Classical Perturbation Research Articles

Smoothed “Slack Cable” Models for Large Amplitude Oscillations of Suspension Bridges

This paper proposes a technique to study the dynamics of suspension bridges in those cases where the use of traditional models is not possible, i.e., when the deformed shape of the structure allows the simultaneous presence of taut and slack hangers. The typical unilateral behavior of these suspension elements, which are unable to transmit compressive forces, introduces a discontinuity in the stiffness that implies strong computational difficulties in the solution of the equations governing the motion; as a consequence, the response of the continuous or discrete models proposed may be evaluated only through step by step time integration. In this paper an alternative technique is proposed, which replaces the distribution of unilateral hangers with a smooth nonlinear regular system made equivalent to the effective one by means of an energy criterion. The proposed technique, referred to as “equivalent nonlinearization,” allows description of the motion through partial differential equations that are nonlinear but regular and, therefore, can be solved through classical perturbation techniques. The paper also describes two models formulated on the basis of the previously mentioned technique and some classes of solutions corresponding to particular characteristics of the forcing action. Dedicated to Prof. Giuliano Augusti for his imminent 70th birthday.

A formulation of a reactivity variation for taking into account spatial and energy collapsing

This paper presents an original method to evaluate the reactivity worth induced by space or energy condensation procedures commonly used by reactor physicists. Exact and approximate formulas are derived, which allow assigning a reactivity worth to each computation region in order to determine its importance. Some examples are shown aiming at highlighting that results obtained using the proposed formulation have a physical meaning and could be used to define an optimised spatial mesh and to analyse in a new way the usual energy collapsing process. This formulation can be considered as complement to the classical perturbation theory definition of a reactivity variation.

Nonlinear Flow Between Permeable Disks Using Computer‐Extended Series Method

The problem of two‐dimensional, steady, nonlinear flow of an incompressible, viscous fluid between two parallel permeable disks is studied using the computer‐extended series solution (CESS). The limitation of the classical regular perturbation technique (RPT) in solving this problem is highlighted and the CESS method in conjunction with Padé approximation is advocated to analyze the problem for much larger values of suction/injection Reynolds number R and to achieve higher accuracy. The skin‐friction coefficient and coefficient of pressure distribution are evaluated for different values of R. The advantages of using CESS method over the RPT and numerical technique are discussed.

Delocalization of quantum wavepacket in coherently perturbed kicked Anderson model

We numerically investigate quantum diffusion of the initially localized wave packet in kicked one-dimensional disordered system (1DDS) with classical coherent perturbation. The wave packet localizes in the unperturbed kicked Anderson model. On the other hand, it diffuses obeying subdiffusion or normal diffusion law by coupling with coherent perturbation consisting of a few frequency components. We call the state “dynamically delocalized state”. We give the perturbation strength dependence of the diffusion rate. Moreover, it is numerically shown that in the subdiffusive cases the corresponding space ( n)–time ( t) distribution functions are characterized by a unified scaling form P( n, t)∝exp{−const(| n|/ t α/2 ) β } ( 0⩽α⩽1, 1⩽β⩽2 ) concerning the space–time indexes ( α, β). It is also shown that an universal relation β=2/(2− α) works on the delocalization in the perturbed kicked Anderson model.

Nonlinear spectral analysis of flow through porous media with isotropic lognormal hydraulic conductivity

We extend previous work on flow through D-dimensional random porous media with isotropic, lognormal and multifractal hydraulic conductivity K to include all (not necessarily multifractal) isotropic lognormal K fields. We use an approximate nonlinear analysis method to obtain the marginal distributions of the hydraulic gradient ∇ H and the specific flow q ̄ , the effective conductivity K eff, and the spectral density tensors of the ∇ H and q̱ fields. We find that the amplitudes of ∇ H and q̱ have lognormal distribution and their common orientation is distributed like Brownian motion on the unit sphere in R D , at a ‘time’ that depends on the variance of ln( K) and the space dimension D. Under ergodicity conditions, we obtain K eff= E[ K]exp{−(1/ D) σ ln( K) 2}, which is the formula conjectured by Matheron [Elements Pour Une Theorie Des Millieux Poreaux, 166 pp]. Our spectral density tensors of ∇ H and q̱ differ from the classical linear perturbation results in that they become more isotropic as the wavenumber amplitude k=| k ̄ | increases. A second difference, which becomes noticeable when log( K) has a broad-band spectrum and high variance, is that the nonlinear spectra decay more slowly with k. The theoretical results are confirmed through two-dimensional simulations.

