Perturbation Analysis Research Articles

On the Computation of Floquet Multipliers for Periodic Solution in Piecewise-smooth Dynamical System

The floquet multiplier is one of the most important indicators for the stability and bifurcation analysis for periodic solutions in nonlinear dynamical systems. Different from the well-established Floquet theory for the perturbation systems of smooth systems, much less has been understood in its counterpart for non-smooth systems. Here in this paper, we will report an unusual and interesting feature of the Floquet multipliers for piecewise-smooth dynamical systems. When the initial condition of the periodic solution is located at the boundary splitting the solution domain, the multipliers would be calculated falsely in certain circumstances, respectively, by a saltation matrix method or a direct numerical integration for the perturbation system. We elucidate the origin of the fake multipliers through perturbation analysis, and furthermore suggest an effective manner to avoid the miscalculation. This finding would be of fundamental significance to both the real-world applications and theory establishment of the Floquet theory in non-smooth systems

Relevant interval perturbation method for uncertainty analysis of structural–acoustic coupling model

The parameters in structural–acoustic coupling models are often coupled with each other, and the independence of the uncertain parameters is a necessary condition for using the traditional interval perturbation method. This article proposes a relevant interval perturbation method for the uncertainty analysis of a structural–acoustic coupling model with uncertain-but-bounded parameters. Aiming at the linear inequality constraint relationship in the uncertain parameters, this method introduces a sensitivity ranking mechanism for the uncertainty parameters, which overcomes the limitation of uncertain parameter independence in the traditional interval model, and effectively restrains the interval expansion phenomenon of the system response in the uncertainty analysis. Through numerical examples, it is demonstrated that the relevant interval perturbation analysis method can effectively solve a constrained problem when there are uncertain parameters, and the obtained response interval radius is smaller than that of the traditional method under constraint conditions.

Development of a mathematical model for the prediction of drill quality in voice-activated ultrasonic vibration assisted drilling process.

In different industries, drilling quality play an important role in the success of operations by producing millions of quality drilled parts. The production of high-quality parts at the lowest possible cost to the manufacturer is the primary goal of contemporary process and assembly industries. To assess the quality index of drilling holes; circularity index and surface roughness are important metrics. In order to produce better drilled parts at lower prices, this study work attempts to determine the key process variables using voice acting modules to investigate the induced vibrations effects that have the greatest impact on surface roughness and circularity index in a vertical CNC drilling machine. In order to forecast the surface roughness and circularity index of drilled holes within a given operational range, a mathematical model has been proposed by adopting a coupled algorithm using design expert and statistical approach. As an experimental design technique, Response Surface Methodology with small central composite design is used to enable the evaluation of various process parameters related with machining and vibration parameters and their effects on surface roughness and circularity index. The created model is improved using statistical tools to ensure the best match to the experimental data. It has been observed with the increase of RPM and vibration frequency the surface roughness increases up to a certain limit and then decrease whereas with the increase feed the surface roughness increases. But in terms of circularity index with the increase of the process parameters the plotted graph trends under perturbation analysis shows the similar pattern CI decreases up to a certain limit and then increases.

Dynamics and thermal stability of the bypass polymerase, DinB homolog (Dbh).

The DinB homolog polymerase (Dbh) is a member of the Y-family of translesion DNA polymerases that can synthesize using a damaged DNA template. Since Dbh comes from the thermophilic archaeon Sulfolobus acidocaldarius, it is capable of functioning over a wide range of temperatures. Existing X-ray structures were determined at temperatures where the protein is least active. Here we use NMR and circular dichroism to understand how the structure and dynamics of Dbh are affected by temperature (2°C-65°C) and metal ion binding in solution. We measured hydrogen exchange protection factors, temperature coefficients, and chemical shift perturbations with and without magnesium and manganese. We report on regions of the protein that become more dynamic as the temperature is increased toward the functional temperature. Hydrogen exchange protection factors and temperature coefficients reveal that both the thumb and finger domains are very dynamic relative to the palm and little-finger (LF) domains. These trends remain true at high temperature with dynamics increasing as temperatures increase from 35°C to 50°C. Notably, NMR spectra show that the Dbh tertiary structure cold denatures beginning at 25°C and increases in denaturation as the temperature is lowered to 5°C with little change observed by CD. Above 35°C, chemical shift perturbation analysis in the presence and absence of magnesium and manganese reveals three ion binding sites, without DNA bound. In contrast, these bound metals are not apparent in any Dbh crystal structures of the protein without DNA. Two ion binding sites are confirmed to be near the active site, as reported in other Y-family polymerases, and we report a novel ion binding site in the LF domain. Thus, the solution-state structure of the Dbh polymerase is distinct from that of the solid-state structures and shows an unusually high cold denaturation temperature.

