Small Parameter Research Articles

Exploring the geometry of the bifurcation sets in parameter space

By studying a nonlinear model by inspecting a p-dimensional parameter space through (p-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p-1)$$\\end{document}-dimensional cuts, one can detect changes that are only determined by the geometry of the manifolds that make up the bifurcation set. We refer to these changes as geometric bifurcations. They can be understood within the framework of the theory of singularities for differentiable mappings and, in particular, of the Morse Theory. Working with a three-dimensional parameter space, geometric bifurcations are illustrated in two models of neuron activity: the Hindmarsh–Rose and the FitzHugh–Nagumo systems. Both are fast-slow systems with a small parameter that controls the time scale of a slow variable. Geometric bifurcations are observed on slices corresponding to fixed values of this distinguished small parameter, but they should be of interest to anyone studying bifurcation diagrams in the context of nonlinear phenomena.

Performance analysis of an underwater wireless optical communication link with Lommel beam

Abstract In order to mitigate the stochastic interference of underwater channels and improve the quality of underwater communication systems, it is essential to study the performance of the underwater wireless optical communication (UWOC) links utilizing vortex beams with unique attributes. In this paper, the analytical formulae for the bit error rate (BER) and the average capacity of the UWOC link with diffraction-free Lommel beam are derived under the Rytov theory. Simulation results demonstrate that the system with a long wavelength, a high system signal-to-noise ratio(SNR), a small asymmetric parameter and receiving aperture diameter achieves a high average capacity and a low BER. Furthermore, in the underwater channel with a larger kinetic energy dissipation rate per unit mass of fluid and inner scale, a smaller mean-squared temperature dissipation rate, temperature salinity contribution ratio and transmission distance, the performance of the communication link can be improved. Meanwhile, it is found that the performance of the link with carrier Lommel beam are less sensitive to the topological charge, the scaling factor of the beam and the turbulent outer scale. These findings provide theoretical support for the design and implementation of an UWOC link utilizing the Lommel beam.

Secondary massive quarks with the Mellin-Barnes expansion

Processes involving only massless or massive quarks at tree-level get corrections from massive (lighter, heavier, or equal-mass) secondary quarks starting at two-loop order, generated by a virtual gluon splitting into a massive quark anti-quark pair. One convenient approach to compute such two-loop corrections is starting with the one-loop diagram considering the virtual gluon massive. Carrying out a dispersive integral with a suitable kernel over the gluon mass yields the desired two-loop result. On the other hand, the Mellin-Barnes representation can be used to compute the expansion of Feynman integrals in powers of a small parameter. In this article we show how to combine these two ideas to obtain the corresponding expansions for large and small secondary quark masses to arbitrarily high orders in a straightforward manner. Furthermore, the convergence radius of both expansions can be shown to overlap, being each series rapidly convergent. The advantage of our method is that the Mellin representation is obtained directly for the full matrix element from the same one-loop computation one needs in large-β0 computations, therefore many existing results can be recycled. With minimal modifications, the strategy can be applied to compute the expansion of the one-loop correction coming from a massive gauge boson. We apply this method to a plethora of examples, in particular those relevant for factorized cross sections involving massless and massive jets, recovering known results and obtaining new ones. Another bonus of our approach is that, postponing the Mellin inversion, one can obtain the small- and large-mas expansions for the RG-evolved jet functions. In many cases, the series can be summed up yielding closed expressions.

Pitchfork bifurcation along a slow parameter ramp: Coherent structures in the critical scaling

AbstractWe investigate the slow passage through a pitchfork bifurcation in a spatially extended system, when the onset of instability is slowly varying in space. We focus here on the critical parameter scaling, when the instability locus propagates with speed , where is a small parameter that measures the gradient of the parameter ramp. Our results establish how the instability is mediated by a front traveling with the speed of the parameter ramp, and demonstrate scalings for a delay or advance of the instability relative to the bifurcation locus depending on the sign of , that is on the direction of propagation of the parameter ramp through the pitchfork bifurcation. The results also include a generalization of the classical Hastings–McLeod solution of the Painlevé‐II equation to Painlevé‐II equations with a drift term.

A BDF2 method for a singularly perturbed transport equation

A singularly perturbed transport equation is considered. A variable two-step backward differentiation formulas (BDF2) on a Shishkin-type mesh is used to discrete the first-order derivatives of the singularly perturbed transport equation. The stability and error analysis are derived by using the discrete orthogonal convolution kernels. It is proved that the scheme is second-order uniformly convergent with respect to the small parameter, which improves previous results. Numerical experiments are presented to support the theoretical result.

