tools to harvest sugar beets without damaging them.

The characterization of the domain of the Friedrichs extension as a restriction of the maximal domain is well known. It depends on principal solutions. Here we establish a characterization as an extension of the minimal domain. Our proof is different and closer in spirit to the Friedrichs construction. It starts with the assumption that the minimal operator is bounded below and does not directly use oscillation theory.

This study uses a high-speed visualization technique to investigate the breakup process and flow behavior of low pressure water jets issued from non-circular orifices including square, triangular, and rectangular shapes. These orifices have approximately the same sectional areas. Stability curve and Ohnesorge chart are employed to make a comparison with circular jets discharged from a circular orifice of the same sectional area. The analysis is carried out for gauge pressures varying from 0.1 psi to 70 psi with small pressure steps corresponding to a range from 0.7 kPa to 482.6 kPa in metric units. Axis-switching phenomenon is observed and analyzed through calculating the axis-switching wavelength and oscillation frequency for rectangular jets. It is found that results for circular jets agreed well with classic theory. Non-circular jets demonstrate enhanced instabilities as a whole compared to circular jets. The different behaviors of non-circular jets are reasonably explained by Rayleigh’s oscillation theory. Axis-switching and aspect-ratio effect in rectangular jets is found to slow down the increase of breakup-length in the Rayleigh breakup regime. Square and triangular jets are more susceptible to wind effects and they are more unstable especially at higher pressure conditions. This can be concluded from the shorter breakup-length and narrower transitional region from the Rayleigh regime to the wind-induced regime as compared to the circular and rectangular jets. Axis-switching wavelength of the rectangular jets is found to increase linearly with increasing jet velocity and oscillation frequency decreases correspondingly.

By theorems of Sturm and Lyapunov, the L ∞ and L 1 norms of the coefficient q in a differential equation u ″ + q ( t ) u = 0 limit the oscillation of nontrivial solutions. This paper establishes similar results for the L p norms when 1 p ∞ .

In this work we establish the theory of dichotomies for generalized ordinary differential equations, introducing the concepts of dichotomies for these equations, investigating their properties and proposing new results. We establish conditions for the existence of exponential dichotomies and bounded solutions. Using the correspondences between generalized ordinary differential equations and other equations, we translate our results to measure differential equations and impulsive differential equations. The fact that we work in the framework of generalized ordinary differential equations allows us to manage functions with many discontinuities and of unbounded variation.

Oscillation theory locates the spectrum of a differential equation by counting the zeros of its solutions. We present a version of this theory for canonical systems $Ju'=-zHu$ and then use it to discuss semibounded operators from this point of view. Our main new result is a characterization of systems with purely discrete spectrum in terms of the asymptotics of their coefficient functions; we also discuss the exponential types of the transfer matrices.

We suggest a new constructive approach for the solvability analysis and approximate solution of general non-local boundary value problems for non-linear systems of ordinary differential equations with locally Lipschitzian non-linearities. The practical application of the techniques is explained on a numerical example.

We discuss the existence of nontrivial solutions to the boundary value problems for a coupled system of second-order nonlinear difference equations by using the critical point theory. The nontrivial solutions where neither of the components is identically zero are achieved under some sufficient conditions.

Exact solutions of the equations of vibrations of elastic bodies can be obtained, as a rule, for homogeneous bodies with a simple geometry.

1 Department of Mathematics, Shanghai Normal University, Shanghai 200234, China 2Department of Applied Mathematics, University of Western Ontario, London, ON, Canada N6A 5B7 3 Center for Applied Mathematics andTheoretical Physics and Faculty of Natural Science, University of Maribor, Krekova 2, Maribor 2000, Slovenia 4Department of Mathematics, Swinburne University of Technology, Melbourne Vic 3122, Australia

Made available in DSpace on 2020-12-12T00:54:25Z (GMT). No. of bitstreams: 0 Previous issue date: 2020-01-01 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) In this paper, we consider one-parameter ( λ>0) families of Liénard differential equations. We are concerned with the study on the asymptotic behavior of periodic solutions for small and large values of λ>0. To prove our main result we use the relaxation oscillation theory and a topological version of the averaging theory. More specifically, the first one is appropriate for studying the periodic solutions for large values of λ and the second one for small values of λ. In particular, our hypotheses allow us to establish a link between these two theories. Universidade Estadual Paulista (UNESP) Faculdade de Engenharia, Ilha Solteira Universidade Estadual de Campinas (UNICAMP) Instituto de Matemática Estatística e Computação Científica Campinas Universidade Estadual Paulista (UNESP) Faculdade de Engenharia, Ilha Solteira FAPESP: 2013/24541-0 FAPESP: 2018/13481-0 FAPESP: 2018/16430-8 FAPESP: 2019/00976-4 FAPESP: 2019/10269-3 CNPq: 306649/2018-7 CNPq: 438975/2018-9 CAPES: 88881.030454/2013-0

We establish in this paper the equivalence between a Volterra integral equation of second kind and a singular ordinary differential equation of third order with two initial conditions and an integral boundary condition, with a real parameter. This equivalence allow us to obtain the solution to some problems for nonclassical heat equation, the continuous dependence of the solution with respect to the parameter and the corresponding explicit solution to the considered problem., 12 pages

In this work a generalised Lienard type system will be considered. We study the problem whether all trajectories of this system intersect the vertical isocline, which is very important in the global asymptotic stability of the origin, oscillation theory, and existence of periodic solutions. Under quite general assumptions we obtain sufficient conditions which are very sharp. We present some new conditions under which the solutions of this system are oscillatory. Some examples are provided to illustrate our results.

