Finite Perturbations Research Articles

Stability of a Queue Fed by Scheduled Traffic at Critical Loading

Performance of a queueing system with scheduled arrivals A scheduled arrival sequence is one in which customers are scheduled to arrive at constant interarrival times, but each customer’s actual arrival time is perturbed from her scheduled arrival time by a random perturbation. In “Stability of a Queue Fed by Scheduled Traffic at Critical Loading”, V.F. Araman and P.W. Glynn consider a single server queue with deterministic service times in which customers arrive following a scheduled arrival process. Unlike a queue fed by renewal traffic, this queue is shown to be stable even when the utilization is equal to one. It is also shown that for finite mean perturbations, a necessary and sufficient condition for stability is when the positive part of the perturbation has bounded support, with no requirement on the negative part of the perturbation. Perhaps surprisingly, this criterion is not reversible, in the sense that such a queue can be stable for a scheduled traffic process in forward time, but unstable for the time-reversal of the same traffic process.

Stability analysis of viscous multi-layer shear flows with interfacial slip

Abstract One of the most fundamental interfacial instabilities in ideal, immiscible, incompressible multifluid flows is the celebrated Kelvin–Helmholtz (KH) instability. It predicts short-wave instabilities that, in the absence of other mollifying physical mechanisms (e.g. surface tension, viscosity), render the nonlinear problem ill-posed and lead to finite-time singularities. The crucial driving mechanism is the jump in tangential velocity across the liquid–liquid interface, i.e. interfacial slip, that can occur since viscosity is absent. The purpose of the present work is to analyse analogous instabilities for viscous flows at small or moderate Reynolds numbers as opposed to the infinite Reynolds numbers that underpin KH instabilities. The problem is physically motivated by both experiments and simulations. The fundamental model considered consists of two superposed viscous, incompressible, immiscible fluid layers sheared in a plane Couette flow configuration, with slip present at the deforming liquid–liquid interface. The origin of slip in viscous flows has been observed in experiments and molecular dynamics simulations, and can be modelled by employing a Navier-slip boundary condition at the liquid–liquid interface. The emerging novel instabilities are studied in detail here. The linear stability of the system is addressed asymptotically for long- and short-waves, and for arbitrary wavenumbers using a combination of analytical and numerical calculations. Slip is found to be capable of destabilising perturbations of all wavelengths. In regimes where the flow is stable to perturbations of all wavelengths in the absence of slip, its presence can induce a Turing-type instability by destabilization of a small band of finite wavenumber perturbations. In the case where the underlying layer is asymptotically thin, the results are found to agree with the linear properties of a weakly non-linear asymptotic model that is also derived here. The weakly nonlinear model extends previous work by the authors that had a thin overlying layer that produces a different evolution equation.

Oscillons in gapless theories

We show that large scale oscillons, i.e., quasiperiodic, long-living particlelike solutions, may exist in massless theories, too. Their existence is explained using an effective (smeared) mass threshold which takes into account nonlinear (finite) perturbations. Published by the American Physical Society 2024

Transition Threshold for the 3D Couette Flow in a Finite Channel

In this paper, we study nonlinear stability of the 3D plane Couette flow ( y , 0 , 0 ) (y,0,0) at high Reynolds number R e {Re} in a finite channel T × [ − 1 , 1 ] × T \mathbb {T}\times [-1,1]\times \mathbb {T} . It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. This work shows that if the initial velocity v 0 v_0 satisfies ‖ v 0 − ( y , 0 , 0 ) ‖ H 2 ≤ c 0 R e − 1 \|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1} for some c 0 > 0 c_0>0 independent of R e Re , then the solution of the 3D Navier-Stokes equations is global in time and does not transit away from the Couette flow in the L ∞ L^\infty sense, and rapidly converges to a streak solution for t ≫ R e 1 3 t\gg Re^{\frac 13} due to the mixing-enhanced dissipation effect. This result confirms the threshold result obtained by Chapman via an asymptotic analysis(JFM 2002). The most key ingredient of the proof is the resolvent estimates for the full linearized 3D Navier-Stokes system around the flow ( V ( y , z ) , 0 , 0 ) (V(y,z), 0,0) , where V ( y , z ) V(y,z) is a small perturbation(but independent of R e Re ) of y y .

