Asymptotic Domain Research Articles

Global Asymptotic Solutions for Magnetohydrodynamic Jets and Winds

We propose a self-consistent and self-explanatory picture of the acceleration/collimation process of magnetohydrodynamic (MHD) outflows in the asymptotic domain. With the criticality and current-closure conditions properly taken into account in the global solutions, the most crucial step is the correct usage of the transfield force balance equation, which determines the curvature radius R of poloidal field lines. The sign of R determines the collimation or decollimation of each field streamline, and without referring to the current-closure condition, magnetic self-collimation cannot be discussed—the anticollimation theorem. The location of the fast magnetosonic points are fixed at the innermost distances of the asymptotic domain, and hence, the asymptotic domain is nothing but the superfast, accelerating domain. The curvature 1/R has so far mistakenly been regarded as negligibly small, leading to the pseudo-force-free state in the far asymptotic domain and thereby producing the wrong concept that full MHD acceleration—meaning the conversion of all the Poynting flux into kinetic energy flux—is implausible. It cannot be stressed enough that this global picture has been reached by combining Sakurai's numerical results with Heyvaerts & Norman's analytic asymptotic formalism.

A modified PID controller ( PIIσβD)

The purpose of this study is to modify the traditional PID controller in order to improve its performance (stability and tracking) by changing the length of integration interval. The performance of the traditional PID controller was improved by changing the length of integration interval to make the most of the returns of the PID and PI σ D controllers. The asymptotic stability domain, in terms of the feedback gains, is derived for systems of second order using the modified controller which will be identified as PII σ β D. Comparing this controller with the traditional PID controller and PI σ D controller proposed in [1], it proves that it is more accurate and more stable. For illustration and comparison, two examples have been simulated to evaluate the performance of the modified controller. All simulation results indicate that the modified controller is better than the traditional PID controller and the PI σ D controller from the accuracy and stability point of view.

Magnetohydrodynamic Acceleration of the Crab Pulsar Wind

It is argued that ideal magnetohydrodynamic processes can realize the full acceleration of pulsar winds. The fast magnetosonic surface must be situated at the innermost points of the asymptotic domain far outside the Alfvenic distances. Ongoing field-flow interactions make plasma particles more and more ballistic, with the field structure becoming increasingly radial, in the asymptotic superfast domain. The condition of transfield force balance itself requires gradual conversion of Poynting flux to kinetic energy flux, thereby approaching a quasi-conical structure. This will be helpful to the resolution of the notorious discrepancy between the Crab pulsar wind theory and the Rees-Gunn model for the Crab Nebula.

Lyapunov stability of multimachine power systems considering the voltage regulator

A new decomposition–aggregation approach is developed, in the paper, to perform transient stability analysis of an N-machine power system. Considering the generator flux decay effect, non-uniform mechanical damping, transfer conductances and the automatic voltage regulator action, the system mathematical model is derived. It is obtained a system aggregation (square) matrix of the order ( N−1)/2, stability of this matrix implies asymptotic stability of the system equilibrium state. As an illustrative example, the developed approach is applied to a 7-machine, 10-bus power system, and it is determined an estimate for the system asymptotic stability domain. A 3-phase short circuit fault (with successful reclosure) is assumed to occur close to a generator bus. Isolating the faulted line clears the fault. The developed approach is used to determine the critical time for reclosing the open line. It is found that the developed approach is powerful (the obtained critical time is about 90% of the exact time computed by using the step-by-step method) and may be used for stability studies of large-scale power systems.

Magnetized centrifugal winds

It is argued that for steady, axisymmetric, non-relativistic magneto-centrifugal winds, not only the boundary and criticality conditions but also the current-closure condition are of crucial significance as global conditions in resolving the acceleration-collimation problem. In Sakurai's numerical models, the split-monopole field adopted at the surface of the source provided the most favourable condition for global collimation of the flow, by making the domain of anti-collimating flow with outgoing electric current degenerate into an infinitely thin boundary layer at the equator, and hence suppressing the explicit appearance of the current-closure condition. For more general or realistic boundary conditions at the source, it is shown that the current-closure condition yields a two-component structure (with the return current at least in part in a volume current, not totally a sheet current) as a natural consequence of the transfield equation in the asymptotic domain. This equation, combined with the Bernoulli (and other) integrals, requires the wind to tend asymptotically to a ‘quasi-conical’ structure, as a natural consequence of the flow particles’ becoming more and more ballistic as a result of the magnetohydrodynamic (MHD) acceleration. This is a result that the Poynting energy flux diminishes to zero along each field line. The criticality problem is solved for magneto-centrifugal winds, to give the eigenvalues of the Alfvénic distance and other quantities at the fast magnetosonic surface, situated somewhere between the subasymptotic and asymptotic domains.