Asymptotic Analysis of an Elastic Layer under Light Fluid Loading

Asymptotic analysis for an elastic layer under light fluid loading was developed. The ratio of fluid and solid densities was chosen as the main small parameter determining a novel scaling. The leading- and next-order approximations were derived from the full dispersion relation corresponding to long-wave, low-frequency, antisymmetric motions. The asymptotic plate models, including the equations of motion and the impenetrability condition, motivated by the aforementioned shortened dispersion equations, were derived for a plane-strain setup. The key findings included, in particular, the necessity of taking into account transverse plate inertia at the leading order, which is not the case for heavy fluid loading. In addition, the transverse shear deformation, rotation inertia, and a number of other corrections appeared at the next order, contrary to the previous asymptotic developments for fluid-loaded plates not assuming a light fluid loading scenario.

Experiment and simulation study on the functional fatigue of shape memory alloy beam actuators

Abstract The functional fatigue behavior of shape memory alloy (SMA) beam actuators is gaining importance as their utilization in engineering applications becomes more widespread. However, research on the functional fatigue behavior of SMA beam actuators under bending conditions is not as extensive as that on SMA wires. In this paper, an experimental study and theoretical analysis of the functional fatigue behavior of SMA beam actuators were conducted. A measuring method for bending deflection and an automatic thermal cyclic test bench was designed. A series of functional fatigue tests were conducted on SMA beam actuators under different bias load conditions and the functional fatigue patterns were obtained. The material damage factor is defined and calculated through specimen tests. By incorporating the damage factor to modify the existing constitutive model, a model considering functional fatigue and tension-compression asymmetry is obtained. Finite Element Analysis (FEA) is performed based on this modified model to simulate the actuating performance of SMA beam actuators and to compute the deflection at different numbers of cycles. Additionally, a small parameter study on the actuator shape is conducted using FEA. By comparing the FEA results with functional fatigue experimental data, the effectiveness of the modified model is validated, demonstrating its capability to describe functional fatigue and predict actuator lifespan.

Asymptotic analysis of the nonsteady micropolar fluid flow through a system of thin pipes

In this paper, we analyze the time‐dependent flow of an incompressible micropolar fluid in a multiple‐pipe system. Motivated by the applications, we assume that the pipes have circular cross‐section and that the ratio between pipes' thickness and its length is small, denoted by the parameter . Far from the junction, the fluid exhibits different behavior depending on the magnitude of the viscosity coefficients with respect to the small parameter . Focusing on the critical case described by the strong coupling between velocity and microrotation, the complete asymptotic expansion of the solution (up to an arbitrary order) is built. To improve the accuracy of the asymptotic approximation, we introduce the boundary layer correctors near the pipes' ends and take into account the interior layer correction in the vicinity of the junction as well. The convergence is also proved via error estimates, providing the rigorous justification of the proposed effective model.

Serum neurofilament light chain levels correlate with small fiber related parameters in patients with hereditary transthyretin amyloidosis with polyneuropathy (ATTRv-PN).

Recent evidence suggests that both serum neurofilament light chain (sNfL) levels and small fiber related diagnostic variables may be valuable disease biomarkers of hereditary transthyretin amyloidosis with polyneuropathy (ATTRv-PN). Our study aimed to explore the relations between sNfL and small fiber related skin biopsy and quantitative sensory testing (QST) parameters in a cohort of ATTRv-PN patients and pre-symptomatic carriers. We retrospectively analyzed data from 13 ATTRv patients and 21 pre-symptomatic carriers who underwent sNfL dosage, skin biopsy, and QST, and analyzed correlations between sNFL, intraepidermal nerve fiber density (IENFD), and cold (CDT) and warm detection thresholds (WDT). Both sNfL and small fiber related parameters significantly differed between carriers and patients (sNfL: p < 0.0001; IENFD: p = 0.0008; CDT, WDT: < 0.0001). sNFL levels were normal in all carriers, altered in 85% of patients, negatively correlated with distal IENFD (r = -0.47, p = 0.005), and significantly correlated with CDT (r = -0.68; p < 0.0001) and WDT (r = 0.57; p < 0.0001). Our study showed that sNfL reliably discriminates symptomatic ATTRv-PN patients from pre-symptomatic carriers, and found significant relations between sNfL, skin biopsy, and QST small fiber related parameters, suggesting that sNfL might be a valuable biomarker of peripheral nerve involvement in ATTRv-PN and a supportive criterion for symptomatic disease transition.

