Linear Stability Analysis Research Articles

Emergence of the reproduction matrix in epidemic forecasting.

During the recent COVID-19 pandemic, the instantaneous reproduction number, R(t), has surged as a widely used measure to target public health interventions aiming at curbing the infection rate. In analogy with the basic reproduction number that arises from the linear stability analysis, R(t) is typically interpreted as a threshold parameter that separates exponential growth (R(t) > 1) from exponential decay (R(t) < 1). In real epidemics, however, the finite number of susceptibles, the stratification of the population (e.g. by age or vaccination state), and heterogeneous mixing lead to more complex epidemic courses. In the context of the multidimensional renewal equation, we generalize the scalar R(t) to a reproduction matrix, [Formula: see text], which details the epidemic state of the stratified population, and offers a concise epidemic forecasting scheme. First, the reproduction matrix is computed from the available incidence data (subject to some a priori assumptions), then it is projected into the future by a transfer functional to predict the epidemic course. We demonstrate that this simple scheme allows realistic and accurate epidemic trajectories both in synthetic test cases and with reported incidence data from the COVID-19 pandemic. Accounting for the full heterogeneity and nonlinearity of the infection process, the reproduction matrix improves the prediction of the infection peak. In contrast, the scalar reproduction number overestimates the possibility of sustaining the initial infection rate and leads to an overshoot in the incidence peak. Besides its simplicity, the devised forecasting scheme offers rich flexibility to be generalized to time-dependent mitigation measures, contact rate, infectivity and vaccine protection.

Neural Network-Based Hypersonic Crossflow Transition Model

A model is proposed for predicting crossflow transition in hypersonic flows. The model is based on transport equations for the crossflow amplification factor and a modified intermittency. The instability onset is estimated with a correlation function. The form of the transport equations is identical to that of typical transport equations for Reynolds-averaged Navier–Stokes (RANS) turbulence models, thus facilitating an easy integration into existing RANS codes. The crossflow amplification factor is estimated from local flow quantities using a neural network. The neural network is trained with a comprehensive database of linear stability theory data. The sensitivity of the amplification factor growth rate prediction to different dimensionless input parameters is evaluated. The input parameters that provide the most-accurate growth rate predictions are selected for the final form of the model. Based on the volume of the training data set, the model is expected to perform reliably for Mach numbers between 5 and 11 and within given upper bounds of the stagnation temperature and pressure. The model is validated for the HiFiRE-5 geometry, a circular cone at angle of attack, and a blunt swept flat plate. The amplification factor distributions obtained from the model are in good agreement with experimental data and linear stability analysis results.

Prey group defense and hunting cooperation among generalist-predators induce complex dynamics: a mathematical study.

Group defense in prey and hunting cooperation in predators are two important ecological phenomena and can occur concurrently. In this article, we consider cooperative hunting in generalist predators and group defense in prey under a mathematical framework to comprehend the enormous diversity the model could capture. To do so, we consider a modified Holling-Tanner model where we implement Holling type IV functional response to characterize grazing pattern of predators where prey species exhibit group defense. Additionally, we allow a modification in the attack rate of predators to quantify the hunting cooperation among them. The model admits three boundary equilibria and up to three coexistence equilibrium points. The geometry of the nontrivial prey and predator nullclines and thus the number of coexistence equilibria primarily depends on a specific threshold of the availability of alternative food for predators. We use linear stability analysis to determine the types of hyperbolic equilibrium points and characterize the non-hyperbolic equilibrium points through normal form and center manifold theory. Change in the model parameters leading to the occurrences of a series of local bifurcations from non-hyperbolic equilibrium points, namely, transcritical, saddle-node, Hopf, cusp and Bogdanov-Takens bifurcation; there are also occurrences of global bifurcations such as homoclinic bifurcation and saddle-node bifurcation of limit cycles. We observe two interesting closed 'bubble' form induced by global bifurcations due to change in the strength of hunting cooperation and the availability of alternative food for predators. A three dimensional bifurcation diagram, concerning the original system parameters, captures how the alternation in model formulation induces gradual changes in the bifurcation scenarios. Our model highlights the stabilizing effects of group or gregarious behaviour in both prey and predator, hence supporting the predator-herbivore regulation hypothesis. Additionally, our model highlights the occurrence of "saltatory equilibria" in ecological systems and capture the dynamics observed for lion-herbivore interactions.

