Non-resonant Conditions Research Articles

Response solutions for elliptic-hyperbolic equations with nonlinearities and periodic external forces

In this paper, we focus on the existence of response solutions, i.e. periodic solutions with the same frequencies as the external forces, for elliptic-hyperbolic partial differential equations with nonlinearities and periodic forces. The main tools are Lyapunov–Schmidt reduction and Nash–Moser iteration scheme, both of which have demonstrated success in hyperbolic scenarios. At each step of the iteration, the Galerkin approximation of the equation is solved. The new issue is that the spectral theory of the generalized Sturm–Liouville problem is employed, which also introduces new difficulties for estimations at each step. Under appropriate non-resonance conditions on the frequency, the existence of response solutions for the model will be established.

Non-Resonant Conditions for the Klein – Gordon Equation on the Circle

Non-Resonant Conditions for the Klein – Gordon Equation on the Circle

Effect of Reynolds Number and Aeroelastic Scaling Upon Launch-Vehicle Ground-Wind Loads

NASA conducted a launch vehicle ground-wind-loads investigation at the NASA Langley Transonic Dynamics Tunnel to investigate wind-induced oscillations (WIOs) of a launch vehicle when exposed to ground winds before launch. Previous publications from this effort have documented the effects of an atmospheric-boundary-layer profile on WIO response and the correlation process between model-scale and full-scale wind characteristics and resulting structural loads. This paper will focus on the importance of aeroelastic scaling and the impact of Reynolds number on WIO response. As described in the literature and confirmed in the present investigation, aeroelastic effects can significantly increase the magnitude of measured loads. Additionally, vortex shedding is sensitive to nuances of the flow in the shear layer, which is governed by Reynolds number. Many wind-tunnel facilities are not capable of producing flight Reynolds numbers for the ground-wind-loads problem. At very low Reynolds numbers, laminar shear layers exhibit different behavior, resulting in different vortex frequencies, oscillating lift magnitudes, and motion sensitivities. This investigation demonstrated that low Reynolds number testing can yield substantially lower dynamic loads with less aeroelastic coupling than those acquired at flight-representative Reynolds numbers for a resonant WIO event. Additionally, a resonant response phenomenon present at flight Reynolds number was absent at low Reynolds number. Conversely, for nonresonant WIO response conditions, similar dynamic load coefficients were obtained for similar test velocities at either Reynolds number condition. These findings impact many large launch vehicles, including the NASA Space Launch System series of vehicles.

Exponential convergence of the weighted Birkhoff average

In this paper, we consider the polynomial and exponential convergence rates of the weighted Birkhoff averages of irrational rotations on tori. It is shown that these can be achieved for finite and infinite dimensional tori which correspond to the quasiperiodic and almost periodic dynamical systems respectively, under certain balance between the nonresonant condition and the decay rate of the Fourier coefficients. Diophantine rotations with finite and infinite dimensions are provided as examples. For the first time, we prove the universality of exponential convergence and arbitrary polynomial convergence in the quasiperiodic case and almost periodic case under analyticity respectively.

Rotating solutions to the incompressible Euler–Poisson equation with external particle

We consider a two-dimensional, incompressible fluid body, together with self-induced interactions. The body is perturbed by an external particle with small mass. The whole configuration rotates uniformly around the common center of mass. We construct solutions, which are stationary in a rotating coordinate system, using perturbative methods. In addition, we consider a large class of internal motions of the fluid. The angular velocity is related to the position of the external particle and is chosen to satisfy a non-resonance condition.

Onthe Reducibility of a Class Nonlinear Almost Periodic Hamiltonian Systems

Inthis paper, we consider the reducibility of a class of nonlinear almost periodic Hamiltonian systems. Under suitable hypothesis of analyticity, non-resonant conditions and non-degeneracy conditions, by using KAM iteration, it is shown that the considered Hamiltonian system is reducible to an almost periodic Hamiltonian system with zero equilibrium points for most small enough parameters. As an example, we discuss the reducibility and stability of an almost periodic Hill’s equation.