On a nonlocal integral operator commuting with the Laplacian and the Sturm–Liouville problem: Low rank perturbations of the operator

We reformulate all general real coupled self-adjoint boundary value problems as integral operators and show that they are all finite rank perturbations of the free space Green’s function on the real line. This free space Green’s function corresponds to the nonlocal boundary value problem proposed earlier by Saito [Appl. Comput. Harmon. Anal. 25, 68–97 (2008)]. We prove these perturbations to be polynomials of rank up to 4. They encapsulate in a fundamental way the corresponding boundary conditions.

Finite rank perturbations of normal operators: hyperinvariant subspaces and a problem of Pearcy

Finite rank perturbations of normal operators: hyperinvariant subspaces and a problem of Pearcy

The peak model for finite rank supersingular perturbations

The peak model for finite rank supersingular perturbations

Nonlinear analysis of coupled neutronic-thermohydraulic stability characteristics of supercritical water-cooled reactor

The Supercritical Water-Cooled Reactor (SCWR) exhibits significant changes in thermophysical properties of the coolant, but its dynamic characteristics are different from existing reactors, due to the supercritical conditions of the coolant. This necessitates the study of the stability behaviour of SCWR, which requires a dynamic model of the reactor. In this work, a simple unsteady lumped parameter model (LPM) for SCWR has been developed. The LPM includes point reactor kinetics for neutron balance and a two-region model for fuel and coolant thermal hydraulics. The two regions are separated from each other by a boundary that depends on the pseudocritical temperature of the coolant. Regions of stable and unstable operation are identified in the parameter space by linear stability analysis. Bifurcation analysis is carried out to capture the non-linear dynamics of the system. Generalised Hopf (GH) points are located, and parameter ranges for supercritical (soft and safe) and subcritical (hard and dangerous) Hopf bifurcation are identified. The type of transient behaviour of the system for finite (though small) perturbations is predicted based on this analysis. The existence of predicted limit cycles is confirmed by numerical simulation of the nonlinear ODEs, using the shooting technique in case of unstable limit cycles. Furthermore, the effects of geometric, control and neutronics parameters on the stability of the system are studied.

Tamed stability of finite difference schemes for the transport equation on the half-line

In this paper, we prove that, under precise spectral assumptions, some finite difference approximations of scalar leftgoing transport equations on the positive half-line with numerical boundary conditions are ℓ 1 \ell ^1 -stable but ℓ q \ell ^q -unstable for any q > 1 q>1 . The proof relies on the accurate description of the Green’s function for a particular family of finite rank perturbations of Toeplitz operators whose essential spectrum belongs to the closed unit disk and with a simple eigenvalue of modulus 1 1 embedded into the essential spectrum.

Finite rank perturbations of heavy-tailed Wigner matrices

One-rank perturbations of Wigner matrices have been closely studied: let [Formula: see text] with [Formula: see text] symmetric, [Formula: see text] i.i.d. with centered standard normal distributions, and [Formula: see text] It is well known [Formula: see text] the largest eigenvalue of [Formula: see text] has a phase transition at [Formula: see text]: when [Formula: see text] [Formula: see text] whereas for [Formula: see text] [Formula: see text] Under more general conditions, the limiting behavior of [Formula: see text] appropriately normalized, has also been established: it is normal if [Formula: see text] or the convolution of the law of [Formula: see text] and a Gaussian distribution if [Formula: see text] is concentrated on one entry. These convergences require a finite fourth moment, and this paper considers situations violating this condition. For symmetric distributions [Formula: see text] heavy-tailed with index [Formula: see text] the fluctuations are shown to be universal and dependent on [Formula: see text] but not on [Formula: see text] whereas a subfamily of the edge case [Formula: see text] displays features of both the light- and heavy-tailed regimes: two limiting laws emerge and depend on whether [Formula: see text] is localized, each presenting a continuous phase transition at [Formula: see text] respectively. These results build on our previous work which analyzes the asymptotic behavior of [Formula: see text] in the aforementioned subfamily.