Convective Overstability in Radially Global Protoplanetary Disks. I. Pure Gas Dynamics

Protoplanetary disks are prone to several hydrodynamic instabilities. One candidate, convective overstability (COS), can drive radial semiconvection that may influence dust dynamics and planetesimal formation. However, the COS has primarily been studied in local models. This paper investigates the COS near the midplane of radially global disk models. We first conduct a global linear stability analysis, which shows that linear COS modes exist only radially inward of their Lindblad resonance (LR). The fastest-growing modes have LRs near the inner radial domain boundary with effective radial wavelengths that can be a substantial fraction of the disk radius. We then perform axisymmetric global simulations and find that the COS’s nonlinear saturation is similar to previous incompressible shearing box simulations. In particular, we observe the onset of persistent zonal and elevator flows for sufficiently steep radial entropy gradients. In full 3D, nonaxisymmetric global simulations, we find the COS produces large-scale, long-lived vortices, which induce the outward radial transport of angular momentum via the excitation of spiral density waves. The corresponding α-viscosity values of order 10−3 agree well with those found in previous 3D compressible shearing box simulations. However, in global disks, significant modifications to their radial structure are found, including the formation of pressure bumps. Interestingly, the COS typically generates an outward radial mass transport, i.e., decretion. We briefly discuss the possible implications of our results for planetesimal formation and for interpreting dust rings and asymmetries observed in protoplanetary disks.

Novel exact solutions to the perturbed Gerdjikov–Ivanov equation

This paper introduces the Nucci reduction method, a novel and efficient approach for deriving exact solutions to the perturbed Gerdjikov–Ivanov equation, offering a significant advancement in the field. The suggested technique involves transforming the equation into real and imaginary components prior to application. We successfully obtained four distinct exact and explicit solutions, along with the corresponding first integrals. Explanations and presentations of solutions are given in a logical manner. We derive an analytical expression for the instability gain and examine its key features using linear stability analysis. Finally, we compare the correctness of the analytical and numerical solutions. We demonstrate the robustness and stability of solitary waves through numerical simulations.

Inversion of fluid-release rates from episodic tremor and slip signals in subduction zones via a coarse-grained reaction diffusion model

Episodic Tremor and Slip (ETS) events showcase dynamic interactions of oscillatory slow slips and tremors deep within subduction zones and offer a window into Earth's internal dynamics. However, the exact mechanisms driving these events remain unresolved. This study proposes a novel approach that goes beyond traditional explanations focused on fluid pressure from mineral dehydration. Existing models often neglect the intricate interplay between fluid and rock pressures across various depths and potential fluid sources. This calls for a more comprehensive understanding of how fluid release from reactions interacts with rock deformation. The present formulation captures the interplay between fluid and solid pressures providing a more rigorous picture of ETS events. It employs a minimalistic and efficient approach based on integrating dehydration reactions. The model thereby develops a generic framework for mineral dehydration, offering an enhanced perspective of the underlying processes without the need to trace down to specific minerals. It allows a refined fit to GPS data by including high-frequency components from linear and nonlinear stability analyses, giving rise to improved correlation coefficients. Through the inclusion of the dynamic interplay between fluid and rock pressure diffusion within subduction zones, we propose a unified model of ETS events.

Transient Amplification of Broken Symmetry in Elastic Snap-Through.

A snap-through bifurcation occurs when a bistable structure loses one of its stable states and moves rapidly to the remaining state. For example, a buckled arch with symmetrically clamped ends can snap between an inverted and a natural state as the ends are released. A standard linear stability analysis suggests that the arch becomes unstable to asymmetric perturbations. Surprisingly, our experiments show that this is not always the case: symmetric transitions are also observed. Using experiments, numerics, and a toy model, we show that the symmetry of the transition depends on the rate at which the ends are released, with sufficiently fast loading leading to symmetric snap-through. Our toy model reveals that this behavior is caused by a region of the system's state space in which any initial asymmetry is amplified. The system may not enter this region when loaded fast (hence remaining symmetric), but will traverse it for some interval of time when loaded slowly, causing a transient amplification of asymmetry. Our toy model suggests that this behavior is not unique to snapping arches, but rather can be observed in dynamical systems where both a saddle-node and a pitchfork bifurcation occur in close proximity.

