Non-resonant Conditions Research Articles

Quasi-Periodic Motion and Hopf Bifurcation of a Two-Dimensional Aeroelastic Airfoil System in Supersonic Flow

In this article, the complex dynamic behavior of a nonlinear aeroelastic airfoil model with cubic nonlinear pitching stiffness is investigated by applying a theoretical method and numerical simulation method. First, through calculating the Jacobian of the nonlinear system at equilibrium, we obtain necessary and sufficient conditions when this system has two classes of degenerated equilibria. They are described as: (1) one pair of purely imaginary roots and one pair of conjugate complex roots with negative real parts; (2) two pairs of purely imaginary roots under nonresonant conditions. Then, with the aid of center manifold and normal form theories, we not only derive the stability conditions of the initial and nonzero equilibria, but also get the explicit expressions of the critical bifurcation lines resulting in static bifurcation and Hopf bifurcation. Specifically, quasi-periodic motions on 2D and 3D tori are found in the neighborhoods of the initial and nonzero equilibria under certain parameter conditions. Finally, the numerical simulations performed by the fourth-order Runge–Kutta method provide a good agreement with the results of theoretical analysis.

Local electron and ion density control using passive resonant coils in inductively coupled plasma

Control of local electron and ion density using passive resonant coils is experimentally investigated in an inductive argon discharge. Four passive resonant coils are installed under a powered coil; each coil has a fan shape and good azimuthal symmetry. Electron energy probability functions and two-dimensional ion density profiles were measured under both resonant and non-resonant regimes. At non-resonance, almost all of the current flows through the powered coil located in the center of the reactor, and the profiles of the electron and ion density are convex. However, at resonance, a large current flows through the passive resonant coil, and dramatic changes are observed in the electron and ion density profiles. At resonance, the electron and ion densities near the passive resonant coil are increased by 300% compared to the non-resonant condition, and radial distributions become almost flat. Experimental results show that the electron and ion density profiles can be effectively controlled by a passive resonant coil at both low pressure (5 mTorr) and high pressure (50 mTorr). These changes in electron and ion density profiles can be understood by the changes of the electron heating and ionization regions.

Rotating periodic integrable solutions for second-order differential systems with nonresonance condition

In this paper, by using Parseval's formula and Schauder's fixed point theorem, we prove the existence and uniqueness of rotating periodic integrable solution of the second-order system $x''+f(t,x)=0$ with $x(t+T)=Qx(t)$ and $\int_{(k-1)T}^{kT}x(s)ds=0$, $k\in Z^+$ for any orthogonal matrix $Q$ when the nonlinearity $f$ satisfies nonresonance condition.

Are charged tips driving TERS-resolution? A full quantum chemical approach.

Experimental evidence suggests an extremely high, possibly even sub-molecular, spatial resolution of tip-enhanced Raman spectroscopy (TERS). While the underlying mechanism is currently still under discussion, two main contributions are considered: The involved plasmonic particles are able to highly confine light to small spatial regions in the near-field, i.e., the electromagnetic effect and the chemical effect due to altered molecular properties of the sample in close proximity to the plasmonic tip. Significant theoretical effort is put into the modeling of the electromagnetic contribution by various groups. In contrast, we previously introduced a computational protocol that allows for the investigation of the local chemical effect-including non-resonant, resonant, and charge transfer contributions-on a plasmonic hybrid system by mapping the sample molecule with a metallic tip model at the (time-dependent) density functional level of theory. In the present contribution, we evaluate the impact of static charges localized on the tip's frontmost atom, possibly induced by the tip geometry in the vicinity of the apex, on the TERS signal and the lateral resolution. To this aim, an immobilized molecule, i.e., tin(II) phthalocyanine (SnPc), is mapped by the plasmonic tip modeled by a single positively vs negatively charged silver atom. The performed quantum chemical simulations reveal a pronounced enhancement of the Raman intensity under non-resonant and resonant conditions with respect to the uncharged reference system, while the contribution of charge transfer phenomena and of locally excited states of SnPc is highly dependent on the tip's charge.