Quantum soft waveguides with resonances induced by broken symmetry

We consider two-dimensional, non-relativistic quantum system with asymptotically straight soft waveguide. We show that the local deformation of the symmetric waveguide can lead to the emerging of the embedded eigenvalues living in the continuous spectrum. The main problem of this paper is devoted to the analysis of weak perturbation of the symmetric system. We show that the original embedded eigenvalues turn to the second sheet of the resolvent analytic continuation and constitute resonances. We describe the asymptotics of the real and imaginary components of the complex resonant pole depending on deformation. Finally, we generalize the problem to three dimensional system equipped with a soft layer.

A note on the paper “A note on inversion of Toeplitz matrices” by X.-G. Lv and T.-Z. Huang

In this note, we revisit the result of the Toeplitz matrix inversion formula proposed by Lv and Huang (2007), then we give a new structured perturbation analysis, which is useful for testing the stability of practical algorithms. It has been shown that our new upper bound is much sharper than the one they have given, especially when the size of the Toeplitz matrix is very large. Numerical experiments illustrate the effectiveness of our theoretical results.

Estimating Posterior Sensitivities with Application to Structural Analysis of Bayesian Vector Autoregressions

The inherent feature of Bayesian empirical analysis is the dependence of posterior inference on prior parameters, which researchers typically specify. However, quantifying the magnitude of this dependence remains difficult. This article extends Infinitesimal Perturbation Analysis, widely used in classical simulation, to compute asymptotically unbiased and consistent sensitivities of posterior statistics with respect to prior parameters from Markov chain Monte Carlo inference via Gibbs sampling. The method demonstrates the possibility of efficiently computing the complete set of prior sensitivities for a wide range of posterior statistics, alongside the estimation algorithm using Automatic Differentiation. The method’s application is exemplified in Bayesian Vector Autoregression analysis of fiscal policy in U.S. macroeconomic time series data. The analysis assesses the sensitivities of posterior estimates, including the Impulse response functions and Forecast error variance decompositions, to prior parameters under common Minnesota shrinkage priors. The findings illuminate the significant and intricate influence of prior specification on the posterior distribution. This effect is particularly notable in crucial posterior statistics, such as the substantial absolute eigenvalue of the companion matrix, ultimately shaping the structural analysis.

Unveiling the nature of gravitational-wave emission in core-collapse supernovae with perturbative analysis

Unveiling the nature of gravitational-wave emission in core-collapse supernovae with perturbative analysis

Perturbative analysis of quasiperiodic patterning of transmon quantum computers: Enhancement of many-body localization

Perturbative analysis of quasiperiodic patterning of transmon quantum computers: Enhancement of many-body localization

Mechanochemical modeling of morphogenesis in cell polarization for budding yeast

Cell polarization is a multiphysics problem resulting from coupling protein pathways and cell morphological evolution. The most common example is asexual reproduction in budding yeast through Cdc42 and septin signals. However, how the interaction between the Cdc42-septin systems and the mechanical deformation of the cell surface is not well understood. To explore the interaction, we build a three-dimensional mechanochemical coupling framework to investigate the Cdc42-septin systems and elucidate how morphogenesis at the cell scale enhances the robustness of cell polarization in budding yeast. We utilize the viscous active shell model to describe the moving boundary of the cell surface, and the mechanochemical coupling is achieved through the introduction of cell surface mean curvature and active contraction-growth effects. With a normal effect of septin, numerical results demonstrate the budding processes with different transition shapes of budding profiles. On the other hand, the mechanochemical coupling problem with a weak effect of septin drives the emergence of wrinkles on the cell surface. We also apply linear perturbation analysis of the cell surface deformation to show how cell shape evolution is subjected to active contraction-growth effects. Our results show that a 3D coupled mechanochemical model can reproduce the budding morphology in yeast. Such a model provides a more comprehensive portrait of cell polarity and can be used to characterize both wild-type and mutant phenotypes such as septin mutants.

