Journal articles on the topic 'Slow-Fast asymptotic analysis'
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1
Pan, Shing-Tai, Ching-Fa Chen, and Jer-Guang Hsieh. "Stability Analysis for a Class of Singularly Perturbed Systems With Multiple Time Delays." Journal of Dynamic Systems, Measurement, and Control 126, no. 3 (September 1, 2004): 462–66. http://dx.doi.org/10.1115/1.1793172.
Abstract:
The paper is to investigate the asymptotic stability for a general class of linear time-invariant singularly perturbed systems with multiple non-commensurate time delays. It is a common practice to investigate the asymptotic stability of the original system by establishing that of its slow subsystem and fast subsystem. A frequency-domain approach is first presented to determine a sufficient condition for the asymptotic stability of the slow subsystem (reduced-order model), which is a singular system with multiple time delays, and the fast subsystem. Two delay-dependent criteria, ε-dependent and ε-independent, are then proposed in terms of the H∞-norm for the asymptotic stability of the original system. Furthermore, a simple estimate of an upper bound ε* of singular perturbation parameter ε is proposed so that the original system is asymptotically stable for any ε∈0,ε*. Two numerical examples are provided to illustrate the use of our main results.
2
Nave, OPhir, Israel Hartuv, and Uziel Shemesh. "Θ-SEIHRD mathematical model of Covid19-stability analysis using fast-slow decomposition." PeerJ 8 (September 21, 2020): e10019. http://dx.doi.org/10.7717/peerj.10019.
Abstract:
In general, a mathematical model that contains many linear/nonlinear differential equations, describing a phenomenon, does not have an explicit hierarchy of system variables. That is, the identification of the fast variables and the slow variables of the system is not explicitly clear. The decomposition of a system into fast and slow subsystems is usually based on intuitive ideas and knowledge of the mathematical model being investigated. In this study, we apply the singular perturbed vector field (SPVF) method to the COVID-19 mathematical model of to expose the hierarchy of the model. This decomposition enables us to rewrite the model in new coordinates in the form of fast and slow subsystems and, hence, to investigate only the fast subsystem with different asymptotic methods. In addition, this decomposition enables us to investigate the stability analysis of the model, which is important in case of COVID-19. We found the stable equilibrium points of the mathematical model and compared the results of the model with those reported by the Chinese authorities and found a fit of approximately 96 percent.
3
Thomas, Jim. "Resonant fast–slow interactions and breakdown of quasi-geostrophy in rotating shallow water." Journal of Fluid Mechanics 788 (January 8, 2016): 492–520. http://dx.doi.org/10.1017/jfm.2015.706.
Abstract:
In this paper we investigate the possibility of fast waves affecting the evolution of slow balanced dynamics in the regime $Ro\sim Fr\ll 1$ of a rotating shallow water system, where $Ro$ and $Fr$ are the Rossby and Froude numbers respectively. The problem is set up as an initial value problem with unbalanced initial data. The method of multiple time scale asymptotic analysis is used to derive an evolution equation for the slow dynamics that holds for $t\lesssim 1/(fRo^{2})$, $f$ being the inertial frequency. This slow evolution equation is affected by the fast waves and thus does not form a closed system. Furthermore, it is shown that energy and enstrophy exchange can take place between the slow and fast dynamics. As a consequence, the quasi-geostrophic ideology of describing the slow dynamics of the balanced flow without any information on the fast modes breaks down. Further analysis is carried out in a doubly periodic domain for a few geostrophic and wave modes. A simple set of slowly evolving amplitude equations is then derived using resonant wave interaction theory to demonstrate that significant wave-balanced flow interactions can take place in the long-time limit. In this reduced system consisting of two geostrophic modes and two wave modes, the presence of waves considerably affects the interactions between the geostrophic modes, the waves acting as a catalyst in promoting energetic interactions among geostrophic modes.
4
Zhou, Yanli, Shican Liu, Shuang Li, and Xiangyu Ge. "The Correction of Multiscale Stochastic Volatility to American Put Option: An Asymptotic Approximation and Finite Difference Approach." Journal of Function Spaces 2021 (September 17, 2021): 1–14. http://dx.doi.org/10.1155/2021/1217665.
Abstract:
It has been found that the surface of implied volatility has appeared in financial market embrace volatility “Smile” and volatility “Smirk” through the long-term observation. Compared to the conventional Black-Scholes option pricing models, it has been proved to provide more accurate results by stochastic volatility model in terms of the implied volatility, while the classic stochastic volatility model fails to capture the term structure phenomenon of volatility “Smirk.” More attempts have been made to correct for American put option price with incorporating a fast-scale stochastic volatility and a slow-scale stochastic volatility in this paper. Given that the combination in the process of multiscale volatility may lead to a high-dimensional differential equation, an asymptotic approximation method is employed to reduce the dimension in this paper. The numerical results of finite difference show that the multiscale volatility model can offer accurate explanations of the behavior of American put option price.
5
MARVÁ, M., J. C. POGGIALE, and R. BRAVO DE LA PARRA. "REDUCTION OF SLOW–FAST PERIODIC SYSTEMS WITH APPLICATIONS TO POPULATION DYNAMICS MODELS." Mathematical Models and Methods in Applied Sciences 22, no. 10 (August 13, 2012): 1250025. http://dx.doi.org/10.1142/s021820251250025x.
Abstract:
This work deals with the approximate reduction of a nonautonomous two time scales ordinary differential equations system with periodic fast dynamics. We illustrate this technique with the analysis of two models belonging to different fields in ecology. On the one hand, we deal with a two patches periodic predator–prey model with a refuge for prey. Considering migrations between patches to be faster than local interaction allows us to study a three-dimensional system by means of a two-dimensional one. On the other hand, a two time scales periodic eco-epidemic model is addressed by considering two competing species, one of them being affected by a periodic SIR epidemic process which is faster than inter-species interactions. The difference between time scales allows us to study the asymptotic behavior of the four-dimensional system by means of a planar, reduced one. Furthermore, we propose a methodology straightforwardly applicable to a very large class of two time scales periodic systems.
6
Schröders, Simon, and Alexander Fidlin. "Asymptotic analysis of self-excited and forced vibrations of a self-regulating pressure control valve." Nonlinear Dynamics 103, no. 3 (February 2021): 2315–27. http://dx.doi.org/10.1007/s11071-021-06241-5.
Abstract:
AbstractPressure vibrations in hydraulic systems are a widespread problem and can be caused by external excitation or self-exciting mechanisms. Although vibrations cannot be completely avoided in most cases, at least their frequencies must be known in order to prevent resonant excitation of adjacent components. While external excitation frequencies are known in most cases, the estimation of self-excited vibration amplitudes and frequencies is often difficult. Usually, numerical studies have to be executed in order to elaborate parameter influences, which is computationally expensive. The same holds true for the prediction of forced oscillation amplitudes. This contribution proposes asymptotic approximations of forced and self-excited oscillations in a simple hydraulic circuit consisting of a pump, an ideal consumer and a pressure control valve. Two excitation mechanisms of practical interest, namely pump pulsations (forced vibrations) and valve instability (self-excited vibrations), are analyzed. The system dynamics are described by a singularly perturbed third-order differential equation. By separating slow and fast variables in the system without external excitation, a first-order approximation of the slow manifold is computed. The flow on the slow manifold is approximated by an averaging procedure, whose piecewise defined zero-order solution maps the valve’s switching property. A modification of the procedure allows for the asymptotic approximation of the system’s forced response to an external excitation. The approximate solutions are validated within a realistic parameter range by comparison with numerical solutions of the full system equations.
7
Mustafin, Almaz T., and Aliya K. Kantarbayeva. "Clearing function in the context of the invariant manifold method." Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes 19, no. 2 (2023): 185–98. http://dx.doi.org/10.21638/11701/spbu10.2023.205.
Abstract:
Clearing functions (CFs), which express a mathematical relationship between the expected throughput of a production facility in a planning period and its workload (or work-inprogress, WIP) in that period have shown considerable promise for modeling WIP-dependent cycle times in production planning. While steady-state queueing models are commonly used to derive analytic expressions for CFs, the finite length of planning periods calls their validity into question. We apply a different approach to propose a mechanistic model for one-resource, one-product factory shop based on the analogy between the operation of machine and enzyme molecule. The model is reduced to a singularly perturbed system of two differential equations for slow (WIP) and fast (busy machines) variables, respectively. The analysis of this slow-fast system finds that CF is nothing but a result of the asymptotic expansion of the slow invariant manifold. The validity of CF is ultimately determined by how small is the parameter multiplying the derivative of the fast variable. It is shown that sufficiently small characteristic ratio ’working machines: WIP’ guarantees the applicability of CF approximation in unsteady-state operation.
8
Glizer, Valery Y. "Asymptotic Analysis of Spectrum and Stability for One Class of Singularly Perturbed Neutral-Type Time-Delay Systems." Axioms 10, no. 4 (November 30, 2021): 325. http://dx.doi.org/10.3390/axioms10040325.
Abstract:
In this study, a singularly perturbed linear time-delay system of neutral type is considered. It is assumed that the delay is small of order of a small positive parameter multiplying a part of the derivatives in the system. This system is decomposed asymptotically into two much simpler parameter-free subsystems, the slow and fast ones. Using this decomposition, an asymptotic analysis of the spectrum of the considered system is carried out. Based on this spectrum analysis, parameter-free conditions guaranteeing the exponential stability of the original system for all sufficiently small values of the parameter are derived. Illustrative examples are presented.