Perturbation bounds for the joint spectrum of commuting matrices

Using the Clifford algebra techniques of Pryde, Bhatia and Bhattacharyya generalized the classical eigenvalue perturbation bound of Henrici to the joint spectrum of commuting tuples of matrices. Elsner improved on this result. Using the techniques of Pryde and Chu, we show that for small perturbations a smaller bound may be given than that obtained by Elsner. A similar result is then obtained using the technique of Elsner himself.

Nonlinear spectral analysis of flow through porous media with isotropic lognormal hydraulic conductivity

We extend previous work on flow through D-dimensional random porous media with isotropic, lognormal and multifractal hydraulic conductivity K to include all (not necessarily multifractal) isotropic lognormal K fields. We use an approximate nonlinear analysis method to obtain the marginal distributions of the hydraulic gradient ∇H and the specific flow q̄, the effective conductivity Keff, and the spectral density tensors of the ∇H and q̱ fields. We find that the amplitudes of ∇H and q̱ have lognormal distribution and their common orientation is distributed like Brownian motion on the unit sphere in RD, at a ‘time’ that depends on the variance of ln(K) and the space dimension D. Under ergodicity conditions, we obtain Keff=E[K]exp{−(1/D)σln(K)2}, which is the formula conjectured by Matheron [Elements Pour Une Theorie Des Millieux Poreaux, 166 pp]. Our spectral density tensors of ∇H and q̱ differ from the classical linear perturbation results in that they become more isotropic as the wavenumber amplitude k=|k̄| increases. A second difference, which becomes noticeable when log(K) has a broad-band spectrum and high variance, is that the nonlinear spectra decay more slowly with k. The theoretical results are confirmed through two-dimensional simulations.

Erratum: “Secular Evolution of Hierarchical Triple Star Systems” (ApJ, 535, 385 [2000

We derive octupole-level secular perturbation equations for hierarchical triple systems, using classical Hamiltonian perturbation techniques. By extending previous work done to leading (quadrupole) order to octupole level (i.e., including terms of order alpha^3, where alpha = a_1/a_2 < 1 is the ratio of semimajor axes) we obtain expressions that are applicable to a much wider range of parameters. For triple systems containing a close inner binary, we also discuss the possible interaction between the classical Newtonian perturbations and the general relativistic precession of the inner orbit. In some cases we show that this interaction can lead to resonances and a significant increase in the maximum amplitude of eccentricity perturbations. We establish the validity of our analytic expressions by providing detailed comparisons with the results of direct numerical integrations of the three-body problem obtained for a large number of representative cases. We also discuss applications of the theory in the context of several observed triple systems of current interest, including the millisecond pulsar PSR B1620-26 in M4, the giant planet in 16 Cygni, and the protostellar binary TMR-1. [short version]

Delocalization and Sensitivity of Quantum Wavepacket in Coherently Perturbed Kicked Anderson Model

We consider quantum diffusion of the initially localized wavepacket in one-dimensional kicked disordered system with classical coherent perturbation. The wavepacket localizes in the unperturbed kicked Anderson model. However, the wavepacket get delocalized even by coupling with monochromatic perturbation. We call the state "dynamically delocalized state". It is numerically shown that the delocalized wavepacket spread obeying diffusion law, and the perturbation strength dependence of the diffusion rate is given. The sensitivity of the delocalized state is also shown by the time-reversal experiment after random change in phase of the wavepacket. Moreover, it is found that the diffusion strongly depend on the initial phase of the perturbation. We discuss a relation between the "classicalization" of the quantum wave packet and the time-dependence of the initial phase dependence. The complex structure of the initial phase dependence is related to the entropy production in the quantum system.

Secular dynamics of the three-body problem: application to the υ Andromedae planetary system

The discovery of extra-solar planetary systems with multiple planets in highly eccentric orbits (∼0.1–0.6), in contrast with our own Solar System, makes classical secular perturbation analysis very limited. In this paper, we use a semi-numerical approach to study the secular behavior of a system composed of a central star and two massive planets in co-planar orbits. We show that the secular dynamics of this system can be described using only two parameters, the ratios of the semi-major axes and the planetary masses. The main dynamical features of the system are presented in geometrical pictures that allows us to investigate a large domain of the phase space of this three-body problem without time-expensive numerical integrations of the equations of motion, and without any restriction on the magnitude of the planetary eccentricities. The topology of the phase space is also investigated in detail by means of spectral map techniques, which allow us to detect the separatrix of a non-linear secular apsidal resonance. Finally, the qualitative study is supplemented by direct numerical integrations. Three different regimes of secular motion with respect to the secular angle Δ ϖ are possible: they are circulation, oscillation (around 0° and 180°), and high eccentricity libration in a non-linear secular resonance. The first two regimes are a continuous extension of the classical linear secular perturbation theory; the last is a new feature, hitherto unknown, in the secular dynamics of the three-body problem. We apply the analysis to the case of the two outer planets in the υ Andromedae system, and obtain its periodic and ordinary orbits, the general structure of its secular phase space, and the boundaries of its secular stability; we find that this system is secularly stable over a large domain of eccentricities. Applying this analysis to a wide range of planetary mass and semi-major axis ratios (centered about the υ Andromedae parameters), we find that apsidal oscillation dominates the secular phase space of the three-body coplanar system, and that the non-linear secular resonance is also a common feature.