On the role of different nonlinear damping forms in the dynamic behavior of the generalized Beck’s column

The influence of internal and external nonlinear damping forms on the dynamics of a generalized Beck’s column, namely a visco-elastic cantilever beam, subjected to conservative and non-conservative loads at its free end, is investigated. A variational principle provides the equations of motion of the system, which are properly recast into an integro-differential form. The linear stability analysis of the system is then carried out and bifurcation points are detected in the space of parameters associated with the conservative and non-conservative loads. Starting from Hopf’s bifurcation points, a post-critical analysis, based on the Method of Multiple Scales is directly performed on the continuous system, avoiding any a-priori discretization. This method provides the bifurcation equations whose analysis reveals the double nature of nonlinear damping, which can be beneficial or detrimental in terms of stable or unstable bifurcated equilibria. It is found that both the internal and external forms of nonlinear damping can turn a supercritical instability of the system into a subcritical one, thus revealing another destabilizing effect of damping, beyond the very well-known one occurring in the linear field. Numerical simulations, grounded on a Galerkin discretization of the original system, confirm the analytical findings.

Robustness of Turing models and gene regulatory networks with a sweet spot.

Traditional linear stability analysis based on matrix diagonalization is a computationally intensive process for high-dimensional systems of differential equations, posing substantial limitations for the exploration of Turing systems of pattern formation where an additional wave-number parameter needs to be investigated. In this paper, we introduce an efficient and intuitive technique that leverages Gershgorin's theorem to determine upper limits on regions of parameter space and the wave number beyond which Turing instabilities cannot occur. This method offers a streamlined avenue for exploring the phase diagrams of other complex multi-parametric models, such as those found in gene regulatory networks in systems biology. Due to its suitability for the asymptotic limit of infinitely large systems, it predicts the existence of a sweet spot in network size for maximal Jacobian stability.

Insights on phase speed and the critical Reynolds number of falling films.

We revisit the studies of gravity-driven viscous falling films with and without imposed shear stress to provide new perspectives on phase speed and the critical Reynolds number for surface instability. We use the traditional long-wave expansion technique implemented for investigating the linear stability analysis [C.S. Yih, Phys. Fluids 6, 321 (1963)0031-917110.1063/1.1706737]. The principal purpose is to create a unified relationship between the leading-order phase speed and the critical Reynolds number that will hold for falling films on impermeable substrates with or without shear stress acting at the liquid film surface. The analytical result demonstrates that the critical Reynolds number for the onset of surface instability is [5/(2c_{0})]cotθ, where c_{0} is the leading-order phase speed of the surface mode and θ is the angle of inclination with the horizontal. Clearly, the critical Reynolds number of the surface mode is explicitly dependent on the leading-order phase speed. Furthermore, we reveal that the basic parallel flow with or without imposed shear stress is linearly unstable to infinitesimal disturbances if the modified Reynolds number, Re_{M}=(Rec_{0}/cotθ)[ReistheReynoldsnumber,andθ≠π/2], is greater than its critical value of 5/2, which is independent of the shear stress applied at the film surface. In addition, it is demonstrated that Re_{M} controls the surface instability in the long-wave regime for both shear-imposed and non-shear-imposed film flows.