Fatigue Life Prediction and Verification of Railway Fastener Clip Based on Critical Plane Method

Accurately predicting the fatigue life of fastener clips is crucial for improving material processing technology, enhancing the anti-fatigue design level, and guiding the maintenance and repair of track structures. Conventional methods that solely evaluate the fatigue life of fastener clips based on the uniaxial stress index under displacement loads may lead to significant errors. In this paper, the stress field of a type-III clip under displacement loading conditions was numerically simulated based on fatigue test standards, and the fatigue life of the clip was analyzed using the Fatemi–Socie (FS) multiaxial fatigue criterion based on the critical plane method. A comparison with standard fatigue test results revealed that, under non-resonance conditions, the predicted position of the fatigue critical plane of the fastener clip coincided with the fracture surface observed at the middle of the measured small arc, with a life prediction error of 7.3%. To further investigate the predictive capability of the FS multiaxial fatigue criterion for the fatigue life of the fastener clip under resonance conditions, the stress levels of the clip under non-resonance and resonance conditions were compared through on-site testing, and the effects of inertial loads caused by vertical and lateral vibration acceleration on the stress field of the clip were analyzed in numerical simulation according to the results of clip acceleration tests; the fastener clip’s resonance fracture position and fatigue life were also predicted based on the aforementioned multiaxial fatigue criterion. To verify the accuracy of the predicted results, a testing method was proposed that equates the high-frequency resonant inertial load of the fastener clip to a low-frequency additional preload under the premise of consistent stress fields. A comparison with the numerical simulation results shows that considering only vertical inertial loads would result in a discrepancy between the measured fracture location and the actual one. Considering both vertical and lateral inertial loads, the fracture location (at the heel of the small arc) under resonance conditions could be accurately determined, with a life prediction error of 13.8%. Compared to the non-resonant displacement loading condition, the inertial loads caused by acceleration under resonance conditions led to a reduction in fatigue life of approximately 77.8%.

Energy Resolved Mass Spectrometry for Interoperable Non-resonant Collisional Spectra in Metabolomics.

In untargeted metabolomics, the unambiguous identification of metabolites remains a major challenge. This requires high-quality spectral libraries for reliable metabolite identification, which is essential for translating metabolomics data into meaningful biological information. Several attempts have been made to generate reproducible product ion spectra (PIS) under a low collision energy (ELab) regime and nonresonant collisional conditions but have not fully succeeded. We examined the ERMS (energy-resolved mass spectrometry) breakdown curves of two lipo-amino acids and showed the possibility to highlight "singular points", called descriptors hereafter (linked to respective ELab depending on the instrument), for each of the monomodal product ion profiles. Using several instruments based on different technologies, the PIS recorded at these specific ELab sites shows remarkable similarities. The descriptors appeared as being independent of the fragmentation mechanisms and can be used to overcome the main instrumental effects that limit the interoperability of spectral libraries. This proof-of-concept study, performed on two particular lipo-amino acids, demonstrates the high potential of ERMS-derived information to determine the instrument-specific ELab at which PIS recorded in nonresonant conditions become highly similar and instrument-independent, thus comparable across platforms. This innovative but straightforward approach could help remove some of the obstacles to metabolite identification in nontargeted metabolomics, putting an end to a challenging chimera.

Nonlinear model reduction to temporally aperiodic spectral submanifolds.

We extend the theory of spectral submanifolds (SSMs) to general non-autonomous dynamical systems that are either weakly forced or slowly varying. Examples of such systems arise in structural dynamics, fluid-structure interactions, and control problems. The time-dependent SSMs we construct under these assumptions are normally hyperbolic and hence will persist for larger forcing and faster time dependence that are beyond the reach of our precise existence theory. For this reason, we also derive formal asymptotic expansions that, under explicitly verifiable nonresonance conditions, approximate SSMs and their aperiodic anchor trajectories accurately for stronger, faster, or even temporally discontinuous forcing. Reducing the dynamical system to these persisting SSMs provides a mathematically justified model- reduction technique for non-autonomous physical systems whose time dependence is moderate either in magnitude or speed. We illustrate the existence, persistence, and computation of temporally aperiodic SSMs in mechanical examples under chaotic forcing.

Quadratic Growth of Out-of-Time-Ordered Correlators in Quantum Kicked Rotor Model.