9
Chabyshova, Elmira, and Gennady Goloshubin. "Seismic modeling of low-frequency “shadows” beneath gas reservoirs." GEOPHYSICS 79, no. 6 (November 1, 2014): D417—D423. http://dx.doi.org/10.1190/geo2013-0379.1.
Abstract:
P-wave amplitude anomalies below reservoir zones can be used as hydrocarbon markers. Some of those anomalies are considerably delayed relatively to the reflections from the reservoir zone. High P-wave attenuation and velocity dispersion of the observed P-waves cannot justify such delays. The hypothesis that these amplitude anomalies are caused by wave propagation through a layered permeable gaseous reservoir is evaluated. The wave propagation through highly interbedded reservoirs suggest an anomalous amount of mode conversions between fast and slow P-waves. The converted P-waves, which propagated even a short distance as slow P-waves, should be significantly delayed and attenuated comparatively, with the fast P-wave reflections. The amplitudes and arrival time variations of conventional and converted P-wave reflections at low seismic frequencies were evaluated by means of an asymptotic analysis. The calculations confirmed that the amplitude anomalies due to converted P-waves are noticeably delayed in time relatively to fast P-wave reflections. However, the amplitudes of the modeled converted P-waves were much lower than the amplitude anomalies observed from exploration cases.
10
Kathirkamanayagan, M., and G. S. Ladde. "Large scale singularly perturbed boundary value problems." Journal of Applied Mathematics and Simulation 2, no. 3 (January 1, 1989): 139–67. http://dx.doi.org/10.1155/s1048953389000122.
Abstract:
In this paper an alternative approach to the method of asymptotic expansions for the study of a singularly perturbed linear system with multiparameters and multiple time scales is developed. The method consists of developing a linear non-singular transformation that transforms an arbitrary n—time scale system into a diagonal form. Furthermore, a dichotomy transformation is employed to decompose the faster subsystems into stable and unstable modes. Fast, slow, stable and unstable modes decomposition processes provide a modern technique to find an approximate solution of the original system in terms of the solution of an auxiliary system. This method yields a constructive and computationally attractive way to investigate the system.
11
Yu, Yue, Qianqian Wang, Qinsheng Bi, and C. W. Lim. "Multiple-S-Shaped Critical Manifold and Jump Phenomena in Low Frequency Forced Vibration with Amplitude Modulation." International Journal of Bifurcation and Chaos 29, no. 05 (May 2019): 1930012. http://dx.doi.org/10.1142/s021812741930012x.
Abstract:
Motivated by the forced harmonic vibration of complex mechanical systems, we analyze the dynamics involving different waves in a double-well potential oscillator coupling amplitude modulation control of low frequency. The combination of amplitude modulation factor significantly enriches the dynamical behaviors on the formation of multiple-S-shaped manifold and multiple jumping phenomena that alternate between epochs of slow and fast motion. We can conduct bifurcation analysis to identify two harmonic vibrations. One is that the singular orbit makes multiple jumps to a fast trajectory segment from one attracting equilibrium to another as the expression of slow variable by using the DeMoivre formula. With the increase of tuning frequency, the system exhibits relaxation-type oscillations whose small amplitude oscillations are produced by nonlinear local cycles together with a distinct large amplitude cycle oscillation accounting for the Melnikov threshold values. The tuning frequency may not only affect the asymptotic expressions for the solution curves near fold singularities but also allow for the large amplitude orbit vibrations near fold-cycle singularities. Numerical analysis for computing critical manifolds and their intersections is used to detect the dynamical features in this paper.
12
Shen, Lu, Daniel J. Jacob, Mauricio Santillana, Kelvin Bates, Jiawei Zhuang, and Wei Chen. "A machine-learning-guided adaptive algorithm to reduce the computational cost of integrating kinetics in global atmospheric chemistry models: application to GEOS-Chem versions 12.0.0 and 12.9.1." Geoscientific Model Development 15, no. 4 (February 25, 2022): 1677–87. http://dx.doi.org/10.5194/gmd-15-1677-2022.
Abstract:
Abstract. Global modeling of atmospheric chemistry is a great computational challenge because of the cost of integrating the kinetic equations for chemical mechanisms with typically over 100 coupled species. Here we present an adaptive algorithm to ease this computational bottleneck with no significant loss in accuracy and apply it to the GEOS-Chem global 3-D model for tropospheric and stratospheric chemistry (228 species, 724 reactions). Our approach is inspired by unsupervised machine learning clustering techniques and traditional asymptotic analysis ideas. We locally define species in the mechanism as fast or slow on the basis of their total production and loss rates, and we solve the coupled kinetic system only for the fast species assembled in a submechanism of the full mechanism. To avoid computational overhead, we first partition the species from the full mechanism into 13 blocks, using a machine learning approach that analyzes the chemical linkages between species and their correlated presence as fast or slow in the global model domain. Building on these blocks, we then preselect 20 submechanisms, as defined by unique assemblages of the species blocks, and then pick locally and on the fly which submechanism to use in the model based on local chemical conditions. In each submechanism, we isolate slow species and slow reactions from the coupled system of fast species to be solved. Because many species in the full mechanism are important only in source regions, we find that we can reduce the effective size of the mechanism by 70 % globally without sacrificing complexity where/when it is needed. The computational cost of the chemical integration decreases by 50 % with relative biases smaller than 2 % for important species over 8-year simulations. Changes to the full mechanism including the addition of new species can be accommodated by adding these species to the relevant blocks without having to reconstruct the suite of submechanisms.
13
Gassner, Steven, and Carlo Cafaro. "Information geometric complexity of entropic motion on curved statistical manifolds under different metrizations of probability spaces." International Journal of Geometric Methods in Modern Physics 16, no. 06 (June 2019): 1950082. http://dx.doi.org/10.1142/s0219887819500828.
Abstract:
We investigate the effect of different metrizations of probability spaces on the information geometric complexity of entropic motion on curved statistical manifolds. Specifically, we provide a comparative analysis based upon Riemannian geometric properties and entropic dynamical features of a Gaussian probability space where the two distinct dissimilarity measures between probability distributions are the Fisher–Rao information metric and the [Formula: see text]-order entropy metric. In the former case, we observe an asymptotic linear temporal growth of the information geometric entropy (IGE) together with a fast convergence to the final state of the system. In the latter case, instead, we note an asymptotic logarithmic temporal growth of the IGE together with a slow convergence to the final state of the system. Finally, motivated by our findings, we provide some insights on a tradeoff between complexity and speed of convergence to the final state in our information geometric approach to problems of entropic inference.
14
Nave, OPhir, and Manju Sharma. "Singular Perturbed Vector Field (SPVF) Applied to Complex ODE System with Hidden Hierarchy Application to Turbocharger Engine Model." International Journal of Nonlinear Sciences and Numerical Simulation 21, no. 1 (February 25, 2020): 99–113. http://dx.doi.org/10.1515/ijnsns-2019-0024.
Abstract:
AbstractIn this paper, we present the concept of singularly perturbed vector field (SPVF) method and its application to spark ignition turbocharger engine. Given a mathematical/physical model, which consist of hidden multi-scale variables, the SPVF methods transfer (using the change of coordinates) and decompose such system to fast and slow subsystems. This decomposition enables one to apply different asymptotic methods such as the method of the integral manifold, homotopy analysis method, singular perturbation method, etc. The resulting subsystem enables us to understand better the complex system and to simplify the calculations. In addition, we investigated the stability of the equilibrium points of the model. This analysis has been done due to the SPVF method which reduces the complexity of the mathematical model.
15
Zhang, Kexin, Caihui Yu, Hongbin Wang, and Xianghong Li. "Multi-scale dynamics of predator-prey systems with Holling-IV functional response." AIMS Mathematics 9, no. 2 (2024): 3559–75. http://dx.doi.org/10.3934/math.2024174.
Abstract:
<abstract><p>In this paper, we propose a Holling-IV predator-prey system considering the perturbation of a slow-varying environmental capacity parameter. This study aims to address how the slowly varying environmental capacity parameter affects the behavior of the system. Based on bifurcation theory and the slow-fast analysis method, the critical condition for the Hopf bifurcation of the autonomous system is given. The oscillatory behavior of the system under different perturbation amplitudes is investigated, corresponding mechanism explanations are given, and it is found that the motion pattern of the non-autonomous system is closely related to the Hopf bifurcation and attractor types of the autonomous system. Meanwhile, there is a bifurcation hysteresis behavior of the system in bursting oscillations, and the bifurcation hysteresis mechanism of the system is analyzed by applying asymptotic theory, and its hysteresis time length is calculated. The final study found that the larger the perturbation amplitude, the longer the hysteresis time. These results can provide theoretical analyses for the prediction, regulation, and control of predator-prey populations.</p></abstract>
16
Holmes, M. H., W. M. Lai, and V. C. Mow. "Singular Perturbation Analysis of the Nonlinear, Flow-Dependent Compressive Stress Relaxation Behavior of Articular Cartilage." Journal of Biomechanical Engineering 107, no. 3 (August 1, 1985): 206–18. http://dx.doi.org/10.1115/1.3138545.
Abstract:
The dominant mechanism giving rise to the viscoelastic response of articular cartilage during compression is the nonlinear diffusive interaction of the fluid and solid phases of the tissue as they flow relative to one another. The present study is concerned with the role of this interaction under uniaxial stress relaxation in compression. The model is a biphasic mixture of fluid and solid which incorporates the strain-dependent permeability found earlier from permeation experiments. When a ramp-displacement is imposed on the articular surface, simple, but accurate, asymptotic approximations are derived for the deformation and stress fields in the tissue for slow and moderately fast rates of compression. They are shown to agree very well with experiment and they provide a simple means for determining the material parameters. Moreover, they lead to important insights into the role of the flow-dependent viscoelastic nature of articular cartilage and other hydrated biological tissues.
