Dynamics Of Perturbations Research Articles

Spectral Features of Systems With Chaotic Dynamics

Using the Lorentz model and Hamiltonian systems without dissipation as an example, spectral methods for analyzing the dynamics of systems with chaotic behavior are considered. The insufficiency of the traditional approach to the study of perturbation dynamics based on an analysis of the roots of the classical spectral equation is discussed. It is proposed to study nonlinear systems using the method of constructing spectral equations with different eigenvalues, which allows one to take into account the randomness and multiplicity of states. The spectral features of instability and chaos for systems without dissipation are shown by the example of short-wave perturbations of a flow of a weakly ionized plasma gas.

Normal modes of vibrations around Hubble flow in jellium

A macroscopic Coulomb system of identical charged particles with or without a compensating background charge can evolve maintaining spatial homogeneity and isotropy that mimic the cosmological evolution of a universe with repulsive gravity. Here we study dynamics of small perturbations on the background of the corresponding Hubble flow by analyzing its normal modes of vibrations. Arbitrary disturbance of the flow can be resolved into two electroacoustic, two vortical, and one entropic modes whose dynamics is investigated. Specifically, in the zero pressure or long-wavelength limits perturbations of density and velocity evolve in a manner that is independent of the form of the initial disturbance. The same conclusion applies to vortical perturbations of the velocity for arbitrary pressure while entropic perturbations are advected by the Hubble flow. Without the background charge the underlying Hubble flow describes a Coulomb explosion whose stability with respect to small disturbances is also demonstrated.

Nonlinear conditions for instability of the free surface of a conducting liquid in an external electric field in a confined axisymmetric geometry

The behavior of the free surface of a perfectly conducting liquid in an external uniform electric field is considered in the framework of the Hamiltonian formalism for the case of bounded axisymmetric geometry of the system (the fluid is bounded by a cylindrical rigid wall). Taking into account the influence of quadratic nonlinearities, we derive an amplitude equation which describes the evolution of the boundary. Using this equation, we find the condition for the hard excitation of boundary instability that leads to an explosive growth of surface perturbations. The differences in the description of the dynamics of axisymmetric perturbations of the boundary from the cases of plane, square, and hexagonal symmetries of the problem are discussed.

Mathematical Modeling of Spot Dynamics in a Stratified Medium

The investigation of dynamics of mixed fluid spots in a stratified environment is of interest both for the study of the ocean fine structure and for the study of wake dynamics behind moving underwater objects. The paper is devoted to the construction of a physical and mathematical model for this problem. Salinity is used as the stratifying component. This model is described by the Navier–Stokes equations in the Boussinesq approximation. The problem is solved using a recent version of the splitting method by physical factors the finite difference scheme of which has a high approximation order, minimum scheme viscosity and dispersion, and, which is especially important for problems with large gradients of hydrophysical parameters, problems with a free surface and internal waves, the monotonicity property. Numerous test computations for the study of the influence of grid parameters on the results are performed. The results of comparison with analytical estimates, experimental data, and computations of other researchers are presented. By way of example, the dynamics of the salinity perturbation is discussed, which corresponds to the isophase lines describing the behavior of internal waves during the collapse of spots.

Cosmological framework for renormalization group extended gravity at the action level

General relativity (GR) extensions based on renormalization group (RG) flows may lead to scale-dependent couplings with nontrivial effects at large distance scales. Here we develop further the approach in which RG effects at large distance scales are fully encoded in an effective action and we apply it to cosmology. In order to evaluate the cosmological consequences, our main assumption is the use of a RG scale such that the (infrared) RG effects only appear at perturbative order (not at the background level). The emphasis here is on analytical results and qualitative understanding of the implied cosmology. We employ commonly used parametrizations for describing modified gravity in cosmology (as the slip parameter). From them, we describe the dynamics of the first order perturbations and estimate bounds on the single dimensionless parameter (nu ) introduced by this framework. Possible impacts on dark matter and dark energy are discussed. It is also shown here that the nu parameter effects to fsigma _8 are stronger at low redshifts (z<1.5), while different values for nu do not appreciably change fsigma _8 at higher redshifts, thus opening a window to alleviate an issue that is currently faced by Lambda CDM.