Parabolic invariant tori in quasi-periodically forced skew-product maps

We consider the existence of parabolic invariant tori for a class of quasi-periodically forced analytic skew-product maps φ:Rn×Td→Rn×Td:φ(zθ)=(z+ϕ(z)+h(z,θ)+ϵf(z,θ)θ+ω), where ϕ:Rn→Rn is a homogeneous function of degree l with l≥2 and h=O(|z|l+1). We obtain the following results: (a) For n=1, l being odd and ϵ sufficiently small, parabolic invariant tori exist if ω satisfies the Brjuno-Rüssmann's non-resonant condition. (b) For n=1, and ϵ sufficiently small, parabolic invariant tori also exist if one of the following conditions holds: (i) First order average is non-zero, first order non-average part is small enough and the forcing frequency ω does not need any arithmetic condition. (ii) First order average is non-zero and ω satisfies the Brjuno-type weak non-resonant condition; (iii) l=2, first order average is zero, both first and second order non-average parts are small enough and ω satisfying Brjuno-type weak non-resonant condition; (iv) l>2, first order average is zero, the second order average is non-zero, both first and second order non-average parts are small enough and ω satisfies the Brjuno-type weak non-resonant condition. (c) In the case n>1, if first order average belongs to the interior of the range of ϕ, Spec(Dϕ)∩iR=∅, first order non-average part is small enough and the forcing frequency ω does not need any arithmetic condition, then the quasi-periodically forced skew-product maps above admit parabolic invariant tori for ϵ sufficiently small. The main methods of this paper are KAM theory and fixed point theorem, which are finally shown that it can be directly applied to the existence problem of quasi-periodic response solutions of degenerate harmonic oscillators.

Response solutions to the quasi-periodically forced systems with degenerate equilibrium: a simple proof of a result of W Si and J Si and extensions

We give a simple proof of the existence of response solutions in some quasi-periodically forced systems with degenerate fixed points. The same questions were answered by Si and Si (2018 Nonlinearity 31 2361–18) using two versions of Kolmogorov–Arnold–Moser (KAM) theory. Our method is based on reformulating the existence of response solutions as a fixed point problem in appropriate spaces of smooth functions. By algebraic manipulations, the fixed point problem is transformed into a problem dealt with contraction mapping principle. Compared to the KAM method, the present method does not incur a loss of regularity. That is, the solutions we obtain have the same regularity as the forcing. Moreover, the method here applies when problems are only finitely differentiable. It also weakens slightly the non-resonance conditions on the forcing frequencies. Since the method is based on the contraction mapping principle, we also obtain automatically smooth dependence on parameters and, when studying complex versions of the problem we discover the new phenomenon of monodromy. We also present results for higher dimensional systems, but for higher dimensional systems, the concept of degenerate fixed points is much more subtle than in one dimensional systems. To illustrate the power of the method, we also consider two problems not studied by Si and Si (2018 Nonlinearity 31 2361–18): the forcing with zero average and second order oscillators. We show that in the zero average forcing case, the solutions are qualitatively different, but for the second order oscillators are remarkably similar.

A four-field three-phase flow model with both miscible and immiscible components

A three-phase flow model with hybrid miscibility constraints is proposed: three immiscible phases are considered (liquid water, liquid metal and gas) but the gaseous phase is composed with two miscible components (steam water and non-condensable gas). The modelling approach is based on the building of an entropy inequality for the system of partial differential equations: once an interfacial velocity is given by the user, the model is uniquely defined, up to some relaxation time scales, and source terms complying with the second principle of thermodynamics can then be provided. The convective part of the system is hyperbolic when fulfilling a non-resonance condition and classical properties are studied (Riemann invariants, symmetrization). A key property is that the system possesses uniquely defined jump conditions. Last, preservation of thermodynamically admissible states and pressure relaxation are investigated.