17
Pasekov, V. P. "To the analysis of weak two-locus viability selection and quasi linkage equilibrium." Доклады Академии наук 484, no. 6 (May 23, 2019): 781–85. http://dx.doi.org/10.31857/s0869-56524846781-785.
Abstract:
A model of weak viability selection at two multi-allele loci with standardization of approaches through the use of perturbation theory is examined. The estimate of the quasi-equilibrium value for the linkage disequilibrium coefficient D is analyzed, and results in terms of average effects in quantitative genetics and in terms of the theory of singular perturbations in mathematics are obtained. The approximation of a discrete-time model of a random mating population with non-overlapping generations under weak selection by ordinary differential equations is considered. Weak selection is considered as a perturbation of the model without selection. The resulting model is singularly perturbed; that is, fast (D) and slow (allele frequencies) variables can be distinguished. The first approximation equation for quasi-equilibrium of D is obtained using the first terms of the Taylor series expansion of the model functions. It coincides with the corresponding part of the system of the first approximation of the asymptotic series for solving singularly perturbed equations.
18
Halfiani, Vera, Dwi Fadhiliani, Harish Abdillah Mardi, and Marwan Ramli. "Nonlinear Evolution of Benjamin-Bona-Mahony Wave Packet due to an Instability of a Pair of Modulations." International Journal of Differential Equations 2018 (June 3, 2018): 1–10. http://dx.doi.org/10.1155/2018/1716571.
Abstract:
This article discusses the evolution of Benjamin-Bona-Mahony (BBM) wave packet’s envelope. The envelope equation is derived by applying the asymptotic series up to the third order and choosing appropriate fast-to-slow variable transformations which eliminate the resonance terms that occurred. It is obtained that the envelope evolves satisfying the Nonlinear Schrodinger (NLS) equation. The evolution of NLS envelope is investigated through its exact solution, Soliton on Finite Background, which undergoes modulational instability during its propagation. The resulting wave may experience phase singularity indicated by wave splitting and merging and causing amplification on its amplitude. Some parameter values take part in triggering this phenomenon. The amplitude amplification can be analyzed by employing Maximal Temporal Amplitude (MTA) which is a quantity measuring the maximum wave elevation at each spatial position during the observation time. Wavenumber value affects the extreme position of the wave but not the amplitude amplification. Meanwhile, modulational frequency value affects both terms. Comparison of the evolution of the BBM wave packet to the previous results obtained from KdV equation gives interesting outputs regarding the extreme position and the maximum wave peaking.
19
Leibovich, S., S. N. Brown, and Y. Patel. "Bending waves on inviscid columnar vortices." Journal of Fluid Mechanics 173 (December 1986): 595–624. http://dx.doi.org/10.1017/s0022112086001283.
Abstract:
Bending waves, perturbation modes leading to deflections of the vortex centreline, are considered for an infinitely long straight vortex embedded in an irrotational flow of unlimited extent. We first establish the general form of the dispersion relation for long waves on columnar vortices with arbitrary distributions of axial and azimuthal vorticity by a singular perturbation analysis of the Howard-Gupta equation. The asymptotic results are shown to compare favourably with numerical solutions of the Howard-Gupta equation for wavelengths comparable to the vortex core radius and longer. Dispersion relations are then found numerically for specific models of vortex core structures observed experimentally; here no restrictions are placed on wavelength. The linear dispersion relation has an infinite number of branches, falling into two families; one with infinite phase speed at zero wavenumber (which we call ‘fast’ waves), the other with zero phase speed (‘slow’ waves). In the long-wave limit, slow waves have zero group velocity, while the fast waves may have finite non-zero group speeds that depend on the form of the velocity profiles on the axis of rotation. Weakly nonlinear waves are described under most circumstances by the nonlinear Schrödinger equation. Solitons are possible in certain ‘windows’ of wavenumbers of the carrier waves. An example has already been presented by Leibovich & Ma (1983), who compute solitons and soliton windows on a fast-wave branch for a vortex with a particular core structure. Experimental data of Maxworthy, Hopfinger & Redekopp (1985) reveal solitons which appear to be associated with the slow branch, and these are computed for velocity profiles fitting their data. The nonlinear Schrödinger equation is shown to fail for long waves, and to be replaced by a nonlinear integro-differential equation.
20
Aubert, Julien, and Nicolas Gillet. "The interplay of fast waves and slow convection in geodynamo simulations nearing Earth’s core conditions." Geophysical Journal International 225, no. 3 (February 10, 2021): 1854–73. http://dx.doi.org/10.1093/gji/ggab054.
Abstract:
SUMMARY Ground observatory and satellite-based determinations of temporal variations in the geomagnetic field probe a decadal to annual timescale range where Earth’s core slow, inertialess convective motions and rapidly propagating, inertia-bearing hydromagnetic waves are in interplay. Here we numerically model and jointly investigate these two important features with the help of a geodynamo simulation that (to date) is the closest to the dynamical regime of Earth’s core. This model also considerably enlarges the scope of a previous asymptotic scaling analysis, which in turn strengthens the relevance of the approach to describe Earth’s core dynamics. Three classes of hydrodynamic and hydromagnetic waves are identified in the model output, all with propagation velocity largely exceeding that of convective advection: axisymmetric, geostrophic Alfvén torsional waves, and non-axisymmetric, quasi-geostrophic Alfvén and Rossby waves. The contribution of these waves to the geomagnetic acceleration amounts to an enrichment and flattening of its energy density spectral profile at decadal timescales, thereby providing a constraint on the extent of the $f^{-4}$ range observed in the geomagnetic frequency power spectrum. As the model approaches Earth’s core conditions, this spectral broadening arises because the decreasing inertia allows for waves at increasing frequencies. Through non-linear energy transfers with convection underlain by Lorentz stresses, these waves also extract an increasing amount of energy from the underlying convection as their key timescale decreases towards a realistic value. The flow and magnetic acceleration energies carried by waves both linearly increase with the ratio of the magnetic diffusion timescale to the Alfvén timescale, highlighting the dominance of Alfvén waves in the signal and the stabilizing control of magnetic dissipation at non-axisymmetric scales. Extrapolation of the results to Earth’s core conditions supports the detectability of Alfvén waves in geomagnetic observations, either as axisymmetric torsional oscillations or through the geomagnetic jerks caused by non-axisymmetric waves. In contrast, Rossby waves appear to be too fast and carry too little magnetic energy to be detectable in geomagnetic acceleration signals of limited spatio-temporal resolution.
21
Gurevich, Boris. "Effect of fluid viscosity on elastic wave attenuation in porous rocks." GEOPHYSICS 67, no. 1 (January 2002): 264–70. http://dx.doi.org/10.1190/1.1451798.
Abstract:
Attenuation and dispersion of elastic waves in fluid‐saturated rocks due to pore fluid viscosity is investigated using an idealized exactly solvable example of a system of alternating solid and viscous fluid layers. Waves in periodic layered systems at low frequencies can be studied using an asymptotic analysis of Rytov's exact dispersion equations. Since the wavelength of the shear wave in the fluid (viscous skin depth) is much smaller than the wavelength of the shear or compressional waves in the solid, the presence of viscous fluid layers requires a consideration of higher‐order terms in the low‐frequency asymptotic expansions. This expansion leads to asymptotic low‐frequency dispersion equations. For a shear wave with the directions of propagation and of particle motion in the bedding plane, the dispersion equation yields the low‐frequency attenuation (inverse quality factor) as a sum of two terms which are both proportional to frequency ω but have different dependencies on viscosity η: one term is proportional to ω/η, the other to ωη. The low‐frequency dispersion equation for compressional waves allows for the propagation of two waves corresponding to Biot's fast and slow waves. Attenuation of the fast wave has the same two‐term structure as that of the shear wave. The slow wave is a rapidly attenuating diffusion‐type wave, whose squared complex velocity again consists of two terms which scale with iω/η and iωη. For all three waves, the terms proportional to η are responsible for the viscoelastc phenomena (viscous shear relaxation), whereas the terms proportional to η−1 account for the visco‐inertial (poroelastic) mechanism of Biot's type. Furthermore, the characteristic frequencies of visco‐elastic ωV and poroelastic ωB attenuation mechanisms obey the relation ωVωB = AωR2, where ωR is the resonant frequency of the layered system, and A is a dimensionless constant of order 1. This result explains why the visco‐elastic and poroelastic mechanisms are usually treated separately in the context of macroscopic theories that imply ω << ωR. The poroelastic mechanism dominates over the visco‐elastic one when the frequency‐indepenent parameter B = ωB/ωV = 12η2/μsρfhf2 << 1, and vice versa, where hf is the fluid layer thickness, ρf the fluid density, and μs represents the shear modulus of the solid.
22
Beyer, Hans-Georg. "Convergence Analysis of Evolutionary Algorithms That Are Based on the Paradigm of Information Geometry." Evolutionary Computation 22, no. 4 (December 2014): 679–709. http://dx.doi.org/10.1162/evco_a_00132.