Structural determinants for membrane binding of the EGFR juxtamembrane domain.

Overactivation of the epidermal growth factor receptor (EGFR) is critical for the development of multiple cancers. Previous studies have shown that the cell membrane is a key regulator of EGFR kinase activity through its interaction with the EGFR juxtamembrane domain (JM). However, the lipid recognition specificity of EGFR-JM and its interaction details remain unclear. Using lipid strip and liposome pulldown assays, we showed that EGFR-JM could specifically interact with PI(4,5)P2-or phosphatidylserine-containing membranes. We further characterized the JM-membrane interaction using NMR-titration-based chemical shift perturbation and paramagnetic relaxation enhancement analyses, and found that residues I649 - L659 comprised the membrane-binding site. Furthermore, the membrane-binding region contains the predicted dimerization motif of JM, 655LRRLL659, suggesting that membrane binding may affect JM dimerization and, therefore, regulate kinase activation.

Efficient numerical calculation of Lyapunov-exponents and stability assessment for quasi-periodic motions in nonlinear systems

Investigating the stability of stationary motions is a highly relevant aspect when characterizing dynamical systems. For equilibria and periodic motions, well established theories and approaches exist to assess their stability: in both cases stability may be assessed using eigenvalue analyses of small perturbations. When it comes to quasi-periodic motions, such eigenvalue analyses are not applicable, since these motions can not be parameterized on finite time intervals. However, quasi-periodic motions can be densely embedded on finite invariant manifolds with periodic boundaries. In this contribution, a new approach is presented, which exploits this embedding in order to derive a sequence of finite mappings. Based on these mappings, the spectrum of 1st order Lyapunov-exponents is efficiently calculated. If the linearization of the problem is regular in the sense of Lyapunov, these exponents may be used to assess stability of the investigated solution. Beyond the numerical calculation of Lyapunov-exponents, an approach is presented which allows to check Lyapunov-regularity numerically. Together, both methods allow for an efficient numerical stability assessment of quasi-periodic motions. To demonstrate, verify and validate the developed approach, it is applied to quasi-periodic motions of two coupled van-der-Pol oscillators as well as a quasi-periodically forced Duffing equation. Additionally, a “step-by-step application instruction” is provided to increase comprehensibility and to discuss the required implementation steps in an applied context.

The Impact of Ambipolar Diffusion on the Rossby Wave Instability in a Protoplanetary Disk

Recent observational and simulation studies have revealed that ambipolar diffusion is an important phenomenon in the outer regions of a protoplanetary disk (PPD). However, numerous simulation studies have found that ambipolar diffusion suppresses the turbulence caused by the magnetorotational instability (MRI) in these regions of a PPD. The aim of this study is to investigate the impact of ambipolar diffusion on the Rossby wave instability (RWI) at large radii of a PPD. To accomplish this, we examine the occurrence of the RWI in a PPD threaded by the magnetic field in the presence of ambipolar diffusion. Additionally, we scale the ambipolar diffusivity with respect to both the toroidal magnetic field and an important parameter known as the Elsässer number. We obtain the growth rate of unstable RWI modes in the outer regions of a PPD using linear perturbation analysis. In our nonaxisymmetric perturbation analysis, we find that the amplitude of the toroidal field oscillates in various modes for small values of the Elsässer numbers. For small Elsässer numbers, the growth rate of unstable modes associated with the RWI decreases. In other words, ambipolar diffusion suppresses the perturbation caused by the RWI. This effect is similar to the effect of ambipolar diffusion on the MRI. In contrast to the MRI, where there is a range around 1 for the Elsässer numbers, the existence of ambipolar diffusion supports the occurrence of RWI. Finally, we compare our findings with those of simulation studies to emphasize the importance of RWI in the outer regions of a PPD.