Nature of halo in light nuclei

A feature peculiar to light neutron-rich nuclei is that their lowest decay thresholds are only slightly above their ground states. Among them, 6He and 11Li are two most striking examples. The energy needed to break 6He (11Li) into an alpha particle (9Li) and two neutrons is about 1 MeV (300 keV). So small a value prompts one to construct their theory by analogy with the zero-range-nuclear-force approximation previously applied to the deuteron. A more detailed analysis shows, however, that the simple version of this approximation applied to systems that decay through a three-particle channel does not take into account some important features of these systems and requires significant improvements. First, with increasing distance between three particles, the potential energy decreases, in contrast to what is observed for binary systems, in inverse proportion to the hyperradius cubed. Second, the Pauli exclusion principle adds complexity even in the asymptotic domain, and we meet its demands in constructing the 6He and 11Li wave functions in the continuum. An approach is proposed to analyze weakly bound three-cluster systems that takes into account the aforementioned features and which describes correctly the experimentally observed structure of bound and unbound states above the threshold for three-particle decay.

The Multidimensional Asymptotic Stability Domain of Linear Discrete Systems: Its Symmetry and Other Properties

The properties of the closed asymptotic stability domain of a linear discrete system in the space of coefficients of its characteristic polynomial are investigated. An example is given.

Asymptotic Stability, Local Uniqueness, and Domain of Attraction of Two-Dimensional Step Type Contrast Structures

We prove the asymptotic stability of a two-dimensional stationary solution with internal transition layer to a singularly perturbed parabolic problem. We also construct a set of functions belonging to the domain of attraction of such a solution.

Acceleration and collimation of magnetized winds

The acceleration-collimation problem is discussed for stationary, axisymmetric, polytropic, non-relativistic MHD outflows, with causality and the current-closure condition taken into account. To elucidate the properties of physically realizable ‘quasi-conical’ winds, we consider four kinds of rather unphysical flows in contrast, namely ‘radial’, ‘asymptotic’, ‘conical’ and ‘current-free’ flows. ‘Radial’ flows are supposed to possess the radial structure from the source to infinity, thereby not fulfilling the transfield equation, though keeping causal contact with the source. ‘Asymptotic’ flows coincide in the asymptotic domain with the ‘quasi-conical’ winds, and ones extrapolated inwards from them through the subasymptotic domain to the source. Thirdly, ‘conical’ flows are supposed to satisfy the transfield equation in the subasymptotic domain; thus they are not literally conical, but are supposed to satisfy the ‘solvability condition at infinity for the conical structure’. It is, however, argued that there is one difficulty in connecting the asymptotic conical structure causally to the structure upstream. Finally, ‘current-free’ flows with no poloidal and toroidal currents everywhere in the wind zone are treated, but it is pointed out that there is no means of satisfying the current-closure condition in the wind zone. Of physical relevance are the ‘quasi-conical’ winds, for which it is shown that the condition that open field lines in the wind zone can reach infinity leads to the requirement that the Poynting flux, proportional to ζ≡αρϖ2η, is not carried to infinity along these field lines, i.e., ζ0, where α is the angular velocity of field lines, ρ the gas density, and η the mass flux per unit flux tube. While ζ decreases from a value of ζB≡ζA+4πηδα near the coronal base through at the Alfvenic surface to null at infinity, the specific angular momentum of the flow increases up to and the flow energy reaches nearly at infinity, where δ is a constant of the Bernouilli integral, and ϖA is the axial distance of the Alfvenic surface. It is also argued that ‘quasi-conical’ winds with the current-closure condition fulfilled in the wind zone possess the two-componentness of outflow as one of their generic properties.

Do magnetized winds self-collimate?