Investigation of supersonic heat-conductivity hyperbolic waves in radiative ablation flows.

We carry out a numerical investigation of three-dimensional linear perturbations in a self-similar ablation wave in slab symmetry, representative of the shock transit phase of a pellet implosion in inertial confinement fusion (ICF). The physics of ablation is modeled by the equation of gas dynamics with a nonlinear heat conduction as an approximation for radiation transport. Linear perturbation responses of the flow, its external surface, and shock-wave front, when excited by external pressure or heat-flux perturbation pulses, are computed by fully taking into account the flow compressibility, nonuniformity, and unsteadiness. These responses show the effective propagation, at supersonic speeds, of perturbations from the flow external surface through the whole conduction region of the ablation wave, beyond its Chapman-Jouguet point, and up to the ablation front, after the birth of the ablation wave. This supersonic forward propagation of perturbations is evidenced by means of a set of appropriate pseudocharacteristic variables and is analyzed to be associated to the "heat-conductivity" waves previously identified by Clarisse etal. [J. Fluid Mech. 848, 219 (2018)JFLSA70022-112010.1017/jfm.2018.343]. Such heat-conductivity linear waves are found to prevail over heat diffusion as a feedthrough mechanism [Aglitskiy et al., Philos. Trans. R. Soc. A 368, 1739 (2010)PTRMAD1364-503X10.1098/rsta.2009.0131] for perturbations of longitudinal characteristic lengths of the order of-or larger than-the conduction region size, and long transverse wavelengths with respect to this region size, and over time scales shorter to much shorter than the shock transit phase duration. This mechanism, which results from the dependency of the heat conductivity on temperature and density in conjunction with a flow temperature stratification, is expected to occur for other types of nonlinear heat conductions-e.g., electron heat conduction-as well as to be efficient at transmitting large scale perturbations from the surrounding of an ICF pellet to its inner compressed core at later times of its implosion. Besides, the proposed set of pseudocharacteristic variables are recommended for analyzing perturbation dynamics in an ablation flow as it furnishes additional propagation information over the fundamental linear modes of fluid dynamics [Kovásznay, J. Aeronautic. Sci. 20, 657 (1953)1936-995610.2514/8.2793], especially in the flow conduction region.

Attitude control: A key factor during the design of low-thrust propulsion for CubeSats

Orbital capability is a decisive step forward for nanosatellites in general and CubeSats in particular. Although trajectory maneuvers and their implementation have been thoroughly studied for classical satellites, the high level of constraints on CubeSats in terms of mass, volume and power, makes the transition delicate. Orbit, attitude and power control subsystems available for this format limit too optimistic performance available in literature. To verify this hypothesis, we simulate trajectory maneuvers in Earth orbit with representative CubeSat hardware and software. A low-thrust trajectory solver based on classical orbital elements from the literature is adapted to our context. A home-made attitude control simulation tool is coupled to include both control and perturbative dynamics. Increases in time and propellant consumption of more than 100% are caused by thrust direction errors such as misalignments and attitude control limitations, sometimes leading to mission loss. These results highlight an important increase in complexity for the CubeSat format that is not covered by the usual approach. Such limitations should be considered from the very start of the design of a nanosatellite mission with trajectory modification requirements.

Boundary layer collapses described by the two-dimensional intermediate long-wave equation

We study the nonlinear dynamics of localized perturbations of a confined generic boundary-layer shear flow in the framework of the essentially two-dimensional generalization of the intermediate long-wave (2d-ILW) equation. The 2d-ILW equation was originally derived to describe nonlinear evolution of boundary layer perturbations in a fluid confined between two parallel planes. The distance between the planes is characterized by a dimensionless parameter D. In the limits of large and small D, the 2d-ILW equation respectively tends to the 2d Benjamin-Ono and 2d Zakharov-Kuznetsov equations. We show that localized initial perturbations of any given shape collapse, i.e., blow up in a finite time and form a point singularity, if the Hamiltonian is negative, which occurs if the perturbation amplitude exceeds a certain threshold specific for each particular shape of the initial perturbation. For axisymmetric Gaussian and Lorentzian initial perturbations of amplitude a and width σ, we derive explicit nonlinear neutral stability curves that separate the domains of perturbation collapse and decay on the plane (a, σ) for various values of D. The amplitude threshold a increases as D and σ decrease and tends to infinity at D → 0. The 2d-ILW equation also admits steady axisymmetric solitary wave solutions whose Hamiltonian is always negative; they collapse for all D except D = 0. But the equation itself has not been proved for small D. Direct numerical simulations of the 2d-ILW equation with Gaussian and Lorentzian initial conditions show that initial perturbations with an amplitude exceeding the found threshold collapse in a self-similar manner, while perturbations with a below-threshold amplitude decay.