Apparent Diachroneity of Calcareous Nannofossil Datums During the Early Eocene in the High‐Latitude South Pacific Ocean

AbstractThe late Paleocene to early Eocene interval is characterized by a series of carbon perturbations that caused transient warming (hyperthermal) events, of which the Paleocene‐Eocene Thermal Maximum (PETM) was the largest. These hyperthermals can be recognized in the pelagic sedimentary record as paired negative δ13C and δ18O excursions, in addition to decreased calcium carbonate and increased iron content caused by carbonate dissolution. However, current data are predominantly sourced from the equatorial‐to subequatorial regions. Here we present a new high‐latitude late Paleocene—early Eocene record, recovered during International Ocean Discovery Program (IODP) Expedition 378 on the Campbell Plateau off New Zealand, in the southwest Pacific Ocean. To construct an age model, we correlated our chemostratigraphic and biostratigraphic data to existing astronomically‐tuned age models from Walvis Ridge (South Atlantic Ocean) and Demerara Rise (equatorial Atlantic Ocean). Our results indicate that the Site U1553 composite section spans ∼7 million years of the latest Paleocene to early Eocene (50.5–57.5 Ma), and preserves many of the early Eocene hyperthermals; including a PETM interval that is more expanded than elsewhere in this region. However, construction of the age model also revealed discrepancies between the chemostratigraphic and biostratigraphic tie points used for correlation. This is likely due to latitudinal diachroneity in the calcareous nannofossil biostratigraphic datums, which are primarily based on low‐to mid‐latitude assemblages. Therefore, our study highlights the need to establish a revised calcareous nannofossil biozonation that is more appropriate for high‐latitude age models.

TDiff invariant field theories for cosmology

We study scalar field theories invariant under transverse diffeomorphismsin cosmological contexts. We show that in the geometric optics approximation, the corresponding particles move along geodesics and contribute with the same active mass (energy) to the gravitational field as in Diff invariant theories. However, for low-frequency (super-Hubble) modes, the contributions to the energy-momentum tensor differ from that of Diff invariant theories. This opens up a wide range of possibilities for cosmological model building. As an example, we show that the simplest TDiff invariant scalar field theory with only kinetic term could drive inflation and generate a nearly scale invariant (red-tilted) spectrum of density fluctuations. We also present a detailed analysis of cosmological perturbations and show that the breaking of full Diff invariance generically induces new non-adiabatic pressure perturbations. A simple scalar field dark matter model based on a purely kinetic term that exhibits the same clustering properties as standard cold dark matter is also presented.

Conformal transformations and cosmological perturbations in New General Relativity

Abstract We study conformal transformations in the most general parity-preserving models of the New General Relativity type. Then we apply them to analysis of cosmological perturbations in the (simplest) spatially flat cosmologies. Strong coupling issues around Minkowski spacetime are seen for many special cases of these models. At the same time, the behaviour of the most general three-parameter case seems to be very robust, presumably always with only the eight first-class constraints coming from diffeomorphisms. Also the case of the so-called 1-parameter New GR doesn't show any discontinuity between Minkowski and the cosmology, though without showing any deviations from GR which would be observable at this level either.

A theoretical study on the bulk viscosity in a compressible flow of an electrically conducting gas in the presence of magnetoacoustic waves

Only a few studies in the current literature are dedicated to the analysis of the effect of a bulk viscosity (i.e., second viscosity coefficient), and little is known about how to determine this quantity for high frequency compressible flows involving magnetic effect. The dissipation phenomenon associated with rapid expansions in the flow and the consequent attenuation of magnetoacoustic waves by the bulk viscosity has been neglected by several works on gas dynamics at high frequency. In this paper, we present a theoretical study on a compressible flow of an electrically conducting barotropic gas in the presence of a bulk viscosity. The governing equations represent a coupling between hydrodynamics and Maxwell's equations in the context of magnetohydrodynamics (MHD), and the relevant physical parameters of the flow are presented after an appropriate dimensional analysis of these equations. First, an analysis of small perturbations around an equilibrium state of the electrically conducting gas results in a system of linearized equations in the frequency-wavenumber space. The dispersion relation for magnetic waves is determined in terms of magnetic and the bulk viscosity effects. Second, based on the dispersion relation for magnetoacoustic waves, we propose a new expression to estimate the bulk viscosity in terms of the gas barotropic coefficient, the wavenumber, the wave frequency, the orientation of the applied field, and the Euler magnetic parameter, which measures the relative importance between magnetic and thermodynamic pressures. The behavior of the bulk viscosity is examined as a function of the wavenumber for different magnetic Euler numbers and field orientation, revealing how the rate of energy dissipation associated with the bulk viscosity can be controlled by varying the intensity of an applied magnetic field and its orientation. We show that understanding magnetoacoustic waves not only provides a tool for estimating the bulk viscosity in plasma flows in terms of these wave parameters, but also offers a potential pathway to manipulating both the magnitude and orientation of a magnetic field in order to reduce the rate of energy dissipation in most of plasma flows. This rate of dissipation produced by the rapid expansion of a recombining gas can be orders of magnitude larger than the one produced by the standard shear viscosity. These insights into bulk viscosity in compressible MHD flow at high frequency have potential applications in reacting plasmas where magnetoacoustic waves are damped by bulk viscosity, in production and dissipation of turbulence and in phenomenon of sonoluminescence as occurs in liquid containing gas bubble oscillating at fairly high frequency.