In this paper, we propose an efficient numerical method for ordinary differential equation with fractional order, based on Legendre-Gauss-Radau interpolation, which is easy to be implemented and possesses the spectral accuracy. We apply the proposed method to multi-order fractional ordinary differential equation. Numerical results demonstrate the effectiveness of the approach.

We analyse the oscillation and non-oscillation of second-order half-linear differential equations with periodic and asymptotically almost periodic coefficients, where the equations have the so-called Riemann-Weber form. For these equations, we find an explicit oscillation constant. Corollaries and examples are mentioned as well.

The methods of the nonlinear oscillation theory are used to study a system of coupled nonidentical spin-transfer nanooscillators that interact due to spin waves in a common ferromagnetic layer. The effect of nonidentity with respect to parameters on the phase locking of such a system is analyzed in the presence of variations in the distance between nanocontacts. The effect of broadband locking under significant nonidentity of the control parameters related to the domination of one oscillator is studied.

The accuracy of calculating the gravitational constant, G, using a method based on the numerical integration of an equation of oscillations and a method based on the nonlinear oscillation theory is analyzed. Taking account of a higher (the seventh) power at an amplitude of 80 mrad reduces the error of calculating the moment of attraction forces by 47 times. This reduces the error of calculating G from 15 to 0.3 ppm.

This paper is primarily concerned with analyzing some special properties of the characteristic equation of the linear delay differential equation of the first and odd order. The basic concepts and results keen to the oscillatory nature of solutions, as well as stability analysis are re-investigated using the properties of Lambert W function and compared with existing ones. A brief survey of the function origin and possible applications is presented together with a short behavior analysis.

Erbe’s and Hassan’s contributions regarding oscillation criteria are interesting in the development of oscillation theory of dynamic equations on time scales. The objective of this paper is to amend these results.

In the paper we offer criteria for property (A) of the third-order nonlinear functional differential equation with advanced argument $(a(t)(x'(t))^\gamma )'' + p(t)f(x(\sigma (t))) = 0,$ , where $\mathop \smallint \limits^\infty a^{ - 1/\gamma } (s)ds = \infty $ . We establish new comparison theorems for deducing property (A) of advanced differential equations from that of ordinary differential equations without deviating argument. The presented comparison principle fill the gap in the oscillation theory.

Based on the theory of thermal effect, the temperature rise, deformation and the deformation of periodically polarized region are simulated. The theory of nonlinear transformation, the effect of thermal effect on nonlinear effect is analyzed.Based on the oscillation theory of the laser resonator, the laser resonator was proposed to be constructed, and the influence of the laser resonator and the optical parametric oscillation was analyzed to describe the optical parametric oscillation process of Nd:MgO:PPLN crystal construction more accurately.The thermal effect of the original dynamic model of OPO in the inner cavity was modified to better reflect the actual situation of the compound self-optical parametric oscillator. The experimental study of the future self-optical parametric oscillator provided a design idea and laid a theoretical foundation for the subsequent experiments.

For a differential equation of arbitrary even order with operator coefficients, a generalization of Arnold’s theorems of alternation and non-oscillation is obtained. The oscillation theorem for a problem with general boundary conditions is proved.

We present a comprehensive treatment of relative oscillation theory for finite Jacobi matrices. We show that the difference of the number of eigenvalues of two Jacobi matrices in an interval equals the number of weighted sign-changes of the Wronskian of suitable solutions of the two underlying difference equations. Until now only the case of perturbations of the main diagonal was known. We extend the known results to arbitrary perturbations, allow any (half-)open and closed spectral intervals, simplify the proof, and establish the comparison theorem., Comment: 13 pages

In the paper we offer criteria for oscillation of the third order Euler differential equation with delay $$ y'''(t)+\frac {k^2}{t^3}y(ct)=0. $$ We provide detail analysis of the properties of this equation, we fill the gap in the oscillation theory and provide necessary and sufficient conditions for oscillation of equation considered.

In the paper, we present several functional inequalities and offer their application for higher order advanced differential equations of the formx^(^n^)(t)+q(t)x(@s(t))=0,to be oscillatory. The conditions obtained essentially improve, complements and extends many other known results and involves unimprovable constants.

The boundary value problem (BVP) for a class of nonlinear ordinary differential equations is examined. It can be used to describe the eversion problem for a spherical shell composed of a class of transversely isotropic incompressible Mooney-Rivlin materials. The solutions of the nonlinear equation describing the relation among deformed thickness, initial thickness and material parameters of the spherical shell are obtained. The effects of structure parameter and material parameters on the thickness of the everted spherical shell are discussed by numerical examples.

We prove that the existence of the mean values of coefficients is sufficient for second-order half-linear Euler-type differential equations to be conditionally oscillatory. We explicitly find an oscillation constant even for the considered equations whose coefficients can change sign. Our results cover known results concerning periodic and almost periodic positive coefficients and extend them to larger classes of equations. We give examples and corollaries which illustrate cases that our results solve. We also mention an application of the presented results in the theory of partial differential equations.

We establish a fixed point theorem withw-distance for nonlinear contractive mappings in complete metric spaces. As applications of our results, we derive the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions. Here, we need not assume that the equation has a lower solution.