We investigate both theoretically and numerically the dynamics of out-of-time-ordered correlators (OTOCs) in quantum resonance conditions for a kicked rotor model. We employ various operators to construct OTOCs in order to thoroughly quantify their commutation relation at different times, therefore unveiling the process of quantum scrambling. With the help of quantum resonance condition, we have deduced the exact expressions of quantum states during both forward evolution and time reversal, which enables us to establish the laws governing OTOCs' time dependence. We find interestingly that the OTOCs of different types increase in a quadratic function of time, breaking the freezing of quantum scrambling induced by the dynamical localization under non-resonance condition. The underlying mechanism is discovered, and the possible applications in quantum entanglement are discussed.

Coherent superposition of states in degenerate systems using zero-area pulses

The coherent superposition of states in degenerate quantum systems is investigated using detuned laser pulses in which the Rabi frequencies are time-dependent, and the pulse area is zero. In this study, a quantum system with an arbitrary number of degenerate states in the ground set as well as an arbitrary number of degenerate states in the exciting set is considered. We assume that all states in the ground set are coupled to the excited states using laser pulses such that the pulse area of Rabi frequency is zero, and all of them have the same time dependence. It is also assumed that all laser pulses are in a non-resonant condition with Bohr transitions, and all detunings are the same. We will show that by applying the appropriate temporal dependence for the pulses and the appropriate rate for the detunings, the population can be transferred from an arbitrary superposition from the ground states to a desired superposition from the excited states.

Almost Global Existence for Some Hamiltonian PDEs with Small Cauchy Data on General Tori

In this paper we prove a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schrödinger equation with a convolution potential, a beam equation and a quantum hydrodinamical equation. We also apply it to the stability of plane waves in NLS. The main point is that the abstract result is based on a nonresonance condition much weaker than the usual ones, which rely on the celebrated Bourgain’s Lemma which provides a partition of the “resonant sites” of the Laplace operator on irrational tori.

Existence of Solutions for an Impulsive p-laplacian Equation with Nonresonance Conditions

Existence of Solutions for an Impulsive p-laplacian Equation with Nonresonance Conditions

Quantitative Green’s function estimates for lattice quasi-periodic Schrödinger operators

In this paper, we establish quantitative Green's function estimates for some higher dimensional lattice quasi-periodic (QP) Schr\"odinger operators. The resonances in the estimates can be described via a pair of symmetric zeros of certain functions and the estimates apply to the sub-exponential type non-resonant conditions. As the application of quantitative Green's function estimates, we prove both the arithmetic version of Anderson localization and the $(\frac 12-)$-H\"older continuity of the integrated density of states (IDS) for such QP Schr\"odinger operators. This gives an affirmative answer to Bourgain's problem in\cite{Bou00}.

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

On the local Gevrey integrability

This paper uses the Gevrey's smooth normalization theory to investigate the local integrability of vector fields, which have linear parts with one zero eigenvalue and the others non-resonant. First, we explore the general properties of C∞ local integrability. Then we show the same regularity of the Gevrey's smooth local integrability as that of the vector fields under Poincaré's non-resonant condition for the case that the real parts of the eigenvalues are all positive or negative. Lastly, a sharper expression of the loss of the regularity is provided by the lowest order of the resonant terms together with the indices of Gevrey's smoothness and the diophantine condition for the case that the matrix is in the diagonal form. One of the main tools utilized here is the KAM method in the Banach space fixed with a weighted norm.

Modulation of photo-induced Raman enhancement in Ag nanoparticles deposited on nanometer-thick TiO2 films. An interplay between plasmonic properties and irradiation energy

Photo-induced enhanced Raman spectroscopy (PIERS) is an innovative technology that offers additional enhancement in Raman signal compared to surface-enhanced Raman spectroscopy (SERS). In this study, we fabricated nanohybrids consisting of silver nanoparticles on an ultra-thin anatase film using a photoreduction method. This approach allowed for the controllable synthesis of SERS and PIERS nanoplatforms, characterized by oval-shaped nanoparticles, yet varying in size and surface coverage, leading to distinct plasmonic properties. A mere 15-minute UV pre-treatment with low photon density already initiated significant charge-transfer processes followed by Raman spectra under non-resonant conditions of the molecule and estimated by enhancement factor in the range of 12–––17. This phenomenon was observed for a molecular monolayer of a thiol derivative. Not only boosting electron migration appeared. This unique interface of the Ag-anatase composite undoubtedly contributed to extended relaxation times of photo-induced enhancement. Furthermore, we investigated how plasmonic and morphological features of the nanoplatforms, in conjunction with UV and Vis illumination, modulated the migration of photoinduced electrons from the semiconductor to the metal. These findings highlighted the variety of processes contributing to the creation of efficient PIERS materials.