Abstract:
The convergence behaviors of so-called natural evolution strategies (NES) and of the information-geometric optimization (IGO) approach are considered. After a review of the NES/IGO ideas, which are based on information geometry, the implications of this philosophy w.r.t. optimization dynamics are investigated considering the optimization performance on the class of positive quadratic objective functions (the ellipsoid model). Exact differential equations describing the approach to the optimizer are derived and solved. It is rigorously shown that the original NES philosophy optimizing the expected value of the objective functions leads to very slow (i.e., sublinear) convergence toward the optimizer. This is the real reason why state of the art implementations of IGO algorithms optimize the expected value of transformed objective functions, for example, by utility functions based on ranking. It is shown that these utility functions are localized fitness functions that change during the IGO flow. The governing differential equations describing this flow are derived. In the case of convergence, the solutions to these equations exhibit an exponentially fast approach to the optimizer (i.e., linear convergence order). Furthermore, it is proven that the IGO philosophy leads to an adaptation of the covariance matrix that equals in the asymptotic limit—up to a scalar factor—the inverse of the Hessian of the objective function considered.
23
Shaburov, Alexander. "Asymptotic expansion of a solution for one singularly perturbed optimal control problem with a convex integral quality index depends on slow variables and smooth control constraints." Tambov University Reports. Series: Natural and Technical Sciences, no. 125 (2019): 119–36. http://dx.doi.org/10.20310/1810-0198-2019-24-125-119-136.
Abstract:
The paper deals with the problem of optimal control with a convex integral quality index depends on slow variables for a linear steady-state control system with a fast and slow variables in the class of piecewise continuous controls with a smooth control constraints x ε = A 11 x ε + A 12 y ε + B 1 u, εy ε = A 21 x ε + A 22 y ε + B 2 u, J ε u ≔φ x ε T + 0 T u(t) 2 dt→ min, t∈ 0, T , x ε0 = x 0 ,u ≤1, y ε0 = y 0 , where x ε ∈Rn , y ε ∈Rm , u∈Rr ; A ij , B i , i, j =1,2, - are constant matrices of the corresponding dimension, and φ(·) - is the strictly convex and cofinite function that is continuously differentiable in Rn in the sense of convex analysis. In the general case, Pontryagin’s maximum principle is a necessary and sufficient optimum condition for the optimization of a such a problem. The initial vector of the conjugate state l ε is the unique vector, thus determining the optimal control. It is proven that in the case of a finite number of control switching points, the asymptotics of the vector l ε has the character of a power series.
24
Lewi, Jeremy, Robert Butera, and Liam Paninski. "Sequential Optimal Design of Neurophysiology Experiments." Neural Computation 21, no. 3 (March 2009): 619–87. http://dx.doi.org/10.1162/neco.2008.08-07-594.
Abstract:
Adaptively optimizing experiments has the potential to significantly reduce the number of trials needed to build parametric statistical models of neural systems. However, application of adaptive methods to neurophysiology has been limited by severe computational challenges. Since most neurons are high-dimensional systems, optimizing neurophysiology experiments requires computing high-dimensional integrations and optimizations in real time. Here we present a fast algorithm for choosing the most informative stimulus by maximizing the mutual information between the data and the unknown parameters of a generalized linear model (GLM) that we want to fit to the neuron's activity. We rely on important log concavity and asymptotic normality properties of the posterior to facilitate the required computations. Our algorithm requires only low-rank matrix manipulations and a two-dimensional search to choose the optimal stimulus. The average running time of these operations scales quadratically with the dimensionality of the GLM, making real-time adaptive experimental design feasible even for high-dimensional stimulus and parameter spaces. For example, we require roughly 10 milliseconds on a desktop computer to optimize a 100-dimensional stimulus. Despite using some approximations to make the algorithm efficient, our algorithm asymptotically decreases the uncertainty about the model parameters at a rate equal to the maximum rate predicted by an asymptotic analysis. Simulation results show that picking stimuli by maximizing the mutual information can speed up convergence to the optimal values of the parameters by an order of magnitude compared to using random (nonadaptive) stimuli. Finally, applying our design procedure to real neurophysiology experiments requires addressing the nonstationarities that we would expect to see in neural responses; our algorithm can efficiently handle both fast adaptation due to spike history effects and slow, nonsystematic drifts in a neuron's activity.
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Lanzerstorfer, Daniel, and Hendrik C. Kuhlmann. "Global stability of the two-dimensional flow over a backward-facing step." Journal of Fluid Mechanics 693 (November 3, 2011): 1–27. http://dx.doi.org/10.1017/jfm.2011.399.
Abstract:
AbstractThe two-dimensional, incompressible flow over a backward-facing step is considered for a systematic variation of the geometry covering expansion ratios (step to outlet height) from 0.25 to 0.975. A global temporal linear stability analysis shows that the basic flow becomes unstable to different three-dimensional modes depending on the expansion ratio. All critical modes are essentially confined to the region behind the step extending downstream up to the reattachment point of the separated eddy. An energy-transfer analysis is applied to understand the physical nature of the instabilities. If scaled appropriately, the critical Reynolds number approaches a finite asymptotic value for very large step heights. In that case centrifugal forces destabilize the flow with respect to an oscillatory critical mode. For moderately large expansion ratios an elliptical instability mechanism is identified. If the step height is further decreased the critical mode changes from oscillatory to stationary. In addition to the elliptical mechanism, the strong shear in the layer emanating from the sharp corner of the step supports the amplification process of the critical mode. For very small step heights the basic state becomes unstable due to the lift-up mechanism, which feeds back on itself via the recirculating eddy behind the step, resulting in a steady critical mode comprising pronounced slow and fast streaks.
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Leguillon, Dominique. "Comparison of Matched Asymptotics, Multiple Scalings and Averages in Homogenization of Periodic Structures." Mathematical Models and Methods in Applied Sciences 07, no. 05 (August 1997): 663–80. http://dx.doi.org/10.1142/s0218202597000359.
Abstract:
Historically, homogenization of periodic structures has been first investigated by the method of multiple scalings expansions. More recently, an interpretation has been given in terms of averages and effective moduli. Both approaches involve a slow variable at the macroscopic scale, and a fast one at the microscopic level. The periodicity of the solutions with respect to the second variable is a strong assumption made prior to any analysis. Although involving similar calculations, the two approaches differ and it is not so obvious to link them together. The matched asymptotic expansions presented here allow to give a common explanation to the two already mentioned approaches. The first one corresponds to an outer expansion while the second one describes the leading term of an inner expansion. Moreover, no a priori assumption is made, the periodicity of the solutions occurs as a consequence of the structure of the inner problems. The next term (involving a quadratic dependence on the local variable) of the inner expansion can be derived in the same way. The same matched asymptotics process can be used to define homogenized boundary conditions as well as boundary layers. These layers come from a mismatch between the general form of the solution within the domain and the boundary conditions which occur to be a perturbation of the periodicity. Indeed, it is not easy to give an exact definition of the boundary conditions in the original problem, the inner expansion defined on the enlarged domain allows one to give a precise framework for these conditions. They split into two parts, a macroscopic one defined on the smooth (homogenized) boundary and a microscopic periodic fluctuation taking into account the exact shape of the boundary.
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Cai, Yuhua, and Stefan A. H. Geritz. "Resident-invader dynamics of similar strategies in fluctuating environments." Journal of Mathematical Biology 81, no. 4-5 (September 7, 2020): 907–59. http://dx.doi.org/10.1007/s00285-020-01532-8.
Abstract:
Abstract We study resident-invader dynamics in fluctuating environments when the invader and the resident have close but distinct strategies. First we focus on a class of continuous-time models of unstructured populations of multi-dimensional strategies, which incorporates environmental feedback and environmental stochasticity. Then we generalize our results to a class of structured population models. We classify the generic population dynamical outcomes of an invasion event when the resident population in a given environment is non-growing on the long-run and stochastically persistent. Our approach is based on the series expansion of a model with respect to the small strategy difference, and on the analysis of a stochastic fast-slow system induced by time-scale separation. Theoretical and numerical analyses show that the total size of the resident and invader population varies stochastically and dramatically in time, while the relative size of the invader population changes slowly and asymptotically in time. Thereby the classification is based on the asymptotic behavior of the relative population size, and which is shown to be fully determined by invasion criteria (i.e., without having to study the full generic dynamical system). Our results extend and generalize previous results for a stable resident equilibrium (particularly, Geritz in J Math Biol 50(1):67–82, 2005; Dercole and Geritz in J Theor Biol 394:231-254, 2016) to non-equilibrium resident population dynamics as well as resident dynamics with stochastic (or deterministic) drivers.
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Tulin, Marshall P., Yitao Yao, and Pei Wang. "The Generation and Propagation of Ship Internal Waves in a Generally Stratified Ocean at High Densimetric Froude Numbers, Including Nonlinear Effects." Journal of Ship Research 44, no. 03 (September 1, 2000): 197–227. http://dx.doi.org/10.5957/jsr.2000.44.3.197.