Madelung transformation of the quantum bouncer problem

The “quantum bouncer” problem of a quantum particle in a gravitational field, bouncing elastically from a horizontal surface, is mapped via the Madelung transformation into the dynamics of a 1D, compressible, inviscid fluid. Under this transformation the energy states of the quantum bouncer, given in terms of the Airy function, are hydrostatic equilibria states where gravity is balanced by the corresponding vertical pressure gradient force exerted by the fluid. The stability of the hydrostatic states, with respect to small perturbations, is expressed in terms of the positive definite pseudoenergy integral derived here. Pseudoenergy, which is a constant of motion of the linearized perturbation dynamics, is in this case the sum of the perturbation kinetic and available potential energies. The latter is found to be proportional to the square of the vertical derivative of the buoyancy perturbation rather than the square of the buoyancy itself, as is generally the case in classical fluids. For sinusoidal perturbations with vertical wavenumber k, the effective frequency of the buoyancy oscillation is found to be , which is twice the ratio between the gravitational force and the momentum of a free quantum particle.

Oscillating scalar fields and the Hubble tension: A resolution with novel signatures

We present a detailed investigation of a sub-dominant oscillating scalar field ('early dark energy', EDE) in the context of resolving the Hubble tension. Consistent with earlier work, but without relying on fluid approximations, we find that a scalar field frozen due to Hubble friction until ${\rm log}_{10}(z_c)\sim3.5$, reaching $\rho_{\rm EDE}(z_c)/\rho_{\rm tot}\sim10$%, and diluting faster than matter afterwards can bring cosmic microwave background (CMB), baryonic acoustic oscillations, supernovae luminosity distances, and the late-time estimate of the Hubble constant from the SH0ES collaboration into agreement. A scalar field potential which scales as $V(\phi) \propto \phi^{2n}$ with $2\lesssim n\lesssim 3.4$ around the minimum is preferred at the 68% confidence level, and the {\em Planck} polarization places additional constraints on the dynamics of perturbations in the scalar field. In particular, the data prefers a potential which flattens at large field displacements. An MCMC analysis of mock data shows that the next-generation CMB observations (i.e., CMB-S4) can unambiguously detect the presence of the EDE at very high significance. This projected sensitivity to the EDE dynamics is mainly driven by improved measurements of the $E$-mode polarization. We also explore new observational signatures of EDE scalar field dynamics: (i) We find that depending on the strength of the tensor-to-scalar ratio, the presence of the EDE might imply the existence of isocurvature perturbations in the CMB. (ii) We show that a strikingly rapid, scale-dependent growth of EDE field perturbations can result from parametric resonance driven by the anharmonic oscillating field for $n\approx 2$. This instability and ensuing potentially nonlinear, spatially inhomogenoues, dynamics may provide unique signatures of this scenario.

Perturbation dynamics in Keplerian flow under external stochastic forcing

ABSTRACT We investigate the dynamics of linear perturbations in Keplerian flow under external stochastic force. To abstract from the details of flow structure and boundary conditions, we consider the problem in the shearing box approximation. An external force is assumed to have zero mean, even so, induced perturbations form a steady state, which provides angular momentum transfer to the periphery of the flow. The most effective scenario is based on the transient amplification of induced vortices with the following emission of a shearing sound wave, wherein the maximum of the flux linearly depends on Reynolds number. Thus such a mechanism is significant for astrophysical flows, for which enormous Reynolds numbers are typical. At the same time, addressing the problem analytically, we find that for incompressible fluid in the shearing box approximation stochastic forcing does not lead to average angular momentum transfer. Thus the compressibility of the fluid plays an important role here, and one cannot neglect it.