A multiple-timing analysis of temporal ratcheting

We develop a two-timing perturbation analysis to provide quantitative insights on the existence of temporal ratchets in an exemplary system of a particle moving in a tank of fluid in response to an external vibration of the tank. We consider two-mode vibrations with angular frequencies ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} and αω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\omega $$\\end{document}, where α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} is a rational number. If α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} is a ratio of odd and even integers (e.g., 21,32,43\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ frac{2}{1},\\,\ frac{3}{2},\\,\ frac{4}{3}$$\\end{document}), the system yields a net response: here, a nonzero time-average particle velocity. Our first-order perturbation solution predicts the existence of temporal ratchets for α=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =2$$\\end{document}. Furthermore, we demonstrate, for a reduced model, that the temporal ratcheting effect for α=32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =\ frac{3}{2}$$\\end{document} and 43\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ frac{4}{3}$$\\end{document} appears at the third-order perturbation solution. More importantly, we find closed-form formulas for the magnitude and direction of the induced net velocities for these α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} values. On a broader scale, our methodology offers a new mathematical approach to study the complicated nature of temporal ratchets in physical systems.Graphic abstract

Single-cell biclustering for cell-specific transcriptomic perturbation detection in AD progression

The pathogenesis of Alzheimer disease (AD) involves complex gene regulatory changes across different cell types. To help decipher this complexity, we introduce single-cell Bayesian biclustering (scBC), a framework for identifying cell-specific gene network biomarkers in scRNA and snRNA-seq data. Through biclustering, scBC enables the analysis of perturbations in functional gene modules at the single-cell level. Applying the scBC framework to AD snRNA-seq data reveals the perturbations within gene modules across distinct cell groups and sheds light on gene-cell correlations during AD progression. Notably, our method helps to overcome common challenges in single-cell data analysis, including batch effects and dropout events. Incorporating prior knowledge further enables the framework to yield more biologically interpretable results. Comparative analyses on simulated and real-world datasets demonstrate the precision and robustness of our approach compared to other state-of-the-art biclustering methods. scBC holds potential for unraveling the mechanisms underlying polygenic diseases characterized by intricate gene coexpression patterns.

CPADS: a web tool for comprehensive pancancer analysis of drug sensitivity.

Drug therapy is vital in cancer treatment. Accurate analysis of drug sensitivity for specific cancers can guide healthcare professionals in prescribing drugs, leading to improved patient survival and quality of life. However, there is a lack of web-based tools that offer comprehensive visualization and analysis of pancancer drug sensitivity. We gathered cancer drug sensitivity data from publicly available databases (GEO, TCGA and GDSC) and developed a web tool called Comprehensive Pancancer Analysis of Drug Sensitivity (CPADS) using Shiny. CPADS currently includes transcriptomic data from over 29 000 samples, encompassing 44 types of cancer, 288 drugs and more than 9000 gene perturbations. It allows easy execution of various analyses related to cancer drug sensitivity. With its large sample size and diverse drug range, CPADS offers a range of analysis methods, such as differential gene expression, gene correlation, pathway analysis, drug analysis and gene perturbation analysis. Additionally, it provides several visualization approaches. CPADS significantly aids physicians and researchers in exploring primary and secondary drug resistance at both gene and pathway levels. The integration of drug resistance and gene perturbation data also presents novel perspectives for identifying pivotal genes influencing drug resistance. Access CPADS at https://smuonco.shinyapps.io/CPADS/ or https://robinl-lab.com/CPADS.