We discuss the asymptotic behaviour of steady, axisymmetric, non-relativistic, polytropic, perfect magnetohydrodynamic (MHD) winds from rotating magnetized bodies, which have passed smoothly through three critical surfaces. A quantity defined by ζ≡αρϖ2/η plays a crucial role in the asymptotic domain far from the Alfvénic surface. Here α is the angular velocity of field lines, η is the mass flux per unit flux tube and ρ is the mass density. In general, ζ is a function of the magnetic stream function P and the distance s measured along each field line. In the asymptotic domain, ϖBt≈-ζ and the electric current parallel to the poloidal field is j∥∝-(∂ζ/∂P)sBp, while the perpendicular one is j⊥∝-∂ζ/∂s)P. Also ζ(P,s) as a function of s measures the change of the Poynting energy flow along a given field line and as a function of P measures the total electric current passing within the surface of revolution defined by P and with s fixed. The asymptotic transfield equation reduces to ρvp2/R≈j∥Bt/c, where R is the curvature radius of poloidal field lines. This equation indicates that when viewed from the rotation-axis side, the field geometry is convex (R>0) in the range of field lines with ingoing current (j∥<0), whereas it is concave (R<0) in the range of field lines with outgoing current (j∥>0), with a straight, ‘neutral’ field line Pn separating both ranges. For particles to be accelerated by the magnetic stress in both poloidal and toroidal directions, one requires j⊥>0, i.e. (∂ζ/∂s)P<0. When ζ diminishes like ζ∝1/ ln s with no Poynting flux carried to infinity, an ‘exact’causal solution of the transfield equation shows a quasi-radial structure in the asymptotic domain. If the wind zones in both hemispheres have a convex geometry with j∥<0 and Bt<0 in the upper hemisphere and j∥>0 and Bt>0 in the lower, there must exist a large equatorial sheet current that balances the inflow of charges in the wind zones. The conical or cylindrical structure in the whole asymptotic domain, consisting of magnetic surfaces on which finite Poynting flux is carried to infinity with j∥=j⊥=0, imposes severe, probably acausal constraints on the structure in the subasymptotic domains upstream. In summary, the condition of current closure, together with causality, taken into account will not allow all the open field lines emanating from a central object with a realistic source field to bend toward the axis as suggested previously. It is argued that almost all previously developed models or processes claiming magnetic collimation are unsatisfactory with respect to causality and the current-closure condition. Thus magnetized winds as a whole will not collimate the rotation axis without any external help, such as the channelling effects of thick accretion discs and/or confinement resulting from the ambient medium.

Lyapunov-Like Solutions to Stability Problems for Discrete-Time Systems on Asymptotically Contractive Sets

This paper presents new criteria for stability properties of discrete-time non-stationary systems. The criteria are based on the concept of asymptotically contractive sets. As a result, general necessary conditions are established for asymptotic stability of the zero equilibrium state, the instantaneous asymptotic stability domain of which can be either time-invariant or time-varying and then possibly asymptotically contractive. It is shown that the classical Lyapunov stability conditions including the invariance principle by LaSalle cannot be applied to the stability test as soon as the system instantaneous domain of asymptotic stability is asymptotically contractive. In order to investigate asymptotic stability of the zero state in such a case novel criteria are established. Under the criteria the total first time difference of a system Lyapunov function may be non-positive only and still can guarantee asymptotic stability of the zero state. The results are illustrated by examples.

Estimation of the Asymptotic Stability Domain and Asymptotic Behaviour for Nonlinear Neutral Systems

Abstract This paper deals with the estimation of both stability domain and asymptotic behaviours of a class of nonlinear neutral functional differential equations (FDE). The approach for solving such a complex problem is based on comparison methods: the solutions of the original system are compared (by upper-bounding) to the behaviours of some more simple ordinary differential equation (ODE). It is combined with the use of regular vector Lyapunov's functions, that appear to yield less conservative results than usual scalar norms or Lyapunov's functions.

Large-scale simulation studies in image pattern recognition

Many obstacles to progress in image pattern recognition result from the fact that per-class distributions are often too irregular to be well-approximated by simple analytical functions. Simulation studies offer one way to circumvent these obstacles. We present three closely related studies of machine-printed character recognition that rely on synthetic data generated pseudo-randomly in accordance with an explicit stochastic model of document image degradations. The unusually large scale of experiments - involving several million samples that makes this methodology possible have allowed us to compute sharp estimates of the intrinsic difficulty (Bayes risk) of concrete image recognition problems, as well as the asymptotic accuracy and domain of competency of classifiers.

On a modification of the PID controller

In this article we propose a modification of the PID controller. Theexpression for static error for systems with a modified controller isderived. We have obtained the parametric equations describing the boundariesof the asymptotic stability domain in the space of feedback gains. Theasymptotic stability domain for the system of second order with modifiedcontroller is derived. Some numerical results and example arepresented.