Modulation instability and extraction of fractional optical solitons in the presence of generalized Kudryashov’s law and dual form of non-local nonlinearity

This research delves into the study of optical solitons governed by Kudryashov’s law of nonlinear refractive index, incorporating a conformable fractional derivative, perturbed terms, and a quadrupled-power law in optical fibers. Employing a novel approach known as the generalized exp(− S(ζ) ) expansion method, we identify novel optical soliton solutions characterized by exponential, rational, and hyperbolic functions featuring a conformable fractional derivative. These solutions manifest as unified bright-dark, singular, singular periodic, kink, anti-kink, and novel solitary waves. We employ 3D, contour, and 2D plots to analyze the behavior of these solutions across different temporal and fractional-order derivative parameters, offering an understanding of their physical interpretations. Furthermore, we perform modulation instability (MI) analysis that depends on standard linear stability analysis to derive the MI gain spectrum, demonstrating how well our methodology works for nonlinear problems in the natural sciences and engineering domains. This approach provides an effective way to investigate optical solutions in various integer and fractional Schrödinger equations.

Onset of thermal convection in a solid spherical shell with melting at either or both boundaries

SUMMARY Thermal convection in planetary solid (rocky or icy) mantles sometimes occurs adjacent to liquid layers with a phase equilibrium at the boundary. The possibility of a solid–liquid phase change at the boundary has been shown to greatly help convection in the solid layer in spheres and plane layers and a similar study is performed here for a spherical shell with a radius-independent central gravity subject to a destabilizing temperature difference. The solid–liquid phase change is considered as a mechanical boundary condition and applies at either or both horizontal boundaries. The boundary condition is controlled by a phase change number, Φ, that compares the timescale for latent heat exchange in the liquid side to that necessary to build a topography at the boundary. We introduce a numerical tool, available at https://github.com/amorison/stablinrb, to carry out the linear stability analysis of the studied setup as well as other similar situations (Cartesian geometry, arbitrary temperature and viscosity depth-dependent profiles). Decreasing Φ makes the phase change more efficient, which reduces the importance of viscous resistance associated to the boundary and makes the critical Rayleigh number for the onset of convection smaller and the wavelength of the critical mode larger, for all values of the radii ratio, γ. In particular, for a phase change boundary condition at the top or at both boundaries, the mode with a spherical harmonics degree of 1 is always favoured for Φ ≲ 10−1. Such a mode is also favoured for a phase change at the bottom boundary for small (γ ≲ 0.45) or large (γ ≳ 0.75) radii ratio. Such dynamics could help explaining the hemispherical dichotomy observed in the structure of many planetary objects.

Richtmyer-Meshkov instability when a shock wave encounters with a premixed flame from the burned gas

We present a linear stability analysis of the Richtmyer-Meshkov instability that develops when a shock wave reaches a sinusoidally perturbed premixed flame from behind. In the hydrodynamic regime, when acoustic contributions dominate the flame growth rate, the problem is analytically addressed by the direct integration of the sound wave equations at both sides of the flame, which are bounded by the reflected and transmitted shock waves and the flame front that acts as a contact surface in the hydrodynamic limit. The resolution involves: a hyperbolic change of variables to modify the triangular spatio-temporal domain, a transformation in the Laplace variable, the resolution of the functional equations in the frequency domain, and the final inverse Laplace transform. The latter involves a novel resolution method that is proven beneficial for long-time dynamics. Asymptotic analysis is also carried out to describe the early time and late time hydrodynamic response. The nonuniform flow field resulting from distorted oscillating shocks is characterized by acoustic, rotational, and entropic disturbances, each of which exerts a substantial influence on flame dynamics. These disturbances contribute to the intricate interplay of factors shaping the behavior of the flame in response to the nonuniform flow. In particular, the sensitivity of local flame propagation to temperature disturbances is investigated. This work contributes to a deeper understanding of Richtmyer-Meshkov instability dynamics and offers insights into instability reduction through the modulation of temperature disturbances generated by the transmitted shock.