Two-dimensional transient contrasting structure evolution in an inhomogeneous media with the advection

The inner transition layer evolution for two-dimensional quasi-linear initial-boundary value problem for the reaction-advection-diffusion equation in an inhomogeneous media with a small parameter within the high order derivatives is considered. Within the framework of the main (zero order) sum of the asymptotic series, the position of the inner transition layer is described by the Hamilton–Jacobi equation. The potential is calculated as an integrated density function of the reaction sources within the limits of the equilibrium levels. The front line of the transition layer evolves in the same way as the constant-eikonal line (the wavefront line in the other words) for the problem of wave propagation in an inhomogeneous medium in short-wave (geometro-optical) asymptotics. The sum of the asymptotic series of zero order and first order is found, the existence gap of a smooth front line, the time of destruction of the contrasting structure are calculated.

Optimal quantum metrology of two-photon absorption

Abstract Two-photon absorption (TPA) is a nonlinear optical process with wide-ranging applications from spectroscopy to super-resolution imaging. Despite this, the precise measurement and characterisation of TPA parameters are challenging due to their inherently weak nature. We study the potential of single-mode quantum light to enhance TPA parameter estimation through the quantum Fisher information (QFI). Discrete variable (DV) quantum states (defined to be a finite superposition of Fock states) are optimised to maximise the QFI for given absorption, revealing a quantum advantage compared to both the coherent state (classical) benchmark and the single-mode squeezed vacuum state. For fixed average energy ̄n ∈ 2N, the Fock state is shown to be optimal for large TPA parameters, while a superposition of vacuum and a particular Fock state is optimal for small absorption for all ̄n. This differs from single-photon absorption where the Fock state is always optimal. Notably, photon counting is demonstrated to offer optimal or nearly optimal performance compared to the QFI bound for all levels of TPA parameters for the optimised quantum probes, and their quantum advantage is shown to be robust to single-photon loss. Our findings provide insight into known limiting behaviours of Gaussian probes and their different Fisher information (FI) scalings under photon counting (∝ ̄n2 for squeezed vacuum states versus ̄n3 for coherent states). The squeezed state outperforms coherent states for small TPA parameters but underperforms in the intermediate regime, becoming comparable in the large absorption limit. This can be explained through fundamental differences between behaviours of even and odd number Fock states: the former’s QFI diverges in both large and small absorption limits, while the latter diverges only in the small absorption limit, dominating at intermediate scales.

Nonintegrability of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems near homo- and heteroclinic orbits

We consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and study their real-meromorphic nonintegrability in the Bogoyavlenskij sense using a generalized version due to Ayoul and Zung of the Morales-Ramis theory. The perturbation terms are assumed to have finite Fourier series in time, and the perturbed systems are rewritten as higher-dimensional autonomous systems having the small parameter as a state variable. We show that if the Melnikov functions are not constant, then the autonomous systems are not real-meromorphically integrable near homo- and heteroclinic orbits. Our result is not just an extension of previous results for homocliic orbits to heteroclinic orbits and provides a stronger conclusion than them for the case of homoclinic orbits. We illustrate the theory for two periodically forced Duffing oscillators and a periodically forced two-dimensional system.

Periodic perturbations of central force problems and an application to a restricted 3-body problem

We consider a perturbation of a central force problem of the formx¨=V′(|x|)x|x|+ε∇xU(t,x),x∈R2∖{0}, where ε∈R is a small parameter, V:(0,+∞)→R and U:R×(R2∖{0})→R are smooth functions, and U is τ-periodic in the first variable. Based on the introduction of suitable time-maps (the radial period and the apsidal angle) for the unperturbed problem (ε=0) and of an associated non-degeneracy condition, we apply an higher-dimensional version of the Poincaré–Birkhoff fixed point theorem to prove the existence of non-circular τ-periodic solutions bifurcating from invariant tori at ε=0. We then prove that this non-degeneracy condition is satisfied for some concrete examples of physical interest (including the homogeneous potential V(r)=κ/rα for α∈(−∞,2)∖{−2,0,1}). Finally, an application is given to a restricted 3-body problem with a non-Newtonian interaction.