Asymptotic time domain analysis of sound propagation in a deep ocean sound channel

AbstractThis study relates to the time‐domain analysis of sound‐wave propagation in the deep‐ocean sound channel SOFAR (sound fixing and ranging) in the case of a sound source located on or near the channel axis. For asymptotic analysis of response waveforms in the case of an observation point located outside the boundary layer, we used the acoustic ray solution in frequency domain obtained by the acoustic ray tracing method, and derived the acoustic ray expression in time domain. In the case of an observation point located within the boundary layer sound propagation phenomena were clarified that were hard to analyze in terms of the acoustic ray theory. For asymptotic analysis of response waveforms in this case, we proposed a hybrid acoustic ray‐mode expression obtained by combining an acoustic ray with an adiabatic mode. Effectiveness of the proposed asymptotic expression was confirmed by substituting numbers in the acoustic field expressions and completing numerical computations to obtain response waveforms. the results were compared to the reference solutions based on the normal mode theory. It was also demonstrated that numerical computations based on asymptotic expressions are much faster than those based on the normal mode theory. In addition it was found that acoustic field analysis in time domain is more effective than frequency‐domain analysis in terms of physical interpretation of sound wave propagation, since each pulse wave is observed separately.

Phase separation in a three-dimensional, two-phase, hydrodynamic lattice gas

The dynamics of phase separation is explored using an immiscible 3D lattice-gas model. Scaling laws for the growth rate and power spectraS(k) of the growth patterns are computed. For small wavenumbersS(k) shows a crossover fromk2 tok4 behavior. The theoretical prediction for the asymptotic domain growthR≅t2/3 is supported by our results. We discuss the possibility to observe an intermediatet scaling. We show the influence of hydrodynamic forces in symmetric and asymmetric mixtures by comparing simulations with and without momentum conservation. The structure functionS(k) is not significantly modified by hydrodynamics, but the growth rate changes clearly. As a general result, it is shown that, in spite of the unusual thermodynamics of this model, many characteristics of the growth dynamics are surprisingly in agreement with the classical theoretical and experimental results.

Asymptotic Stability via Energy and Power. Part I: New Lyapunov Methodology for Nonlinear Systems

A large class of nonlinear dynamical physical systems is considered. Complete solutions for asymptotic stability of a state and for a set to be the asymptotic stability domain are disco, vered in terms of the stystem power and energy. The power is permitted to be semi-definite. The necessary and sufficient conditions are imposed on the energy. This reflects a new methodology of testing the asymptotic stability. It is essentially inverse relative to the widely used methodology of the Lyapunov method application to nonlinear systems. The first part of this two-part paper presents the theoretical foundation of the new methodology and criteria.

New approach to asymptotic stability: time‐varying nonlinear systems

The results of the paper concern a broad family of time‐varying nonlinear systems with differentiable motions. The solutions are established in a form of the necessary and sufficient conditions for: 1) uniform asymptotic stability of the zero state, 2) for an exact single construction of a system Lyapunov function and 3) for an accurate single determination of the (uniform) asymptotic stability domain. They permit arbitrary selection of a function p(⋅) from a defined functional family to determine a Lyapunov function v(⋅), [v(⋅)], by solving v′(⋅) = −p(⋅) {or equivalently, v′(⋅) = −p(⋅)[1 − v(⋅)]}, respectively. Illstrative examples are worked out.

Time-varying continuous nonlinear systems: uniform asymptotic stability

The framework of the presented research is a large class of time-varying nonlinear systems with continuous motions. The study of the uniform asymptotic stability of the zero equilibrium state developed here goes back to, and relies on, the very foundations of the Lyapunov stability concept and the (second) Lyapunov method. Stability domains are first defined and examined. Then, their qualitative features are used to establish complete solutions to the problem of uniform asymptotic stability of the equilibrium for various subclasses of the systems. The solutions present conditions that are both necessary and sufficient for: (1) the uniform asymptotic stability, (2) an exact direct one-shot construction of a system Lyapunov function and (3) for a direct accurate one-shot determination of the asymptotic stability domain. In addition, the solutions establish a novel Lyapunov-method based approach to the asymptotic stability analysis. This enables an arbitrary selection of a function p(·) from a defined functi...

Exact solutions for asymptotic stability: Non-linear systems

The problem of the necessary and sufficient exact algorithmic conditions for both the construction of a Lyapunov function and determination of the asymptotic stability domain is resolved in the framework of a large class of time-invariant non-linear systems with continuous motions.