Linear temporal stability of Jeffery–Hamel flow of nanofluids

Flow stability plays a key role in transition to turbulence in various systems. This transition initiates with disturbances appearing in the laminar base flow, potentially amplifying over time based on flow and fluid parameters. In response to these amplified disturbances, the flow undergoes successive stages of different laminar flows, ultimately transitioning to turbulence. One influential parameter affecting flow stability is the nanoparticle volume fraction (ϕ) in nanofluids, extensively employed in thermofluid systems like cooling devices to enhance fluid thermal conductivity and the heat transfer coefficient. Focusing on the impact of nanoparticles on Jeffery–Hamel flow stability, this study assumes fluid properties are temperature- and pressure-independent, exclusively examining the momentum transfer aspect. The analysis commences by deriving the base laminar flow solution. Subsequently, linear temporal stability analysis is employed, imposing infinitesimally-small perturbations on the base flow as a modified form of normal modes. A generalized Orr–Sommerfeld equation is derived and solved using a spectral method. Results indicate that, assuming nanofluid viscosity as μnf=μf/(1−ϕ)2.5, nanoparticle effects on momentum transfer and flow stability hinge on the ratio of nano-solid particle density to base fluid density (Rρ=ρs/ρf). For ϕ∈(0,0.1], flow stabilization occurs with ϕ when Rρ<3.5000, while destabilization is observed when Rρ>4.0135. Notably, nanoparticles exhibit a negligible impact on flow stability when 3.5000≤Rρ≤4.0135.

Generalized naming game and Bayesian naming game as dynamical systems.

We study the β model (β-NG) and the Bayesian Naming Game (BNG) as dynamical systems. By applying linear stability analysis to the dynamical system associated with the β model, we demonstrate the existence of a nongeneric bifurcation with a bifurcation point β_{c}=1/3. As β passes through β_{c}, the stability of isolated fixed points changes, giving rise to a one-dimensional manifold of fixed points. Notably, this attracting invariant manifold forms an arc of an ellipse. In the context of the BNG, we propose modeling the Bayesian learning probabilities p_{A} and p_{B} as logistic functions. This modeling approach allows us to establish the existence of fixed points without relying on the overly strong assumption that p_{A}=p_{B}=p, where p is a constant.

Some optical solitons and modulation instability analysis of (3 + 1)-dimensional nonlinear Schrödinger and coupled nonlinear Helmholtz equations

In this study, we acquired some optical solitons of (3 + 1) dimensional nonlinear Schrödinger equation (3DNLSE) and coupled nonlinear Helmholtz equations (CNLHE) by using generalized G′G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left(\\frac{{G}^{\\prime}}{G}\\right)$$\\end{document}-expansion method (GEM). We are able to derive exponential, trigonometric, and hyperbolic solutions. We notice that Mathematica 11.2 offer the equations for these answers. Aside from that, we display some of the solution’s graphic performance. Recently, accurate traveling-wave solutions for nonlinear partial differential equations (NLPDEs) have been acquired using this technique. Through linear stability analysis, we derive an analytical expression for the instability gain and analyze its main characteristics. Finally, we assess the robustness and stability of the solitary waves through numerical simulations and compare the accuracy of the analytical and numerical results. The obtained ultra-short optical solitons can be widely used as a guide for practical applications in telecommunications domain.

Aeroelastic code comparison using the IEA 22MW reference turbine

Reference wind turbine designs and the associated aeroelastic models are widely used in both research and industry. Reference models representing future concepts are of particular interest. Current state of the art aeroelastic tools are relied upon to design the next generation of large wind turbines. However, modelling assumptions may be invalidated by upcoming very large turbines, and different aeroelastic tools may give inconsistent results. A 22MW turbine model has been defined as part of International Energy Agency (IEA) Wind Task 55 on Reference Wind Turbines and Farms to represent future turbines to be deployed in the 2030s. In this study, an aeroelastic model of this turbine has been created in four tools; Bladed, HAWC2, OpenFAST, and QBlade. Code comparisons are presented for steady state operation, linear stability analysis, and time domain power production simulations in steady and turbulent wind. Generally, the codes show a good agreement, but with some differences present in the linear stability analysis, periodic azimuthal variation, and time domain simulations. The models are a good basis for further study with the IEA 22MW turbine, and further code comparison exercises.