Punctuated chaos and the unpredictability of the Galactic Centre S-star orbital evolution

ABSTRACT We investigate the chaotic behaviour of the S-star cluster in the Galactic Centre using precise N-body calculations, free from round-off or discretization errors. Our findings reveal that chaos among the Galactic Centre S-stars arises from close encounters, particularly among pairs and near the massive central body. These encounters induce perturbations, causing sudden changes in the orbital energies of the interacting stars. Consequently, neighbouring solutions experience roughly exponential growth in separation. We propose a theory of ‘punctuated chaos’ that describes the S-star cluster’s chaotic behaviour. This phenomenon results from nearly linear growth in the separation between neighbouring orbits after repeated finite perturbations. Each participating star’s orbit experiences discrete, abrupt changes in energy due to the perturbations. The cumulative effect of these events is further amplified by the steady drift in orbital phase. In the Galactic Centre, perturbations originate from coincidental encounters occurring within a distance of ≲100 au between at least two stars (in some cases, three stars). Our model satisfactorily explains the observed exponential growth in the 27 S-stars cluster. We determine that the S-star system has a Lyapunov time-scale of approximately 462 ± 74 yr. For the coming millennium, chaos in the S-star cluster will be driven mainly by a few of the closest orbiting stars: S2, S5, S6, S8, S9, S14, S18, S31, S21, S24, S27, S29, and S38.

Finite rank perturbations of normal operators: Spectral idempotents and decomposability

We prove that a large class of finite rank perturbations of diagonal operators and, in general, of diagonalizable normal operators of multiplicity one acting boundedly on a separable, infinite dimensional complex Hilbert space are decomposable operators in the sense of Colojoară and Foiaş [1]. Consequently, every operator T in such a class has a rich spectral structure and plenty of non-trivial closed hyperinvariant subspaces which extends, in particular, previous theorems of Foiaş, Jung, Ko and Pearcy [5–7], Fang and J. Xia [3] and the authors [8,9] on an open question posed by Pearcy in the seventies.

Chirped periodic and localized waves of the (1+2)-dimensional chiral nonlinear Schrödinger equation

We study the existence and stability of nonlinearly chirped periodic waves and soliton structures in an optical medium wherein the pulse propagation is governed by the (1+2)-dimensional chiral nonlinear Schrödinger equation. An exact periodic wave solution is presented for the model equation in the presence of all physical processes by using the complex envelope traveling-wave ansatz. A class of optical gray-type solitons is obtained in the special long wave limit. The properties of these structures such as the velocity and wave number are determined by the system parameters. It is found that the frequency chirp accompanying these optical waves is inversely proportional to the intensity of the wave and its amplitude can be controlled by choosing the dispersion parameter appropriately. In addition, the stability of these waveforms is discussed numerically under some initial perturbations. The results show that those nonlinear waves can propagate in a stable fashion in the nonlinear medium under finite initial perturbations, such as amplitude and white noise.

Interaction of Finite Perturbations with the Front and Singularity of the Profile of a Nonlinear Wave in a Dissipative Medium

The interaction of a short pulse with three types of wave profiles in a nonlinear dissipative medium are described: a stationary shock front of finite width, a nonstationary forming front, and a “stationary” wave the profile of which contains a singularity. Exact solutions to the Burgers equation are used, which describe these interactions, as well as the symmetry properties of this equation.

Dispersive estimates for the Schrödinger equation with finite rank perturbations

In this paper, we investigate dispersive estimates for the time evolution of HamiltoniansH=−Δ+∑j=1N〈⋅,φj〉φjinRd,d≥1, where each φj satisfies certain smoothness and decay conditions. We show that, under a spectral assumption, there exists a constant C=C(N,d,φ1,…,φN)>0 such that‖e−itH‖L1−L∞≤Ct−d2,fort>0. As far as we are aware, this seems to provide the first study of L1−L∞ estimates for finite rank perturbations of the Laplacian in any dimension.We first deal with rank one perturbations (N=1). Then we turn to the general case. By using an Aronszajn-Krein type formula for finite rank perturbations, we reduce the problem to the rank one case and solve it in a unified manner. Moreover, we show that in some specific situations, the constant C(N,d,φ1,…,φN) grows polynomially in N. Finally, as an application, we are able to extend the results to N=∞ and deal with some trace class perturbations.