Optical Bistability in an Optomechanical System with N-Type Atoms under Nonresonant Conditions

In this paper, the phenomenon of the optical bistability of a cavity field is theoretically investigated in an optomechanical system containing an N-type atomic ensemble. In this hybrid optomechanical system, the atoms are coupled with two controlling light fields besides coupling with the cavity field. Under the nonresonant condition, we analyze the influences of the coupling strength between cavity and atoms, Rabi frequencies of the controlling light field, the detuning between the controlling light field and atoms, and pump field power on the optical bistable behavior of mean intracavity photon number. The nonlinear distribution of the mean intracavity photon number has a potential application in field optical switches and optical bistable devices.

Completely degenerate responsive tori in Hamiltonian systems**The first author was partially supported by a CSC scholarship and Fundamental Research Funds of Shandong University. The second author was partially supported by NSERC discovery Grant 1257749, PIMS CRG Grant, a faculty development grant from the University of Alberta, and a Scholarship from Jilin University.

We consider the existence of responsive tori for the completely degenerate Hamiltonian system with the following Hamiltonian which is associated with the standard symplectic structure, where λ = ±1 and n > 2, n ⩾ m ⩾ 2 are integers. With P satisfying certain non-degenerate conditions, we obtain the following results: (1) For λ = −1 and ϵ sufficiently small, responsive tori exist for each ω satisfying a weak non-resonant condition; (2) For λ = 1 and ϵ* sufficiently small, there exists a Cantor set with almost full Lebesgue measure such that responsive tori exist for each if ω satisfies a Diophantine condition. Non-existence of responsive tori are also discussed when P fails to satisfy the non-degenerate condition. Our results are directly applicable to the existence problem of quasi-periodic responsive solutions of degenerate harmonic oscillators.

On the reducibility of linear quasi-periodic systems with Liouvillean basic frequencies and multiple eigenvalues

In this paper we consider the linear quasi-periodic systemx˙=(A+ϵP(t))x,x∈Rd, where A is a d×d constant matrix of elliptic type and has multiple eigenvalues, P(t) is analytic quasi-periodic with respect to t with basic frequencies ω=(1,α), where α is irrational, and ϵ is a small perturbation parameter. Under suitable non-resonant condition, non-degeneracy condition and 0≤β(α)<r⁎, where β(α)=limsupn→∞ln⁡qn+1qn, qn is the sequence of denominations of the best rational approximations for α∈R∖Q, 0<r⁎<r, r is the initial radius of analytic strip, it is proved that for most sufficiently small ϵ, this system can be reduced to a constant system x˙=A⁎x, where A⁎ is a constant matrix close to A. As some applications, we apply our results to quasi-periodic Hill's equations, three dimensional skew symmetric systems and n coupled Schrödinger equations to study the Lyapunov stability of the equilibrium and the existence of quasi-periodic solutions.

Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators

In this paper, we consider the Schr\"odinger equation, \begin{equation*} Hu=-u^{\prime\prime}+(V(x)+V_0(x))u=Eu, \end{equation*} where $V_0(x)$ is 1-periodic and $V (x)$ is a decaying perturbation. By Floquet theory, the spectrum of $H_0=-\nabla^2+V_0$ is purely absolutely continuous and consists of a union of closed intervals (often referred to as spectral bands). Given any finite set of points $\{ E_j\}_{j=1}^N$ in any spectral band of $H_0$ obeying a mild non-resonance condition, we construct smooth functions $V(x)=\frac{O(1)}{1+|x|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}_{j=1}^N$. Given any countable set of points $\{ E_j\}$ in any spectral band of $H_0$ obeying the same non-resonance condition, and any function $h(x)>0$ going to infinity arbitrarily slowly, we construct smooth functions $|V(x)|\leq \frac{h(x)}{1+|x|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}$. On the other hand, we show that there is no eigenvalue of $H=H_0+V$ embedded in the spectral bands if $V(x)=\frac{o(1)}{1+|x|}$ as $x$ goes to infinity. We prove also an analogous result for Jacobi operators.