Abstract:
A nonlinear theory for internal wave generation and propagation is derived here for slender ships traveling at high densimetric Froude number (Fh >> 1) in water of small density variation. It is based on an asymptotic equation for the evolution of the internal wave vorticity generated under the ship by a known inviscid ship flow and then self-propagating in the wake. In its numerical implementation, arbitrary pycnoclines and slender ship hulls may be used, and boundary conditions on the ship hull are satisfied; the free surface is treated here as rigid, although this may be relaxed. The theory has been implemented by a suitable numerical method and numerous simulations have been carried out. The results have been compared with earlier OEL experiments. In the near field, emphasis is given to a triple-lobe pattern in the pycnocline, an upwelling along the centerline of motion with a trough on either side, forming close behind the ship. Two distinct types of triple lobes are identified:dominant central lobe and very weak troughs, and;weak central lobe and dominant troughs. The former (a) is shown to result in linear propagation into the far field. The latter (b) results in far-field patterns preceded by a deep trough whose propagation is nonlinear. The comparisons of both simulated trends and actual amplitudes with measurements are good, surprisingly so considering the small scale of the experiment and the asymptotic nature of the theory. The effect of the turbulent wake on the internal waves in the experiments is restricted to a very narrow region behind the ship; the bulk of the wave pattern including the leading waves seem unaffected. Simulations show that under certain conditions of stratification, triple-lobe patterns with abnormally large troughs are generated and lead to strong nonlinear effects; these deep troughs propagate sidewards to large distances aft (over 40 ship lengths) with slow decay, and result in much larger surface currents and strain rates than in the normal case. Correspondingly, fast waves of depression, which decay slowly, were discovered through the simulation of two-dimensional initial value problems, where the initial area of depression was significantly less than required of a true soliton; these "quasi-solitons" are briefly studied here.
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Cheng, Xin, Huashan Liu, and Wenke Lu. "Chattering-Suppressed Sliding Mode Control for Flexible-Joint Robot Manipulators." Actuators 10, no. 11 (October 27, 2021): 288. http://dx.doi.org/10.3390/act10110288.
Abstract:
In this paper, sliding mode tracking control and its chattering suppression method are investigated for flexible-joint robot manipulators with only state measurements of joint actuators. First, within the framework of singular perturbation theory, the control objective of the system is decoupled into two typical tracking aims of a slow subsystem and a fast subsystem. Then, considering lumped uncertainties (including dynamics uncertainties and external disturbances), a composite chattering-suppressed sliding mode controller is proposed, where a smooth-saturation-function-contained reaching law with adjustable saturation factor is designed to alleviate the inherent chattering phenomenon, and a radial basis function neural network (RBFNN)-based soft computing strategy is applied to avoid the high switching gain that leads to chattering amplification. Simultaneously, an efficient extended Kalman filter (EKF) with respect to a new state variable is presented to enable the closed-loop tracking control with neither position nor velocity measurements of links. In addition, an overall analysis on the asymptotic stability of the whole control system is given. Finally, numerical examples verify the superiority of the dynamic performance of the proposed control approach, which is well qualified to suppress the chattering and can effectively eliminate the undesirable effects of the lumped uncertainties with a smaller switching gain reduced by 80% in comparison to that in the controller without RBFNN. The computational efficiency of the proposed EKF increased by about 26%.
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BABIN, A., and A. FIGOTIN. "LINEAR SUPERPOSITION IN NONLINEAR WAVE DYNAMICS." Reviews in Mathematical Physics 18, no. 09 (October 2006): 971–1053. http://dx.doi.org/10.1142/s0129055x06002851.
Abstract:
We study nonlinear dispersive wave systems described by hyperbolic PDE's in ℝd and difference equations on the lattice ℤd. The systems involve two small parameters: one is the ratio of the slow and the fast time scales, and another one is the ratio of the small and the large space scales. We show that a wide class of such systems, including nonlinear Schrodinger and Maxwell equations, Fermi–Pasta–Ulam model and many other not completely integrable systems, satisfy a superposition principle. The principle essentially states that if a nonlinear evolution of a wave starts initially as a sum of generic wavepackets (defined as almost monochromatic waves), then this wave with a high accuracy remains a sum of separate wavepacket waves undergoing independent nonlinear evolution. The time intervals for which the evolution is considered are long enough to observe fully-developed nonlinear phenomena for involved wavepackets. In particular, our approach provides a simple justification for numerically observed effect of almost non-interaction of solitons passing through each other without any recourse to the complete integrability. Our analysis does not rely on any ansatz or common asymptotic expansions with respect to the two small parameters but it uses rather explicit and constructive representation for solutions as functions of the initial data in the form of functional analytic series.
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Yan, Sun-ting, Xiaoli Shen, and Zhijiang Jin. "Static and dynamic symmetric snap-through of non-uniform shallow arch under a pair of end moments considering critical slowing-down effect." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 233, no. 16 (June 10, 2019): 5735–62. http://dx.doi.org/10.1177/0954406219855105.
Abstract:
This paper presents analytical analysis of static and dynamic symmetric snap-through of non-uniform shallow circular shallow arch under a pair of end moments with the same magnitude. The non-uniformity is characterized by dividing the arch into three piecewise constant-stiffness segments. Hamilton's principle is used to derive the governing differential equations by assuming negligible axial inertia. The snap-through load and snap-through criterion are analyzed in detail and by an index plot, the stiffer center case is shown to behave distinctly when stiffer end case is compared. The dynamic snap-through when the moment is slightly higher than the snap-through moment is analyzed analytically by a perturbation method, and a critical slowing effect is observed when the moment is approaching to the snap-through moment. Comparison with dynamic FEA shows a good agreement with the analytical result and analysis on theoretical finite time blow-up phenomenon reveals that when geometric parameters are corresponding to the critical snap-through condition, the initial quadratic phase's motion is slow and relatively blow-up phase's motion is fast. The analytical formulations have been extended to include two limiting cases including rigid end case and rigid center case by using the constrained Hamilton's principle by Lagrangian multipliers. Snap-through criteria analysis reveals closed-form criterion for rigid center case and an asymptotic result for rigid end case. This paper serves to enhance the knowledge on snap-through and critical slowing down for shallow arches with non-uniformity under end moments.
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KUO, ALLEN C., and LORENZO M. POLVANI. "Wave–vortex interaction in rotating shallow water. Part 1. One space dimension." Journal of Fluid Mechanics 394 (September 10, 1999): 1–27. http://dx.doi.org/10.1017/s0022112099005534.
Abstract:
Using a physical space (i.e. non-modal) approach, we investigate interactions between fast inertio-gravity (IG) waves and slow balanced flows in a shallow rotating fluid. Specifically, we consider a train of IG waves impinging on a steady, exactly balanced vortex. For simplicity, the one-dimensional problem is studied first; the limitations of one-dimensionality are offset by the ability to define balance in an exact way. An asymptotic analysis of the problem in the small-amplitude limit is performed to demonstrate the existence of interactions. It is shown that these interactions are not confined to the modification of the wave field by the vortex but, more importantly, that the waves are able to alter in a non-trivial way the potential vorticity associated with that vortex. Interestingly, in this one-dimensional problem, once the waves have traversed the vortex region and have propagated away, the vortex exactly recovers its initial shape and thus bears no signature of the interaction. Furthermore, we prove this last result in the case of arbitrary vortex and wave amplitudes. Numerical integrations of the full one-dimensional shallow-water equations in strongly nonlinear regimes are also performed: they confirm that time-dependent interactions exist and increase with wave amplitude, while at the final state the vortex bears no sign of the interaction. In addition, they reveal that cyclonic vortices interact more strongly with the wave field than anticyclonic ones.
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CAMASSA, ROBERTO, TERRY JO LEITERMAN, and RICHARD M. MCLAUGHLIN. "Trajectory and flow properties for a rod spinning in a viscous fluid. Part 1. An exact solution." Journal of Fluid Mechanics 612 (October 10, 2008): 153–200. http://dx.doi.org/10.1017/s0022112008000918.
Abstract:
An exact mathematical solution for the low-Reynolds-number quasi-steady hydrodynamic motion induced by a rod in the form of a prolate spheroid sweeping a symmetric double cone is developed, and the influence of the ensuing fluid motion upon passive particles is studied. The resulting fluid motion is fully three-dimensional and time varying. The advected particles are observed to admit slow orbits around the rotating rods and a fast epicyclic motion roughly commensurate with the rod rotation rate. The epicycle amplitudes, vertical fluctuations, arclengths and angle travelled per rotation are mapped as functions of their initial coordinates and rod geometry. These trajectories exhibit a rich spatial structure with greatly varying trajectory properties. Laboratory frame asymmetries of these properties are explored using integer time Poincaré sections and far-field asymptotic analysis. This includes finding a small cone angle invariant in the limit of large spherical radius whereas an invariant for arbitrary cone angles is obtained in the limit of large cylindrical radius. The Eulerian and Lagrangian flow properties of the fluid flow are studied and shown to exhibit complex structures in both space and time. In particular, spatial regions of high speed and Lagrangian velocities possessing multiple extrema per rod rotation are observed. We establish the origin of these complexities via an auxiliary flow in a rotating frame, which provides a generator that defines the epicycles. Finally, an additional spin around the major spheroidal axis is included in the exact hydrodynamic solution resulting in enhanced vertical spatial fluctuation as compared to the spinless counterpart. The connection and relevance of these observations with recent developments in nano-scale fluidics is discussed, where similar epicycle behaviour has been observed. The present study is of direct use to nano-scale actuated fluidics.
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NARASIMHA, RODDAM. "THE CONTRIBUTION OF THE BHATNAGAR–GROSS–KROOK MODEL TO THE DEVELOPMENT OF RAREFIED GAS DYNAMICS IN THE EARLY YEARS OF THE SPACE AGE." International Journal of Modern Physics C 25, no. 01 (December 2, 2013): 1340025. http://dx.doi.org/10.1142/s0129183113400251.