Stochastic modeling of complex nonstationary groundwater systems

Despite the intensive research over the past two decades in the field of stochastic subsurface hydrology, a substantial gap remains between theory and application. The most popular stochastic theories are still based on closed-form solutions that apply, strictly speaking, only for statistically uniform flows. In this paper, we present an efficient, nonstationary spectral approach for modeling complex stochastic flows in moderately heterogeneous media. Specifically, we reformulate the governing stochastic equations and introduce a new transfer function to characterize the propagation of system uncertainty. The new transfer function plays a similar role as the commonly used Green’s functions in classical stochastic perturbation methods but is more amenable to numerical solution. The compact transfer function can be used to construct efficiently various spatial statistics of interest, such as covariances, cross-covariances, variances, and mean closure fluxes. We demonstrate the advantages of the proposed approach by applying it to a number of nonstationary problems, including a large, complex problem that is difficult to solve by traditional methods. In particular, we focus in this paper on demonstrating the new approach’s ability to compute efficiently covariances and cross-covariances critical for measurement conditioning, monitoring network analyses, and stochastic transport modeling in the presence of complex mean flow nonstationarities (caused, e.g., by complex trends in aquifer properties, boundary conditions, and sources and sinks). This paper is an extension of our recent work that illustrated the basic approach for modeling nonlocal and nonstationary scale effects and uncertainty propagation in relatively simple situations.

Smoothed ?Slack Cable? Models for Large Amplitude Oscillations of Suspension Bridges

Abstract This paper proposes a technique to study the dynamics of suspension bridges in those cases where the use of traditional models is not possible, i.e., when the deformed shape of the structure allows the simultaneous presence of taut and slack hangers. The typical unilateral behavior of these suspension elements, which are unable to transmit compressive forces, introduces a discontinuity in the stiffness that implies strong computational difficulties in the solution of the equations governing the motion; as a consequence, the response of the continuous or discrete models proposed may be evaluated only through step by step time integration. In this paper an alternative technique is proposed, which replaces the distribution of unilateral hangers with a smooth nonlinear regular system made equivalent to the effective one by means of an energy criterion. The proposed technique, referred to as “equivalent nonlinearization,” allows description of the motion through partial differential equations that are nonlinear but regular and, therefore, can be solved through classical perturbation techniques. The paper also describes two models formulated on the basis of the previously mentioned technique and some classes of solutions corresponding to particular characteristics of the forcing action.

Semiclassical Calculation of Recurrence Spectra of Li Rydberg Atom in Crossed Electric and Magnetic Field**The project supported by National Natural Science Foundation of China (Grant No. 20173031), the Natural Science Foundation of Shandong Province of China, and the Doctoral Foundation of the Ministry of Education of China

Closed-orbit theory is a semiclassical technique for explaining the spectra of Rydberg atoms in external fields. Using the closed-orbit theory and classical perturbation theory, we calculate the scaled recurrence spectra of Lithium atom in magnetic field plus a weak perpendicular electric field. The results show when the crossed electric field is added, the recurrence spectra are weakened greatly. As the scaled electric field f increases, the peaks of the recurrence spectra lose strength. Some recurrences are very sensitive and fall off rapidly as f increases; others persist till much higher f. As the electric field is stronger, some of the peaks revive. This phenomenon, caused by the interference among the electron waves that return to the nucleus, can be computed from the azimuthal dependence of the classical closed orbits.

A Simple Model for the Chaotic Motion Around (433) Eros

The three-dimensional motion in the close vicinity of Asteroid 433 Eros is known to be chaotic for a wide range of inclinations. This asteroid is potato-shaped, clearly distinct from an oblate body and, therefore, classical perturbation theories for motion close to spheroidal bodies cannot be reliably applied. To model the gravity field of elongated bodies like this asteroid, we use a very simple approach, a finite straight segment, rotating uniformly about an axis perpendicular to it. It is known that the potential function of this segment is given in closed-form, avoiding thus both divergency and time consuming in finding derivatives. For this model, we compute three-dimensional periodic orbits, emanating from planar resonant periodic orbits. For these three-dimensional periodic orbits, we compute the stable-unstable transition, which provides a partition in the phase space, thus giving a qualitative explanation for the majority of the relevant dynamical features previously observed-working with either general or special perturbation theories-in the three-dimensional motion around Eros.