Qualitative properties, bifurcations and chaos of a discrete predator–prey system with weak Allee effect on the predator

In this work, the qualitative properties of the fixed points in the non-hyperbolic and degenerate cases, codimension-one bifurcations and Marotto’s chaos of a discrete predator–prey system with weak Allee effect on the predator are investigated. Utilizing the center manifold theorem and the reduction principle, the qualitative property of each fixed point in the non-hyperbolic case is explored. Based on the approximate flow theory, the qualitative property of the boundary fixed point in the degenerate case is investigated. Making use of the center manifold theorem and the bifurcation theory, all the potential codimension-one bifurcation types of the co-existence fixed point are explored, including flip bifurcation and Neimark–Sacker bifurcation. For each type of these bifurcations, not only the concise non-degenerate condition expressed by the system parameters is given, but also the analytical expression of the system orbit caused by the bifurcation is derived. Based on the non-negativity of the bifurcation critical values, the non-resonant and non-degenerate conditions for the occurrence of these bifurcations, the system parameter plane is divided into several regions using tools such as the polynomial complete discriminant system theory, real root separation theory, dichotomy method, implicit function theorem and Cardan’s formula. In each region, the direction and stability of these bifurcations are studied, which helps to theoretically explain the impact of Allee effect on the system. By proving that the co-existence fixed point is a snap-back repeller under appropriate conditions, it is obtained that the system is chaotic. Numerical simulations are completely consistent with all the theoretical analyses. The research results show that the impact of Allee effect on the population system depends not only on the assumption and construction of the system, but also on the system parameter and initial value of the system.

Silver hexacyanoferrate (II) nanocrystals as a new material to improve Raman scattering enhancement during silver surface oxidation

Raman spectroscopy is a powerful analysis technique that shows its full potential when a high amplification of the Raman signal is achieved. In this sense, Surface-Enhanced Raman scattering (SERS) has been the most widely used phenomenon for analysis. SERS provides the amplification of the Raman intensity due to the interaction of molecules with a plasmonic nanostructured surface. The enhancement of the Raman signal can be also obtained during the electrochemical oxidation of a metal electrode; this phenomenon was denoted as Electrochemical-Surface Oxidation Enhanced Raman Scattering (EC-SOERS) and yields a good Raman signal enhancement with high reproducibility. Until now, only chloride and bromide have been employed in EC-SOERS, using a silver electrode to generate silver chloride and silver bromide nanocrystals. In this work, a new EC-SOERS substrate based on the electrogeneration of silver hexacyanoferrate (II) nanocrystals is presented which provides a very sensitive Raman response. The electrogeneration of this new material can be easily followed using spectroelectrochemistry since the characteristic Raman bands of the nanocrystals lie outside of the fingerprint region used for the analysis where the detection of most of the target molecules is performed. Indigo Carmine has been selected as target molecule, obtaining a very good response at nanomolar level under Raman resonance and non-resonance conditions.

Regularity of time-periodic solutions to autonomous semilinear hyperbolic PDEs

This paper concerns autonomous boundary value problems for 1D semilinear hyperbolic PDEs. For time-periodic classical solutions, which satisfy a certain non-resonance condition, we show the following: If the PDEs are continuous with respect to the space variable x and C∞-smooth with respect to the unknown function u, then the solution is C∞-smooth with respect to the time variable t, and if the PDEs are C∞-smooth with respect to x and u, then the solution is C∞-smooth with respect to t and x. The same is true for appropriate weak solutions. Moreover, we show examples of time-periodic functions, which do not satisfy the non-resonance condition, such that they are weak, but not classical solutions, and such that they are classical solutions, but not C∞-smooth, neither with respect to t nor with respect to x, even if the PDEs are C∞-smooth with respect to x and u. For the proofs we use Fredholm solvability properties of linear time-periodic hyperbolic PDEs and a result of E. N. Dancer about regularity of solutions to abstract equivariant equations.