Abstract:
The advent of the space age in 1957 was accompanied by a sudden surge of interest in rarefied gas dynamics (RGD). The well-known difficulties associated with solving the Boltzmann equation that governs RGD made progress slow but the Bhatnagar–Gross–Krook (BGK) model, proposed three years before Sputnik, turned out to have been an uncannily timely, attractive and fruitful option, both for gaining insights into the Boltzmann equation and for estimating various technologically useful flow parameters. This paper gives a view of how BGK contributed to the growth of RGD during the first decade of the space age. Early efforts intended to probe the limits of the BGK model showed that, in and near both the continuum Euler limit and the collisionless Knudsen limit, BGK could provide useful answers. Attempts were therefore made to tackle more ambitious nonlinear nonequilibrium problems. The most challenging of these was the structure of a plane shock wave. The first exact numerical solutions of the BGK equation for the shock appeared during 1962 to 1964, and yielded deep insights into the character of transitional nonequilibrium flows that had resisted all attempts at solution through the Boltzmann equation. In particular, a BGK weak shock was found to be amenable to an asymptotic analysis. The results highlighted the importance of accounting separately for fast-molecule dynamics, most strikingly manifested as tails in the distribution function, both in velocity and in physical space — tails are strange versions or combinations of collisionless and collision-generated flows. However, by the mid-1960s Monte-Carlo methods of solving the full Boltzmann equation were getting to be mature and reliable and interest in the BGK waned in the following years. Interestingly, it has seen a minor revival in recent years as a tool for developing more effective algorithms in continuum computational fluid dynamics, but the insights derived from the BGK for strongly nonequilibrium flows should be of lasting value.
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Bonforte, Matteo, Gabriele Grillo, and Juan Luis Vázquez. "Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold." Archive for Rational Mechanics and Analysis 196, no. 2 (July 1, 2009): 631–80. http://dx.doi.org/10.1007/s00205-009-0252-7.
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WEBB, G. M., A. R. ZAKHARIAN, M. BRIO, and G. P. ZANK. "Parametric instabilities and wave coupling in Alfvén simple waves." Journal of Plasma Physics 66, no. 3 (September 2001): 167–212. http://dx.doi.org/10.1017/s0022377801001283.
Abstract:
A wave coupling formalism for magnetohydrodynamic (MHD) waves in a non-uniform background flow is used to study parametric instabilities of the large-amplitude, circularly polarized, simple plane Alfvén wave in one Cartesian space dimension. The large-amplitude Alfvén wave (the pump wave) is regarded as the background flow, and the seven small-amplitude MHD waves (the backward and forward fast and slow magnetoacoustic waves, the backward and forward Alfvén waves, and the entropy wave) interact with the pump wave via wave coupling coefficients that depend on the gradients and time dependence of the background flow. The dispersion equation for the waves D(k,ω) = 0 obtained from the wave coupling equations reduces to that obtained by previous authors. The general solution of the initial value problem for the waves is obtained by Fourier and Laplace transforms. The dispersion function D(k,ω) is a fifth-order polynomial in both the wavenumber k and the frequency ω. The regions of instability and the neutral stability curves are determined. Instabilities that arise from solving the dispersion equation D(k,ω) = 0, both in the form ω = ω(k), where k is real, and in the form k = k(ω), where ω is real, are investigated. The instabilities depend parametrically on the pump wave amplitude and on the plasma beta. The wave interaction equations are also studied from the perspective of a single master wave equation, with multiple wave modes, and with a source term due to the entropy wave. The wave hierarchies for short- and long-wavelength waves of the master wave equation are used to discuss wave stability. Expanding the dispersion equation for the different long-wavelength eigenmodes about k = 0 yields either the linearized Korteweg–deVries equation or the Schrödinger equation as the generic wave equation at long-wavelengths. The corresponding short-wavelength wave equations are also obtained. Initial value problems for the wave interaction equations are investigated. An inspection of the double-root solutions of the dispersion equation for k, satisfying the equations D(k,ω) = 0 and ∂D(k,ω) = ∂k = 0 and pinch point analysis shows that the solutions of the wave interaction equations for hump or pulse-like initial data develop an absolute instability. Fourier solutions and asymptotic analysis are used to study the absolute instability.
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Spiliopoulos, Konstantinos. "Fluctuation analysis and short time asymptotics for multiple scales diffusion processes." Stochastics and Dynamics 14, no. 03 (May 29, 2014): 1350026. http://dx.doi.org/10.1142/s0219493713500263.
Abstract:
We consider the limiting behavior of fluctuations of small noise diffusions with multiple scales around their homogenized deterministic limit. We allow full dependence of the coefficients on the slow and fast motion. These processes arise naturally when one is interested in short time asymptotics of multiple scale diffusions. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. Depending on the order of interaction between the fast scale and the size of the noise, we get different behavior. In certain cases additional drift terms arise in the limiting process, which are explicitly characterized. These results provide a better approximation to the limiting behavior of such processes when compared to the law of large numbers homogenization limit.
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Galí, Martí, Marcus Falls, Hervé Claustre, Olivier Aumont, and Raffaele Bernardello. "Bridging the gaps between particulate backscattering measurements and modeled particulate organic carbon in the ocean." Biogeosciences 19, no. 4 (March 1, 2022): 1245–75. http://dx.doi.org/10.5194/bg-19-1245-2022.
Abstract:
Abstract. Oceanic particulate organic carbon (POC) is a small but dynamic component of the global carbon cycle. Biogeochemical models historically focused on reproducing the sinking flux of POC driven by large fast-sinking particles (LPOC). However, suspended and slow-sinking particles (SPOC, here < 100 µm) dominate the total POC (TPOC) stock, support a large fraction of microbial respiration, and can make sizable contributions to vertical fluxes. Recent developments in the parameterization of POC reactivity in PISCES (Pelagic Interactions Scheme for Carbon and Ecosystem Studies model; PISCESv2_RC) have improved its ability to capture POC dynamics. Here we evaluated this model by matching a global 3D simulation and 1D simulations at 50 different locations with observations made from biogeochemical (BGC-) Argo floats and satellites. Our evaluation covers globally representative biomes between 0 and 1000 m depth and relies on (1) a refined scheme for converting particulate backscattering at 700 nm (bbp700) to POC, based on biome-dependent POC / bbp700 ratios in the surface layer that decrease to an asymptotic value at depth; (2) a novel approach for matching annual time series of BGC-Argo vertical profiles to PISCES 1D simulations forced by pre-computed vertical mixing fields; and (3) a critical evaluation of the correspondence between in situ measurements of POC fractions, PISCES model tracers, and SPOC and LPOC estimated from high vertical resolution bbp700 profiles through a separation of the baseline and spike signals. We show that PISCES captures the major features of SPOC and LPOC across a range of spatiotemporal scales, from highly resolved profile time series to biome-aggregated climatological profiles. Model–observation agreement is usually better in the epipelagic (0–200 m) than in the mesopelagic (200–1000 m), with SPOC showing overall higher spatiotemporal correlation and smaller deviation (typically within a factor of 1.5). Still, annual mean LPOC stocks estimated from PISCES and BGC-Argo are highly correlated across biomes, especially in the epipelagic (r=0.78; n=50). Estimates of the SPOC / TPOC fraction converge around a median of 85 % (range 66 %–92 %) globally. Distinct patterns of model–observations misfits are found in subpolar and subtropical gyres, pointing to the need to better resolve the interplay between sinking, remineralization, and SPOC–LPOC interconversion in PISCES. Our analysis also indicates that a widely used satellite algorithm overestimates POC severalfold at high latitudes during the winter. The approaches proposed here can help constrain the stocks, and ultimately budgets, of oceanic POC.
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Chen, Nawei, Shenglong Chen, Xiaoyu Li, and Zhiming Li. "Modelling and analysis of the HIV/AIDS epidemic with fast and slow asymptomatic infections in China from 2008 to 2021." Mathematical Biosciences and Engineering 20, no. 12 (2023): 20770–94. http://dx.doi.org/10.3934/mbe.2023919.
Abstract:
<abstract><p>The aim of this paper is to investigate the spread of the HIV/AIDS epidemic in China during 2008–2021. A new mathematical model is proposed to study the dynamics of HIV transmission with acute infection, fast asymptomatic infections, and slow asymptomatic infections. The basic reproduction number is obtained by the next-generation matrix method. A quantitative analysis of the model, including the local behavior, global behavior, and permanence, is performed. Numerical simulations are presented to enhance the results of these analyses. The behavior or the model's parameters are estimated from real data. A sensitivity analysis shows that the proportion of asymptomatic infections co-infected with other diseases significantly affects the basic reproduction number. We further analyze the impact of implementing single and multiple measure(s) in parallel with the epidemic. The study results conclude that multiple measures are more effective in controlling the spread of AIDS compared to just one. The HIV epidemic can be effectively curbed by reducing the contact rate between fast asymptomatic infected individuals and susceptible populations, increasing the early diagnosis and screening of HIV-infected individuals co-infected with other diseases, and treating co-infected patients promptly.</p></abstract>
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Liu, Yang, and Haixia Wu. "Analysis on Global Asymptotical Stability of Genetic Regulatory Networks with Time-Varying Delays via Convex Combination Method." Mathematical Problems in Engineering 2015 (2015): 1–5. http://dx.doi.org/10.1155/2015/303918.
Abstract:
The global asymptotical stability analysis for genetic regulatory networks with time delays is concerned. By using Lyapunov functional theorem, LMIs, and convex combination method, a new delay-dependent stability criterion has been presented in terms of LMIs to guarantee the delayed genetic regulatory networks to be asymptotically stable. The restriction that the derivatives of the time-varying delays are less than one is removed. Our result is applicable to both fast and slow time-varying delays. The stability criterion has less conservative and wider application range. Experimental result has been used to demonstrate the usefulness of the main results and less conservativeness of the proposed method.