Semiclassical evaluation of quantum fidelity.

We present a numerically feasible semiclassical (SC) method to evaluate quantum fidelity decay (Loschmidt echo) in a classically chaotic system. It was thought that such evaluation would be intractable, but instead we show that a uniform SC expression not only is tractable but it also gives remarkably accurate numerical results for the standard map in both the Fermi-golden-rule and Lyapunov regimes. Because it allows Monte Carlo evaluation, the uniform expression is accurate at times when there are 10(70) semiclassical contributions. Remarkably, it also explicitly contains the "building blocks" of analytical theories of recent literature, and thus permits a direct test of the approximations made by other authors in these regimes, rather than an a posteriori comparison with numerical results. We explain in more detail the extended validity of the classical perturbation approximation and show that within this approximation, the so-called "diagonal approximation" is automatic and does not require ensemble averaging.

A geometric analysis of the Lagerstrom model problem

Lagerstrom's model problem is a classical singular perturbation problem which was introduced to illustrate the ideas and subtleties involved in the analysis of viscous flow past a solid at low Reynolds number by the method of matched asymptotic expansions. In this paper the corresponding boundary value problem is analyzed geometrically by using methods from the theory of dynamical systems, in particular invariant manifold theory. As an essential part of the dynamics takes place near a line of non-hyperbolic equilibria, a blow-up transformation is introduced to resolve these singularities. This approach leads to a constructive proof of existence and local uniqueness of solutions and to a better understanding of the singular perturbation nature of the problem. In particular, the source of the logarithmic switchback phenomenon is identified.

The influence of the oscillating electric field on the recurrence spectra of a Li Rydberg atomin a strong magnetic field

Using first order classical perturbation theory, we generalize the closed orbit theory from atime-independent system to a time-dependent one and calculate the recurrencespectra of a Li Rydberg atom in a static magnetic plus an oscillating electric field.The results show that when the oscillating field was added, the strength of therecurrence spectra was greatly decreased. The recurrence spectra whose periodsare integer multiples or near-integer multiples of the period of the perturbatingoscillating field survive, while those that are not integer multiples of the periodof the oscillating field are weakened or vanish. As the scaled electric fieldf increases,the peaks of the recurrence spectra lose strength. Some recurrences are dropped rapidly and disappear asf increases; others existeven at much higher f.The results in this paper are very useful for future experimental research.

A computationally practical method for stochastic groundwater modeling

Stochastic theories of subsurface flow and transport have changed the way we think about heterogeneity but have not had much impact on practical groundwater modeling. Most numerical models still provide no information on prediction uncertainty. This gap between theory and practice is due largely to the excessive computational demands of available numerical methods for solving stochastic problems. The two primary alternatives, Monte Carlo and classical perturbation methods, both require solving large numbers of equations on fine grid spacings that are smaller than the log conductivity correlation length, no matter how smooth the mean flow and variance dynamics of interest are. This results in computational times that are many orders-of-magnitude greater than those required for conventional deterministic simulations. In addition, classical perturbation methods need much more real memory than comparable deterministic simulations. In this paper, we present an innovative approach for stochastic groundwater modeling––one that finally allows taking advantage of the scale disparity between the mean distributions and small-scale process. The new approach, which is called the nonstationary spectral method, implicitly separates the problem into rapidly varying random (stationary) part and a slowly varying deterministic (nonstationary) part and only the latter needs to be solved numerically. A general analysis and a specific numerical example both demonstrate that the nonstationary spectral method is much more efficient than the available alternatives. The methodological innovation dramatically increases the size and range of the problems that can be solved using stochastic methods and represents a major step forward in our effort to make stochastic uncertainty analysis a routine part of groundwater modeling.

A unified framework for input-to-state stability in systems with two time scales

This paper develops a unified framework for studying robustness of the input-to-state stability (ISS) property and presents new results on robustness of ISS to slowly varying parameters, to rapidly varying signals, and to generalized singular perturbations. The common feature in these problems is a time-scale separation between slow and fast variables which permits the definition of a boundary layer system like in classical singular perturbation theory. To address various robustness problems simultaneously, the asymptotic behavior of the boundary layer is allowed to be complex and it generates an average for the derivative of the slow state variables. The main results establish that if the boundary layer and averaged systems are ISS then the ISS bounds also hold for the actual system with an offset that converges to zero with the parameter that characterizes the separation of time-scales. The generality of the framework is illustrated by making connection to various classical two time-scale problems and suggesting extensions.

A classical perturbation technique that works even when the linear part of restoring force is zero

A classical perturbation technique that works even when the linear part of restoring force is zero