Interaction of Finite Perturbations with the Front and Singularity of the Profile of a Nonlinear Wave in a Dissipative Medium

The interaction of a short pulse with three types of wave profiles in a nonlinear dissipative medium are described: a stationary shock front of finite width, a nonstationary forming front, and a “stationary” wave the profile of which contains a singularity. Exact solutions to the Burgers equation are used, which describe these interactions, as well as the symmetry properties of this equation.

Propagation of chirped gray solitons in weakly nonlocal media with parabolic law nonlinearity and spatio-temporal dispersion

In our research, we have examined the existence and stability of chirped periodic and solitary waves in a weakly nonlocal nonlinear medium, which exhibits various types of effects, including inter-modal dispersion, nonlinear dispersion, detuning, spatio-temporal, and parabolic law nonlinearity. By studying the nonlinear Schrödinger equation that describes the field dynamics in this system, a class of nonlinearly chirped periodic waves is derived in the presence of all physical processes. In addition, we have obtained solitary waves of the gray type in the long-wave limit of these nonlinear waveforms. We have found that the frequency chirp associated with these optical waves depends on their intensity and its magnitude can be controlled by manipulating the nonlinear dispersion parameter. Furthermore, we have numerically studied the stability of the gray soliton solution under finite initial perturbations. Our results indicate that the nonlinear waves we have identified represent new types of extremely robust chirped localized structures in weakly nonlocal nonlinear parabolic law media.

Finite Perturbations by Yield Stress of the Constitutive Relations of Nonlinear Viscous Media

Finite Perturbations by Yield Stress of the Constitutive Relations of Nonlinear Viscous Media

Spontaneous phase coordination and fluid pumping in model ciliary carpets

Ciliated tissues, such as in the mammalian lungs, brains, and reproductive tracts, are specialized to pump fluid. They generate flows by the collective activity of hundreds of thousands of individual cilia that beat in a striking metachronal wave pattern. Despite progress in analyzing cilia coordination, a general theory that links coordination and fluid pumping in the limit of large arrays of cilia remains lacking. Here, we conduct in silico experiments with thousands of hydrodynamically interacting cilia, and we develop a continuum theory in the limit of infinitely many independently beating cilia by combining tools from active matter and classical Stokes flow. We find, in both simulations and theory, that isotropic and synchronized ciliary states are unstable. Traveling waves emerge regardless of initial conditions, but the characteristics of the wave and net flows depend on cilia and tissue properties. That is, metachronal phase coordination is a stable global attractor in large ciliary carpets, even under finite perturbations to cilia and tissue properties. These results support the notion that functional specificity of ciliated tissues is interlaced with the tissue architecture and cilia beat kinematics and open up the prospect of establishing structure to function maps from cilium-level beat to tissue-level coordination and fluid pumping.

Chain sequences and zeros of polynomials related to a perturbed [formula omitted] type recurrence relation

In this manuscript, new algebraic and analytic aspects of the orthogonal polynomials satisfying RII type recurrence relation given by Pn+1(x)=(x−cn)Pn(x)−λn(x2+ω2)Pn−1(x),n≥0,ω∈R∖{0},with P−1(x)=0 and P0(x)=1 are investigated when the recurrence coefficients are perturbed. Specifically, representation of new perturbed polynomials (co-polynomials of RII type) in terms of original ones with the interlacing and monotonicity properties of zeros are given. For finite perturbations, a transfer matrix approach is used to obtain new structural relations. Further, it is shown that the transfer matrix approach is computationally more efficient than the classical approach. The consequences of such perturbations on the unit circle are analysed. A particular perturbation in the chain sequence called complementary chain sequences and its effect on the corresponding reflection coefficients is also studied. An illustration involving complementary Romanovski–Routh polynomials is presented.