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Cheng, Chih-Hsiu, Kwan-Hwa Lin, Jiu-Jenq Lin, and Jaw-Lin Wang. "CERVICAL ELECTROMYOGRAPHIC ACTIVITIES DURING NECK MOVEMENTS AT DIFFERENT SPEEDS IN HEALTHY SUBJECTS: VOLUNTARY RESPONSE INDEX ANALYSIS." Biomedical Engineering: Applications, Basis and Communications 19, no. 06 (December 2007): 349–57. http://dx.doi.org/10.4015/s1016237207000458.
Abstract:
The assessment of cervical muscle control patterns is important for the diagnosis of cervical dysfunction. Voluntary response index (VRI), including the similarity index (SI) and the magnitude (MAG), provides quantitative analysis of the surface electromyography (sEMG) pattern and total muscle voluntary activities respectively. This study was to investigate the effect of movement directions and speeds of the VRI response of cervical muscles in healthy subjects. The sEMG of bilateral sternocleidomastoid, semispinalis capitis, and splenius capitis were measured in thirteen asymptomatic young subjects. The subjects performed voluntary neck movements in flexion, extension, left and right side bending at fast, medium, and slow speeds. The results showed that SI ranged from 1–0.8 and MAG was generally less than 40 μV. The SI was significantly smallest and the MAG was largest at fast speed. The MAG was also significantly different among directions but the effect of direction on SI was only significant at fast speed. In conclusion, the movement speed and direction could affect the magnitude and control pattern of cervical muscles, such that both the speed and direction of the examined tasks should be carefully monitored during the assessment of cervical muscle activation.
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McAuliffe, Seán, Ariane Tabuena, Karen McCreesh, Mary O'Keeffe, John Hurley, Tom Comyns, Helen Purtill, Seth O'Neill, and Kieran O'Sullivan. "Altered Strength Profile in Achilles Tendinopathy: A Systematic Review and Meta-Analysis." Journal of Athletic Training 54, no. 8 (August 1, 2019): 889–900. http://dx.doi.org/10.4085/1062-6050-43-18.
Abstract:
Background Persistent strength deficits secondary to Achilles tendinopathy (AT) have been postulated to account for difficulty engaging in tendon-loading movements, such as running and jumping, and may contribute to the increased risk of recurrence. To date, little consensus exists on the presence of strength deficits in AT. Consequently, researchers are uncertain about the appropriate methods of assessment that may inform rehabilitation in clinical practice. Objective To evaluate and synthesize the literature investigating plantar-flexion (PF) strength in individuals with AT. Study Selection Two independent reviewers searched 9 electronic databases using an agreed-upon set of key words. Data Extraction Data were extracted from studies comparing strength measures (maximal, reactive, and explosive strength) between individuals with AT and healthy control participants or between the injured and uninjured sides of people with AT. The Critical Appraisal Skills Programme Case-Control Study Checklist was used to assess the risk of bias for the included studies. Data Synthesis A total of 19 studies were eligible. Pooled meta-analyses for isokinetic dynamometry demonstrated reductions in maximal strength (concentric PF peak torque [PT] slow [Hedges g = 0.52, 44% deficit], concentric PF PT fast [Hedges g = 0.61, 38% deficit], and eccentric PF PT slow [Hedges g = 0.26, 18% deficit]). Reactive strength, particularly during hopping, was also reduced (Hedges g range = 0.32–2.61, 16%–35% deficit). For explosive strength, reductions in the rate of force development (Hedges g range = 0.31–1.73, 10%–21% deficit) were observed, whereas the findings for ground reaction force varied but were not consistently altered. Conclusions Individuals with AT demonstrated strength deficits compared with the uninjured side or with asymptomatic control participants. Deficits were reported across the strength spectrum for maximal, reactive, and explosive strength. Clinicians and researchers may need to adapt their assessment of Achilles tendon function, which may ultimately help to optimize rehabilitation outcomes.
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Lewis, Joanna, Malcolm J. Price, Paddy J. Horner, and Peter J. White. "P003 Genital c. trachomatisinfections last longer in men than women, but are less likely to become established." Sexually Transmitted Infections 93, Suppl 1 (June 2017): A18.1—A18. http://dx.doi.org/10.1136/sextrans-2017-053232.49.
Abstract:
IntroductionRigorous estimates for the duration of untreated chlamydia infection are important for understanding its epidemiology and designing control interventions, but are only available for women. We have estimated the duration of untreated infection in men.MethodsData came from published studies in which untreated, chlamydia-infected men were re-tested at a later date. We used analysis methods that had previously been applied to data from women, which allow for a new infection to take one of multiple courses, each clearing at a different rate. We determined the optimal number of possible courses. Parameter estimates were obtained using a Bayesian statistical framework.ResultsThe best-fitting model had two different courses of infection: ‘slow-’ and ‘fast-clearing’, as had been the case for women. In men only 68% (57%–78%) (median sample; 95% credible interval) of incident infections were ‘slow-clearing’, compared with 77% (69%–84%) in women. The posterior median estimate for the mean infection duration in men was 2.84 (0.87-18.79) years, compared with 1.35 (1.13–1.63 years) in women.DiscussionOur estimated infection duration in men is longer than has previously been assumed. Male infections are less likely to become established (slow-clearing) than those in women but once established, tend to last longer. Long-term, asymptomatic infections in men – in whom chlamydia screening rates are lower – could be sustaining chlamydia prevalence in both sexes. This study provides an improved description of chlamydia’s natural history to better inform public health decision-making. We advocate further data collection to reduce uncertainty in estimates.
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Machado Reyes, Diego, Hanqing Chao, Juergen Hahn, Li Shen, and Pingkun Yan. "Identifying Progression-Specific Alzheimer’s Subtypes Using Multimodal Transformer." Journal of Personalized Medicine 14, no. 4 (April 15, 2024): 421. http://dx.doi.org/10.3390/jpm14040421.
Abstract:
Alzheimer’s disease (AD) is the most prevalent neurodegenerative disease, yet its current treatments are limited to stopping disease progression. Moreover, the effectiveness of these treatments remains uncertain due to the heterogeneity of the disease. Therefore, it is essential to identify disease subtypes at a very early stage. Current data-driven approaches can be used to classify subtypes during later stages of AD or related disorders, but making predictions in the asymptomatic or prodromal stage is challenging. Furthermore, the classifications of most existing models lack explainability, and these models rely solely on a single modality for assessment, limiting the scope of their analysis. Thus, we propose a multimodal framework that utilizes early-stage indicators, including imaging, genetics, and clinical assessments, to classify AD patients into progression-specific subtypes at an early stage. In our framework, we introduce a tri-modal co-attention mechanism (Tri-COAT) to explicitly capture cross-modal feature associations. Data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) (slow progressing = 177, intermediate = 302, and fast = 15) were used to train and evaluate Tri-COAT using a 10-fold stratified cross-testing approach. Our proposed model outperforms baseline models and sheds light on essential associations across multimodal features supported by known biological mechanisms. The multimodal design behind Tri-COAT allows it to achieve the highest classification area under the receiver operating characteristic curve while simultaneously providing interpretability to the model predictions through the co-attention mechanism.
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Dave, A., K. E. Sprecher, K. K. Lui, M. G. Chappel-Farley, I. Y. Chen, K. Blennow, H. Zetterberg, et al. "0422 Apocalypse Tau: The Relationship Between Inflammaging and Local Sleep Disruption in Older Adults is Mediated by Tau Burden." Sleep 43, Supplement_1 (April 2020): A161—A162. http://dx.doi.org/10.1093/sleep/zsaa056.419.
Abstract:
Abstract Introduction Chronic inflammation in aging is independently associated with tau burden and sleep disruption, though the mechanism linking inflammation with sleep disruption remains unknown. Recent evidence associates tau burden with deficits in local expression of sleep spindles and slow wave activity (SWA). Here we test the hypothesis that age-related central inflammation disrupts local sleep by influencing tau pathology. Methods Cognitively asymptomatic older adults from the Wisconsin Alzheimer’s Disease Research Center underwent overnight polysomnography with high-density electroencephalography (hdEEG; 256 channels) at the University of Wisconsin-Madison (n=33, 61.9±6.7 years, 23 female). EEG data were subjected to multitaper spectral analysis (0.5-40Hz) to yield topographic maps of SWA (SWA1:0.5-1Hz, SWA2:1-4.5Hz) and spindle (sigma1:11-13Hz; sigma2:13-16Hz) power during NREM sleep. Cerebrospinal fluid assay-based measurements of YKL-40 (indicating glial activation), phosphorylated tau (Ptau), and total tau (Ttau), were correlated with SWA and sigma topographical power employing Holm-Bonferroni correction. Multiple linear regression models were implemented controlling for age, apnea-hypopnea index (AHI), and sex at significant derivations. Finally, Sobel testing was employed to assess whether tau burden mediated YKL-40-sleep associations. Results Age was associated with YKL-40 (r=0.53, p=0.002), and YKL-40 was associated with both Ptau (r=0.66, p<0.001) and Ttau (r=0.68, p<0.001). Correlations between sigma2 activity and both Ptau and Ttau were detected at 14 derivations, 12 of which remained significant after controlling for age, sex, and AHI. YKL-40 was associated with sigma2 power (r=-0.39, p=0.025) across derivations expressing peak significance with tau. Sobel mediation analyses indicated that both Ptau (t=-2.15, p=0.031) and Ttau (t=-2.36, p=0.018) mediated the relationship between YKL-40 and sigma2 activity at these derivations. SWA was not associated with Ttau, Ptau, or YKL-40. Conclusion These results suggest that age-related increases in central glial activation may disrupt local expression of fast spindles by increasing tau burden, highlighting a potential role for chronic inflammation in sleep deficits observed in aging and Alzheimer’s disease. Support Supported by R56 AG052698, P50AG033514
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Yao, Lijun, Tianjiao Wang, Kazuhiro Sato, Reyka Jayasinghe, I.-Ling Chiang, Darwin D'souza, William Pilcher, et al. "Abstract 1729: Single-cell transcriptome profiling of multiple myeloma bone marrow samples suggests that disease progression interplays with tumor and tumor microenvironment in The MMRF CoMMpass Study." Cancer Research 82, no. 12_Supplement (June 15, 2022): 1729. http://dx.doi.org/10.1158/1538-7445.am2022-1729.