Nonlinear statics of magneto-electro-elastic nanoplates considering flexomagnetoelectric effect based on nonlocal strain gradient theory

The nonlinear static behavior of hygrothermal magnetoelectroelastic(MEE) nanoplates considering the flexomagnetoelectric (FME) effect is researched by engaging the first-order shear deformation theory (FSDT). The constitutive equations of MEE nanoplates take into account the FME effect and hygrothermal effect. Leveraging the FSDT, von Karman's nonlinear equation and Hamilton's principle, the nonlinear control equation for hygrothermal MEE nanoplates can be derived by using the variational approach. The nonlocal strain gradient theory (NSGT) has application in the size effect of MEE nanoplates. The nonlocal nonlinear term in the control equation is handled by employing the Airy stress function, and then the non-linear mechanical model is solved by the Galerkin method. Thus, the nonlinear load-deflection curves (NLDC) of the MEE nanoplate are obtained via the introduction of material parameters, enabling an examination of the impact of various factors such as the FME effect, two small size parameters of NSGT and other parameters on the nonlinear bending behavior of MEE nanoplates. In conclusion, this study offers a theoretical foundation for taking into account the FME effect in the development of nanodevices, thereby contributing to advancements in this field.

Numerical method for second order singularly perturbed delay differential equations with fractional order in time via fitted computational method

The numerical solution of time fractional parabolic differential equations with singular perturbations and delay is the subject of this article. An arbitrarily small perturbation parameter ɛ(0<ɛ<<1) is multiplied the highest order derivative term. The problem’s solution shows an exponential boundary layer on the right side of the spatial domain for small values of ɛ. The solution and its derivatives are discussed together with their properties and bounds. The Crank–Nicolson approach for time discretization and the fitted operator non-standard finite difference method for space discretization are used to solve the time fractional singularly perturbed time delay problem under consideration. The comparison principle and solution bound are used to examine and analyze the scheme’s stability. The scheme’s uniform convergence is examined and demonstrated. The proposed scheme has a uniform convergence order of Mx−1+(Δt)2−γ. Two numerical illustrations for various values of ɛ are taken into consideration to validate the theoretical analysis of the scheme.The constructed scheme provides a better order of convergence and more accurate results than methods found in the literature.

Imaging Rovibrational Excitation of Scattered YO Molecules in Inelastic Collisions with Kr and Ne.

Energy transfer between atoms and molecules is fundamental to many physical and chemical processes, and understanding the mechanisms and outcomes of energy transfer is crucial for various applications in physics and chemistry. Here, the rovibrational excitation of YO(X 2Σ+) molecules with the collision of Kr and Ne has been studied in the laser-ablation crossed beam and time-sliced ion velocity map imaging setup in combination with the resonance enhanced multiphoton ionization scheme. Significant changes in the angular distribution for different rovibrational excitations of YO molecules are observed with the collision of Kr. The sharp forward distribution for low rovibrational excitation of YO(v' = 0, 1) molecules suggest that the weak attractive potential between Kr and YO is dominant at large impact parameters. Comparatively, the strong sideway distribution for highly rovibrationally excited YO(v' = 1, 2, 3, and 5) is due to rainbow scattering from the stronger attractive potential of Kr···YO at relatively small impact parameters. The more isotropic angular distribution in the highly rovibrationally excited YO(v' = 11) indicates the formation of a short-lived complex. A change in the angular distribution of scattered YO with different rovibrational excitations was also observed in the collisions of Ne. For YO as a heteronuclear diatomic molecule, collisions of the Y- and the O-end of YO with rare gases would affect the contribution of inelastic processes at different impact parameters.

A machine vision approach with temporal fusion strategy for concrete vibration quality monitoring

In concrete construction, ensuring the quality of vibration is paramount for maintaining the strength, durability, and quality of structures. This study proposed a method for monitoring the vibration quality of vibrating robots to replace subjective human judgment. The method employed an improved EfficientNetV2 model to classify the vibration quality into four levels: unqualified, middle, qualified, and over-vibration. A temporal fusion strategy was introduced, employing time-domain probability fusion to enhance the accuracy and stability of the results. Additionally, image patching was applied to reduce computational complexity while preserving feature integrity. Experimental results demonstrate that the proposed method outperforms common mainstream models, achieving an accuracy of 96.47%, with a relatively small parameter size of 13.8 M. Compared to non-temporal fusion strategy, the accuracy is improved by 2.06%. This research has been successfully applied in practical engineering, providing a reliable means of quality assurance for concrete structures and demonstrating potential application prospects in the field of engineering construction.