Anisotropy of Isobaric Instability in a Partially Ionized Plasma in a Magnetic Field

Linear equations describing the dynamics of small perturbations in partially ionized heat-releasing plasma in an external magnetic field with an arbitrary induction vector are obtained. Using these equations, a dispersion relation for condensation modes is derived, and their time increments are calculated in the case of isobaric instability in the presence of anisotropy of the plasma thermal conductivity.

Benchmarking semiclassical and perturbative methods for real-time simulations of cavity-bound emission and interference

We benchmark a selection of semiclassical and perturbative dynamics techniques by investigating the correlated evolution of a cavity-bound atomic system to assess their applicability to study problems involving strong light-matter interactions in quantum cavities. The model system of interest features spontaneous emission, interference, and strong coupling behavior and necessitates the consideration of vacuum fluctuations and correlated light-matter dynamics. We compare a selection of approximate dynamics approaches including fewest switches surface hopping (FSSH), multitrajectory Ehrenfest dynamics, linearized semiclassical dynamics, and partially linearized semiclassical dynamics. Furthermore, investigating self-consistent perturbative methods, we apply the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy in the second Born approximation. With the exception of fewest switches surface hopping, all methods provide a reasonable level of accuracy for the correlated light-matter dynamics, with most methods lacking the capacity to fully capture interference effects.

Instability of Surface Quasigeostrophic Spatially Periodic Flows

AbstractThe surface quasigeostrophic (SQG) model is developed to describe the dynamics of flows with zero potential vorticity in the presence of one or two horizontal boundaries (Earth surface and tropopause). Within the framework of this model, the problems of linear and nonlinear stability of zonal spatially periodic flows are considered. To study the linear stability of flows with one boundary, two approaches are used. In the first approach, the solution is sought by decomposing into a trigonometric series, and the growth rate of the perturbations is found from the characteristic equation containing an infinite continued fraction. In the second approach, few-mode Galerkin approximations of the solution are constructed. It is shown that both approaches lead to the same dependence of the growth increment on the wavenumber of perturbations. The existence of instability with a preferred horizontal scale on the order of the wavelength of the main flow follows from this dependence. A similar result is obtained within the framework of the SQG model with two horizontal boundaries. The Galerkin method with three basis trigonometric functions is also used to study the nonlinear dynamics of perturbations, described by a system of three nonlinear differential equations similar to that describing the motion of a symmetric top in classical mechanics. An analysis of the solutions of this system shows that the exponential growth of disturbances at the linear stage is replaced by a stage of stable nonlinear oscillations (vacillations). The results of numerical integration of full nonlinear SQG equations confirm this analysis.

On Mean Field Games for Agents With Langevin Dynamics

Mean field games (MFG) have emerged as a viable tool in the analysis of large-scale self-organizing networked systems. In particular, MFGs provide a game-theoretic optimal control interpretation of the emergent behavior of noncooperative agents. The purpose of this paper is to study MFG models in which individual agents obey multidimensional nonlinear Langevin dynamics, and analyze the closed-loop stability of fixed points of the corresponding coupled forward-backward partial differential equation (PDE) systems. In such MFG models, the detailed balance property of the reversible diffusions underlies the perturbation dynamics of the forward–backward system. We use our approach to analyze closed-loop stability of two specific models. Explicit control design constraints, which guarantee stability, are obtained for a population distribution model and a mean consensus model. We also show that static state feedback using the steady-state controller can be employed to locally stabilize an MFG equilibrium.

Excitation of Non-Wave Modes by Sound of Arbitrary Frequency in a Chemically Reacting Gas

The nonlinear phenomena in the field of high intensity sound propagating in a gas with a chemical reaction, are considered. A chemical reaction of A → B type is followed by dispersion and attenuation of sound which may be atypical during irreversible thermodynamic processes under some conditions. The first and second order derivatives of heat produced in the chemical reaction evaluated at the equilibrium temperature, density and mass fraction of reagent A, are taken into account. The instantaneous equations are derived which govern dynamics of perturbations in non-acoustic modes, and conclusions of the efficiency of their nonlinear excitation by sound are drawn. The advantage of this study is accurate description of dispersion. Acoustic perturbations of any characteristic duration as compared to the duration of chemical reaction are considered, along with periodic, aperiodic perturbations and impulses. The conclusions concern also acoustically active gases.