Phonon spectra of pure and acceptor doped BaZrO3 investigated with visible and UV Raman spectroscopy

We report results from visible and UV Raman spectroscopy studies of the phonon spectra of a polycrystalline sample of the prototypical perovskite type oxide BaZrO3 and a 500 nm thick film of its Y-doped, proton conducting, counterpart BaZr0.8Y0.2O2.9. Analysis of the Raman spectra measured using different excitation energies (between 3.44 eV and 5.17 eV) reveals the activation of strong resonance Raman effects involving all lattice vibrational modes. Specifically, two characteristic energies were identified for BaZrO3, one around 5 eV and one at higher energy, respectively, and one for BaZr0.8Y0.2O2.9, above 5 eV. Apart from the large difference in spectral intensity between the non-resonant and resonant conditions, the spectra are overall similar to each other, suggesting that the vibrational spectra of the perovskites are stable when investigated using an UV laser as excitation source. These results encourage further use of UV Raman spectroscopy as a novel approach for the study of lattice vibrational dynamics and local structure in proton conducting perovskites, and open up for, e.g., time-resolved experiments on thin films targeted at understanding the role of lattice vibrations in proton transport in these kinds of materials.

Resonant and non-resonant coupling of one-dimensional microcavity mode and optical Tamm state

Coupling between the optical Tamm states (OTS) and microcavity mode in one-dimensional photonic crystal (1DPhC) structures is investigated experimentally and numerically with the help of transfer matrix method in resonant and non-resonant conditions, as a function of the distance between the 1DPhC-metal interface and the cavity layer (in terms of number of bilayers, Ntop, between the Tamm cavity and the microcavity). The system under study comprised of twelve bi-layers of alternating SiO2/TiO2 thin films, a half-wave thick TiO2 microcavity region and a layer of silver metal on top of the structure. It is observed that the microcavity mode and the OTS repel each other when Ntop is reduced indicating strong coupling between the two modes. In the resonant condition, wherein the individual structure consisting of either microcavity alone or Tamm cavity alone are tuned for the configuration Ntop = 2, the coupled modes in the hybrid structures are simultaneously localized at metal-1DPhC interface and at the microcavity. In the non-resonant condition when the microcavity mode and the OTS in the individual structure are not tuned, the coupling between the modes for the hybrid structure with Ntop = 2 leads to an enhanced electric field intensity of the microcavity mode compared to that from the uncoupled microcavity mode. Consequently, four different configurations corresponding to the non-resonant condition, having different Ntop are studied experimentally. The last four TiO2 layers and the microcavity layer of all the fabricated samples were incorporated with carbon quantum dots (CQDs). Comparison of photoluminescence enhancement as a function of Ntop in coupled and uncoupled conditions is discussed.

Multiple Solutions for a Kirchhoff-Type Equation

In this paper, we study a class of Kirchhoff-type equation with asymptotically linear right-hand side and compute the critical groups at a point of mountain pass type under suitable Hilbert space. The existence results of three nontrivial solutions under the resonance and non-resonance conditions are established by using the minimax method and Morse theory.

On the Reducibility of Quasiperiodic Linear Hamiltonian Systems and Its Applications in Schrödinger Equation

In this paper, we consider the reducibility of the quasiperiodic linear Hamiltonian system ẋ=A+εQt, where A is a constant matrix with possible multiple eigenvalues, Qt is analytic quasiperiodic with respect to t, and ε is a small parameter. Under some nonresonant conditions, it is proved that, for most sufficiently small ε, the Hamiltonian system can be reduced to a constant coefficient Hamiltonian system by means of a quasiperiodic symplectic change of variables with the same basic frequencies as Qt. Applications to the Schrödinger equation are also given.