Hydromagnetic Thermosolutal Instability of an Oldroyd Nanofluid in a Porous Medium

The onset of double-diffusive convection of an Oldroyd nanofluid fluid saturated in a porous medium, heated and soluted from below, is examined both analytically and numerically under the linear stability analysis in the presence of a horizontal magnetic field and physically realistic boundary condition on the volume fraction of nanoparticles. The resulting eigenvalue problem is solved numerically by employing Galerkin method. We recover the important works of Umavathi and Prathap Kumar (J. C. Umavathi and J. Prathap Kumar, J. Heat Transfer 139, 012401 (2017)), Jaimala et al. (R. Jaimala, Singh, and V. K. Tyagi, Int. J. Heat Mass Transfer 111, 451 (2017); Jaimala, R. Singh, and V. K. Tyagi, Int. J. Heat Mass Transfer 125, 290 (2018)) and Kuznetsov and Nield (A. V. Kuznetsov and D. A. Nield, International Journal of Thermal Sciences 77, 126 (2014)) as special cases. The important findings include: (i) the independence of thermal Rayleigh-Darcy number for stationary convection upon relaxation and retardation parameters, (ii) the parameters Rn, Rs, Le, NA and λ1 (absent in case of stationary convection) enhance both the stationary and oscillatory convections and the parameter Q, ε and λ2 (absent in case of stationary convection) delay the occurrence of both convections, (iii) instability first sets in as oscillatory convection and (iv) a number of sufficient conditions for the existence and also for the non-existence of oscillatory convection are obtained analytically but only stated (without proof).

Heat effects on supersonic jet screech: a linear stability analysis based on parabolized stability equations

In this paper, we use the parabolized stability equations (PSEs) to investigate heat effects on supersonic jet screech. The PSE is derived from the linear Euler equations, and the computations are performed on an empirical mean flow profile. Employing PSE, we can examine several important characteristics of the shear-layer instability waves, including the convection velocity, the growth rate, and the location where the instability waves attain the maximal total amplification. Equipped with these knowledge, we further explore their impacts on the jet screech. Specifically, using a newly proposed iteration scheme, we first identify a more suitable convective Mach number to predict the screech frequency in both heated and cold jets. Second, we find that the transition of the screech mode from the axisymmetric to the helical mode may be explained by the shift in the most amplified instability modes. Using this criterion, we can predict a transition Mach number for the mode staging. Third, by assuming that the effective source location is the position where the instability wave attains its maximal amplitude and using the newly obtained convective Mach number, we can predict the screech mode staging in cold and heated jets using Gao's model. The predictions agree well with the experimental data. Finally, we investigate the heat effects on screech amplitude. It is found that compared to cold jets, the difference between the frequency of the most amplified instability wave and the frequency of the jet screech is much bigger in heated jets, which may lead to a lower screech amplitude.

Bifurcation analysis in the regularized four-sided cavity flow: Equilibrium states in a D2-symmetric fluid system

In this study, we investigate a handful of equilibrium states and their hydrodynamic stability for the incompressible flow in a square cavity driven by the tangential simultaneous motion of all of its lids at the same speed. This problem has been investigated in the recent past, albeit only partially and always disregarding the singular boundary conditions at the corners. The implications of the system symmetries have not been rigorously addressed, either. In our study, we introduce a regularized version of the boundary conditions to avoid the corner singularities, in a way that preserves the main features of the original setup. In addition, the Navier–Stokes equations are discretized by means of highly accurate Chebyshev spectral methods that provide exponential convergence of all computed flows and consistently eliminate any potential source of structural instability of the bifurcation scenario. We employ Newton–Krylov solvers, implemented within continuation algorithms, to accurately compute equilibrium solutions. Linear stability analysis of both the primary symmetric base flow and the secondary asymmetric states uncovers new branches of fully asymmetric steady states. The analysis has allowed identification of six previously undetected bifurcations, all of which are associated with the disruption of either the rotational invariance or the reflection symmetry. Some of these bifurcations have been found to be quite clustered in some regions of the parameter space, which points at the underlying action of higher codimension mechanisms. Notably, all the bifurcations reported occur within the range of low to moderate Reynolds numbers, making the regularized four-sided lid-driven cavity flow a reliable benchmark for assessing different numerical schemes in the context of Navier–Stokes equivariant bifurcation theory.