Parameter analysis of on-orbit volumetric measurement using cavity resonance

Propellant volumetric measurement based on the cavity acoustic resonance may be a promising alternate in space microgravity environment. The present paper concentrates on the influences of the hemispherical structure of the tank, steady/unsteady non-flat continuous propellant-gas interface, and propellant's compressibility on the acoustic resonant frequency of the system. A one dimensional mathematical acoustic resonant model, where acoustic perturbation dynamics only in the axial coordinate is considered, based on conservations of mass and momentum is provided and compared with numerical simulations by COMSOL software and experiments conducted by the previous research. Results show that analytical predictions by the present model keep consistent with numerical simulations and experiments. While the structure of a hemispherical tank can influence the resonant frequency, the present model can optimizationally compensate such effect. The continuous non-flat propellant-gas interface changes the resonant frequency. A flatter interface leads to smaller influence. On the other hand, the unsteady low-frequency low-amplitude wave-like movement of the propellant-gas interface imposes negligible effluence on the resonant frequency of the tank. Finally, the propellant compressibility is quantitatively shown to be negligible for most propellants.

Dynamics of perturbations at the critical points between modulation instability and stability regimes.

We study numerically the evolutions of perturbations at critical points between modulational instability and stability regimes. It is demonstrated that W-shaped solitons and rogue waves can be both excited from weak resonant perturbations at the critical points. The rogue wave excitation at the critical points indicates that rogue wave comes from modulation instability with resonant perturbations, even when the baseband modulational instability is absent. The perturbation differences for generating W-shaped solitons and rogue waves are discussed in detail. These results can be used to generate W-shaped solitons and rogue waves controllably from weak perturbations.

Slow-roll inflation in generalized scalar-torsion gravity

We study the dynamics of inflation in a generalized scalar-torsion gravity scenario by assuming a canonical scalar field non-minimally coupled to torsion with a Galileon-type self-interaction. After obtaining the field equations for a flat FRW background, we derive the second order action for both scalar and tensor perturbations to compute the power spectra of primordial fluctuations. As particular models, we studied at first, a power-law form of coupling function F(x)=1+ξ x2/2, with x≡ ϕ/Mpl, and a monomial scalar field potential V(x)=λ xn/n which is ruled out by current observational data for n⩾ 2. Under slow-roll approximation we obtain analytical expressions for the background as well as perturbative dynamics, and we show that the predictions of the model are consistent with current Planck 2018 constraints on the spectral index ns and the tensor-to-scalar ratio r through the ns−r plane. Accordingly, this model is in agreement with current observational bounds only within the 95% C.L. region in the case of chaotic quadratic inflation (n=2), whereas that for the other monomial potentials such as n=4/3, n=1 and n=2/3, it is found that they are even more favoured, overlapping their results with the 68% C.L. region from last Planck data. Secondly, we studied a model in which the presence of both non-minimal coupling to gravity and the Galileon non-linear self-interaction γ (∂ ϕ)2 □ϕ leads to a suppression of the tensor-to-scalar ratio compared to those predicted in the standard scenario, then predicting 0.024≲ r≲ 0.069. This result allows us to reconcile chaotic quadratic inflation with current Planck data up to the 68% C.L. region. In despite of this, next generation CMB experiments such as BICEP3 or LiteBIRD are expected to put stronger constraints, making possible to either support this model or rule it out.

Studies on the decays in the perturbative QCD approach**Research exercises for excellent undergraduate students.

The decays are investigated for the first time in the perturbative QCD formalism based on the kT factorization theorem, where the light scalar κ is assumed as a two-quark state. Our numerical results and phenomenological analyses on the CP-averaged branching ratios and CP-violating asymmetries show that: (a) the and decays have large decay rates around , which could be examined by the upgraded large hadron collider beauty and/or Belle-II experiments in the near future; (b) a large decay rate of about appears in the pure annihilation channel, which could provide more evidence to help distinguish different QCD-inspired factorization approaches, even understand the annihilation decay mechanism; (c) the pure penguin modes and would provide a promising ground to search for the possible new physics because of their zero direct and mixing-induced CP violations in the standard model. The examinations with good precision from the future experiments will help to further study the perturbative and/or nonperturbative QCD dynamics involved in these considered decay modes.