Determining the parameters of the functioning for a nonlinear ballistic system in a real external environment

Purpose. Development of an approximate nonlinear model for the solution of the external ballistics problem with determination of the nonlinear system parameters and development of a methodology for calculating the aerodynamic characteristics of the cargo, located on the external suspension of unmanned aerial vehicle (UAV) in order to increase the efficiency of their delivery to the specified landing target by means of asymptotic approach with given parameters of the studied system and external load. Methodology. The development of an effective model was carried out using analytical and numerical research algorithms based on a nonlinear system of differential equations in a general form with time-varying coefficients. In order to obtain a solution to the nonlinear problem of external ballistics in a two-dimensional formulation, the assumption of a significant influence of the projection of the velocity function on the ordinate axis in relation to the component on the abscissa axis is introduced. The problem is reduced to the solution of a related system of differential equations with variable coefficients along the corresponding coordinates using the asymptotic approach for a small parameter of the coefficient of frontal aerodynamic resistance. Applied mathematical analysis and modeling have been used for the problem formulation considering studied environmental parameters. Findings. Analytical dependences of the nonlinear problem of ballistics and application of finite-element analysis (FEA) with respect to the cargo motion from the UAV in the presence of the initial speed and wind load in the plane of motion are proposed. It is shown that the obtained analytical solution is correlated with the direct numerical calculation of the basic differential equation with respect to the ordinate axis. Originality. A mathematical nonlinear model of the dynamic process is proposed, assuming the prevailing influence of the speed function along the ordinate axis compared to the function along the abscissa axis. To obtain an approximate analytical solution of the basic nonlinear system of differential equations with variable coefficients the asymptotic perturbation method is applied. The dependence for the axial displacement function is presented considering actual time-flight parameter. Practical value. The obtained analytical dependencies for estimating the time and distance reaching the target with the initial speed of movement and the presence of wind load can be used in applied problems of mathematical physics and engineering calculations of functional dependencies and control of the cargo delivery process and target reaching from an UAV. The obtained analytical results and the solution algorithm can be integrated into applied problems of mathematical physics and engineering calculations, particularly the development of ballistic system control algorithms.

AEKA: FPGA Implementation of Area-Efficient Karatsuba Accelerator for Ring-Binary-LWE-Based Lightweight PQC

Lightweight PQC-related research and development have gradually gained attention from the research community recently. Ring-Binary-Learning-with-Errors (RBLWE)-based encryption scheme (RBLWE-ENC), a promising lightweight PQC based on small parameter sets to fit related applications (but not in favor of deploying popular fast algorithms like number theoretic transform). To solve this problem, in this article, we present a novel implementation of hardware acceleration for RBLWE-ENC based on Karatsuba algorithm, particularly on the field-programmable gate array (FPGA) platform. In detail, we have proposed an area-efficient Karatsuba Accelerator (AEKA) for RBLWE-ENC, based on three layers of innovative efforts. First of all, we reformulate the signal processing sequence within the major arithmetic component of the KA-based polynomial multiplication for RBLWE-ENC to obtain a new algorithm. Then, we have designed the proposed algorithm into a new hardware accelerator with several novel algorithm-to-architecture mapping techniques. Finally, we have conducted thorough complexity analysis and comparison to demonstrate the efficiency of the proposed accelerator, e.g., it involves 62.5% higher throughput and 60.2% less area-delay product (ADP) than the state-of-the-art design for n =512 (Virtex-7 device, similar setup). The proposed AEKA design strategy is highly efficient on the FPGA devices, i.e., small resource usage with superior timing, which can be integrated with other necessary systems for lightweight-oriented high-performance applications (e.g., servers). The outcome of this work is also expected to generate impacts for lightweight PQC advancement.

Fitted Tension Spline Scheme for a Singularly Perturbed Parabolic Problem With Time Delay

A fitted tension spline numerical scheme for a singularly perturbed parabolic problem (SPPP) with time delay is proposed. The presence of a small parameter ε as a multiple of the diffusion term leads to the suddenly changing behaviors of the solution in the boundary layer region. This results in a challenging duty to solve the problem analytically. Classical numerical methods cause spurious nonphysical oscillations unless an unacceptable number of mesh points is considered, which requires a large computational cost. To overcome this drawback, a numerical method comprising the backward Euler scheme in the time direction and the fitted spline scheme in the space direction on uniform meshes is proposed. To establish the stability and uniform convergence of the proposed method, an extensive amount of analysis is carried out. Three numerical examples are considered to validate the efficiency and applicability of the proposed scheme. It is proved that the proposed scheme is uniformly convergent of order one in both space and time. Further, the boundary layer behaviors of the solutions are given graphically.