Asymptotics solutions of a singularly perturbed integro-differential fractional order derivative equation with rapidly oscillating coefficients

In this paper, the regularization method of S.A.Lomov is generalized to the singularly perturbed integrodifferential fractional-order derivative equation with rapidly oscillating coefficients. The main goal of the work is to reveal the influence of the oscillating components on the structure of the asymptotics of the solution to this problem. The case of the absence of resonance is considered, i.e. the case when an integer linear combination of a rapidly oscillating inhomogeneity does not coincide with a point in the spectrum of the limiting operator at all points of the considered time interval. The case of coincidence of the frequency of a rapidly oscillating inhomogeneity with a point in the spectrum of the limiting operator is called the resonance case. This case is supposed to be studied in our subsequent works. More complex cases of resonance (for example, point resonance) require more careful analysis and are not considered in this work.

A deformed model for N-type four-level atom and a single mode field system in the presence of the Kerr medium

The aim of this paper is to study the interaction between a single mode field and four-level atom in N - configuration under nonlinear medium effect. The non-resonance case and the deformation forms in the coupling interaction between the field and the atom are included. The wave function of the proposed system is obtained when the atom is prepared initially in its excited state while the field is prepared in a coherent state. The effect of the deformation and nonlinear medium on the temporal behavior of collapse-revival, field entropy and geometric phase of the system are examined. The results show that the presence of the intensity of the coupling interaction and the non-linear medium have an important influence on the properties of these phenomena.

On the Existence of Time Delay for Rotating Beam with Proportional–Derivative Controller

A rotating beam at varying speed mathematical model is studied. Multiple time scales method is applied to the nonlinear system of differential equations and investigated the system behavior approximate solution in the instance of resonance case. We studied the system in case of applying the delayed control on the displacement and the velocity with Proportional–derivative (PD) controller. The consistency of the steady state solution in the near-resonance case is reviewed and analyzed using the Routh-Huriwitz approach. The factors on the steady state solution of the various parameters are recognized and discussed. Simulation effects are obtained using MATLAB software package. Different response curves are involved to show and compare controller effects at various system parameters.

High-Order Approximation to Two-Level Systems with Quasiresonant Control

In this paper, we focus on high-order approximate solutions to two-level systems with quasi-resonant control. Firstly, we develop a high-order renormalization group (RG) method for Schrödinger equations. By this method, we get the high-order RG approximate solution in both resonance case and out of resonance case directly. Secondly, we introduce a time transformation to avoid the invalid expansion and get the high-order RG approximate solution in near resonance case. Finally, some numerical simulations are presented to illustrate the effectiveness of our RG method. We aim to provide a mathematically rigorous framework for mathematicians and physicists to analyze the high-order approximate solutions of quasi-resonant control problems.

Generalized Picone inequalities and their applications to (p,q)-Laplace equations

We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the (p,q) -Laplace-type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation -\hspace{-0.25em}{\text{Δ}}_{p}u-{\text{Δ}}_{q}u={f}_{\mu }(x,u,\nabla u) in a bounded domain \text{Ω}\hspace{0.25em}\subset {{\mathbb{R}}}^{N} under certain assumptions on the nonlinearity and with a special attention to the resonance case {f}_{\mu }(x,u,\nabla u)={\lambda }_{1}(p)|u{|}^{p-2}u+\mu |u{|}^{q-2}u , where {\lambda }_{1}(p) is the first eigenvalue of the p-Laplacian.

Torus and Subharmonic Motions of a Forced Vibration System in 1 : 5 Weak Resonance

The Neimark-Sacker bifurcation of a forced vibration system is considered in this paper. The series solution to the motion equation is obtained, and the Poincaré map is established. The fixed point of the Poincaré map is guaranteed by the implicit function theorem. The map is transformed into its normal form at the fifth-order resonance case. For some parameter values, there exists the torus T1. Furthermore, the phenomenon of phase locking on the torus T1 is investigated and the parameter condition under which there exists subharmonic motion on the torus T1 is determined.

On the Nonlinear Mode-Coupling in Ultra Precision Manufacturing Machines: Experimental and Analytical Analyses

Recent studies in passively-isolated systems have shown that mode coupling is desirable for best vibration suppression, thus refuting the long-standing rule of mode decoupling. However, these studies have ignored the non-linearities in the isolators. In this work, we consider stiffness nonlinearity from pneumatic isolators and study the nonlinear free undamped vibrations of a passively-isolated ultra-precision manufacturing (UPM) machine. Experimental analysis is conducted to guide the mathematical formulation. The system comprises linearly and nonlinearly coupled in-plane horizontal and rotational motion of the UPM machine with quadratic nonlinear stiffness from the isolators. We present closed-form expressions using the method of multiple scales for two cases viz. the non-resonant case and the bounded internal resonance case. We validate our theoretical findings through direct numerical simulations. For the non-resonant case, we show that the system behaves similar to a linear system. However, for the nearly internal resonance case, we demonstrate strong energy exchange between the modes stemming from nonlinear mode coupling. We further study the effect of nonlinear mode coupling on the vibration isolation performance and demonstrate that mode coupling is not always desirable.

Dynamic Response of a Single Pile Embedded in Sand Including the Effect of Resonance

In this paper, responses of a single pile embedded in sand soil (loose and dense) under dynamic loading (sinusoidal dynamic vibrations of 0.1 g to 0.5 g) have been investigated by two-dimensional analysis using the finite element method (FEM). Viscous (dashpot) boundaries have been used for taking the boundary effects of far-field into account. The applicability and accuracy of site responses of two-dimensional analysis due to the FEM modelling have been well verified with one-dimensional site responses. The results indicate that the relative density of sand (loose, dense) becomes prominent for the displacements of the pile, specifically under the frequency effects of resonance. While the pile in loose sand causes the displacements of 0.1 m to 0.5 m, the pile in dense sand leads to the displacements of 0.05 m to 0.25 m, proportionally with the dynamic loads from 0.1 g to 0.5 g. Moreover, the displacements reach their peak value at the frequency ratio of the resonance case. Viscous boundaries are found sufficient for modelling excessive displacements due to dynamic loading. However, the displacements reveal that high vibrations (> 0.1 g for loose sand, > 0.2 g for dense sand) influencing the pile deformations are critical for the issues of settlements. This is more significant for the resonance case in order for ensuring sufficient design. Consequently, the findings from the study are promising good contributions for pile design under the dynamic effect.