Parametric Study of Driver and Reflector of Single Axis Acoustic Levitator using Finite Element Method

Total radiation force on a spherical levitating object, which is placed between a single axis acoustic levitator, is obtained using finite element simulation. Variation in the total radiation force on the spherical levitating object with respect to the position of the object between the driver and the reflector is studied in resonance as well as non-resonance condition. Simulation results are verified with experimental results available in the literature. Further, a parametric study has been performed on the radius of curvature of driver and reflector. Three different cases have been considered. (1) Curved driver surface with flat reflector surface. (2) Curved reflector surface with flat driver surface. (3) Both driver and reflector having curved surfaces. It is observed that the case with both driver and reflector surfaces being curved results in maximum radiation force on the spherical levitating object. The values of radius of curvature for maximum radiation force for all three cases are also obtained. Total radiation forces for all three cases (with optimum value of radius of curvature) as well as the flat surfaced driver-reflector arrangement are compared.

Development and analysis of a stick-slip rotary piezoelectric positioner achieving high velocity with compact structure

A high-velocity rotary stick-slip piezoelectric positioner working in non-resonant condition driven by piezo-stacks is proposed in this paper. For improving working performance through the parasitic motion principle, a compact flexible mechanism with rotational symmetric structure is designed. Its special structure can realize self-centering function, thus there is no need to adopt a bearing for the rotor. And in order to reduce the amounts of needed piezo-stacks and achieve synchronous action of four driving feet, a connecting ring is designed to connect four driving feet. A prototype is manufactured and corresponding experiments are carried out, experimental results show the proposed positioner has a good working performance. Its maximum velocity reaches 151.4 mrad/s, which is greater than most existing non-resonant stick-slip piezoelectric positioners driven by piezo-stacks. And a dynamic model is established and calculated in MATLAB/Simulink, a series of simulation are carried out to further investigate influential factors of the proposed positioner.

Advanced Raman spectroscopy of Cs2AgBiBr6 double perovskites and identification of Cs3Bi2Br9 secondary phases

Cs2AgBiBr6 double perovskites are intensely investigated as lead-free, stable alternatives to organic lead halide perovskites. In this contribution, we revise the vibrational modes of Cs2AgBiBr6 and show that Raman spectroscopy is well suited for the detection of common secondary phases such as Cs3Bi2Br9. The combined Raman analysis at pre-, near- and non-resonant conditions allows us to extract the four vibrational modes of Cs2AgBiBr6 to unprecedented detail, the most intense one at 177.3 cm-1. The experimental results are in good agreement with theoretical density functional theory calculations. In addition, we monitor the decomposition of Cs2AgBiBr6 into Cs3Bi2Br9 at temperatures above 255 °C.

Investigations on nonlinear optical properties of gold nanoparticles doped fluoroborate glasses for optical limiting applications

Gold nanoparticles (NPs) exhibit strong surface plasmon resonance (SPR) in the visible spectral region. Lithium Zinc Calcium Fluoroborate glasses embedded with Au NPs were synthesized by conventional melt quench technique. The gold NPs embedded glasses exhibited dichroic nature. A small kink around 44.61∘ in the X-ray diffraction (XRD) pattern suggests the presence of Au (2 0 0) in the matrix. High resolution-transmission electron microscopic (HR-TEM) measurements further confirmed the existence of spherical and ellipsoidal Au NPs. Thermogravimetric analysis (TGA) reveals the role of network modifiers in the matrix as the glasses that contained higher concentration of lithium oxide exhibits lower Tg thereby resulting in creation of more non bridging oxides (NBO's). The Z-Scan studies were performed under non-resonant condition (800 nm) to analyze the application of these glasses for optical limiting devices.

The existence of almost-periodic solutions for 1-dimensional nonlinear Schrödinger equation with quasi-periodic forcing

In this paper, we prove that there exist almost-periodic solutions for the Schrödinger equation with quasi-periodic forcing under periodic boundary conditions. This extends the result of Cong et al. [J. Differ. Equations 264, 4504–4563 (2018)] to the case with quasi-periodic forcing. The main difficulty is to ensure the positive lower bound of the non-resonant condition.