Abstract:
Abstract Multiple Myeloma (MM) is a hematologic malignancy marked by uncontrolled clonal expansion of plasma cells. Previous research has examined single-cell transcriptome profiles of Monoclonal gammopathy of undetermined significance (MGUS) and MM tumor microenvironment (TME) and found that natural killer (NK) cell abundance is elevated in the early stages and correlated with altered chemokine receptor expression. This study suggested the critical role of immune cells on myeloma progression from asymptomatic MGUS to symptomatic MM. Up to date, however, there are no published studies comprehensively comparing tumor and immune populations differences between MM NON-progressors (NPs) and FAST-progressors (FPs) and investigating how clonal plasma cells affect disease progression in a large cohort. Therefore, understanding how tumor and immune cells influence disease progression within symptomatic MM is of great interest.Here, we subjected CD138-negative Bone Marrow Mononuclear cells (BMMC) samples from 418 MM patients to scRNA-seq. From MMRF CoMMpass study (NCT01454297), we also have whole exome sequencing (WES) and bulk RNA-seq from CD138-positive fraction of BMMC samples. Based on time to progressive disease (TTPD), we classified patients into 2 categories. Patients with TTPD less than 18 months were classified as FAST-progressors (FPs), whereas patients with TTPD more than 5 years were classified as NON-progressors (NPs). By analyzing patient genomic alterations and its association with progression, we found that there was a significant association of slow MM progression with t(11;14) (p = 0.048), consistent with previous study. In our preliminary analysis, we profiled 83 CD138-sorted MM bone marrow samples using scRNA-seq. Interestingly, we found plasma cells from samples with the same genetic alterations tend to cluster together, highlighting the important role of genetic drivers in transcriptome profiles of plasma cells. Moreover, integrated analysis of bone marrow samples from 83 MM patients and 4 healthy donors revealed an atypical naïve-B cell subset with enrichment of cells from fast-progressors and partial expression of MS4A1. Differentially expressed genes for this naïve-B cell subset includes KLF9, BCL2L11, JOSD1, and IRS2, etc. Overall, as part of MMRF immune profiling research, this study will help to interrogate how genetic alterations and disease progression interplay MM tumor and TME and provide a sufficiently broad and valuable dataset for systematically characterizing MM at single-cell resolution. Hopefully, this study could identify novel targets for MM immunotherapies, and ultimately identify patients with high risk of fast progression for early intervention in the clinic. Citation Format: Lijun Yao, Tianjiao Wang, Kazuhiro Sato, Reyka Jayasinghe, I-Ling Chiang, Darwin D'souza, William Pilcher, Edgar Gonzalez-Kozlova, Yered Pita-Juarez, Taxiarchis Kourelis, Deon Bryant Doxie, Beena Thomas, Brian Lee, Swati Sharma Bhasin, Upadhyaya Bhaskar, Mark Fiala, Julie Fortier, Travis Dawson, John Leech, Shaji Kumar, Hearn Cho, Seunghee Kim-Schulze, Bee Raj, Stephen Oh, the MMRF Immune Profiling Research Team, John Dipersio, Ravi Vij, Adeeb Rahman, Ionnis Vlachos, Shaadi Mehr, Mark Hamilton, Daniel Auclair, Surendra Dasari, David Avigan, Madhav Dhodapkar, Sacha Gnjatic, Manoj Bhasin, Li Ding. Single-cell transcriptome profiling of multiple myeloma bone marrow samples suggests that disease progression interplays with tumor and tumor microenvironment in The MMRF CoMMpass Study [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2022; 2022 Apr 8-13. Philadelphia (PA): AACR; Cancer Res 2022;82(12_Suppl):Abstract nr 1729.
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Bourguin, Solesne, Siragan Gailus, and Konstantinos Spiliopoulos. "Typical dynamics and fluctuation analysis of slow–fast systems driven by fractional Brownian motion." Stochastics and Dynamics, October 13, 2020, 2150030. http://dx.doi.org/10.1142/s0219493721500301.
Abstract:
This paper studies typical dynamics and fluctuations for a slow–fast dynamical system perturbed by a small fractional Brownian noise. Based on an ergodic theorem with explicit rates of convergence, which may be of independent interest, we characterize the asymptotic dynamics of the slow component to two orders (i.e. the typical dynamics and the fluctuations). The limiting distribution of the fluctuations turns out to depend upon the manner in which the small-noise parameter is taken to zero relative to the scale-separation parameter. We study also an extension of the original model in which the relationship between the two small parameters leads to a qualitative difference in limiting behavior. The results of this paper provide an approximation, to two orders, to dynamical systems perturbed by small fractional Brownian noise and incorporating multiscale effects.
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Carter, Paul, and Alan R. Champneys. "Wiggly canards: Growth of traveling wave trains through a family of fast-subsystem foci." Discrete & Continuous Dynamical Systems - S, 2022, 0. http://dx.doi.org/10.3934/dcdss.2022036.
Abstract:
<p style='text-indent:20px;'>A class of two-fast, one-slow multiple timescale dynamical systems is considered that contains the system of ordinary differential equations obtained from seeking travelling-wave solutions to the FitzHugh-Nagumo equations in one space dimension. The question addressed is the mechanism by which a small-amplitude periodic orbit, created in a Hopf bifurcation, undergoes rapid amplitude growth in a small parameter interval, akin to a <i>canard explosion</i>. The presence of a saddle-focus structure around the slow manifold implies that a single periodic orbit undergoes a sequence of folds as the amplitude grows. An analysis is performed under some general hypotheses using a combination ideas from the theory of canard explosion and Shilnikov analysis. An asymptotic formula is obtained for the dependence of the parameter location of the folds on the singular parameter and parameters that control the saddle focus eigenvalues. The analysis is shown to agree with numerical results both for a synthetic normal-form example and the FitzHugh-Nagumo system.</p>
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Chini, Gregory P., Guillaume Michel, Keith Julien, Cesar B. Rocha, and Colm-cille P. Caulfield. "Exploiting self-organized criticality in strongly stratified turbulence." Journal of Fluid Mechanics 933 (December 23, 2021). http://dx.doi.org/10.1017/jfm.2021.1060.
Abstract:
A multiscale reduced description of turbulent free shear flows in the presence of strong stabilizing density stratification is derived via asymptotic analysis of the Boussinesq equations in the simultaneous limits of small Froude and large Reynolds numbers. The analysis explicitly recognizes the occurrence of dynamics on disparate spatiotemporal scales, yielding simplified partial differential equations governing the coupled evolution of slow large-scale hydrostatic flows and fast small-scale isotropic instabilities and internal waves. The dynamics captured by the coupled reduced equations is illustrated in the context of two-dimensional strongly stratified Kolmogorov flow. A noteworthy feature of the reduced model is that the fluctuations are constrained to satisfy quasilinear (QL) dynamics about the comparably slowly varying large-scale fields. Crucially, this QL reduction is not invoked as an ad hoc closure approximation, but rather is derived in a physically relevant and mathematically consistent distinguished limit. Further analysis of the resulting slow–fast QL system shows how the amplitude of the fast stratified-shear instabilities is slaved to the slowly evolving mean fields to ensure the marginal stability of the latter. Physically, this marginal stability condition appears to be compatible with recent evidence of self-organized criticality in both observations and simulations of stratified turbulence. Algorithmically, the slaving of the fluctuation fields enables numerical simulations to be time-evolved strictly on the slow time scale of the hydrostatic flow. The reduced equations thus provide a solid mathematical foundation for future studies of three-dimensional strongly stratified turbulence in extreme parameter regimes of geophysical relevance and suggest avenues for new sub-grid-scale parametrizations.
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Grundy, R. E. "The behaviour of a forced spherical pendulum operating in a weightless environment." Quarterly Journal of Mechanics and Applied Mathematics, November 29, 2023. http://dx.doi.org/10.1093/qjmam/hbad008.
Abstract:
Summary In this article, we show that by subjecting the pivot of a simple inextensible pendulum to small amplitude high frequency rectilinear oscillations it is possible to make it operate in a weightless environment. The axis of vibration of the pivot defines a preferred direction in space and a consequential dynamical structure which is completely absent when the pivot is fixed. Using spherical polar coordinates centred at the pivot, we show that the motion of such a pendulum has fast and slow-scale components which we analyse using the method of multiple scales. The slow scale equation for the polar angle is autonomous, and a phase plane analysis reveals the essential orbital structure including the existence of conical solutions analogous to the terrestrial fixed pivot conical pendulum. In the absence of an azimuthal velocity component, its behaviour can provide a direct simulation of a plane terrestrial simple fixed pivot pendulum with a correspondingly simple form for the small amplitude period. We can also use a two-scale analysis to examine the effects of damping. Here, the slow scale polar equation has two asymptotically stable states, and we employ a combination of numerical and asymptotic analyses to elicit the slow scale orbital trajectories.