THE THEOREM OF THE STABILITY OF NONLINEAR NONAUTONOMOUS SYSTEMS UNDER THE FREQUENTLY-ACTING PERTURBATION—LIAPUNOV’S INDIRECT METHOD

In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied. This study is a forward development of the study of the stability in the Liapunov sense; furthermore, it is of significance in practice since perturbations are often not single in the time domain. Malkin proved a general theorem about thesubject. To apply the theorem, however, the user has to construct a Liapunov function which satisfies specified conditions and it is difficult to find such a function for nonlinear nonautonomous systems. In the light of the principle of Liapunov’s indirect method, which is an effective method to decide the stability of nonlinear systems in the Liapunov sense, the authors have achieved several important conclusions expressed in the form of theorems to determine the stability of nonlinear nonautonomous systems under the frequently-acting perturbation.     

THE THEOREM OF THE STABILITY OF NONLINEAR NONAUTONOMOUS SYSTEMS UNDER THE FREQUENTLY-ACTING PERTURBATION—LIAPUNOV’S INDIRECT METHOD

In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied.This study is a forward development of the study of thestability in the Liapunov sense;furthermore,it is of significance in practice sinceperturbations are often not single in the time domain.Malk in proved a general theoremabout thesubject.To apply the theorem,however,the user has to construct a Liapunovfunction which satisfies specified conditions and it is difficult to find such a function fornonlinear nonautonomous systems.In the light of the principle of Liapunov’s indirectmethod,which is an effective method to decide the stability of nonlinear systems in theLiapunov sense,the authors have achieved several important conclusions expressed in theform of theorems to determine the stability of nonlinear nonautonomous systems under thefrequently-acting perturbation.     

Nonlinear stability of a convective motion in a porous layer driven by a horizontally periodic temperature gradient

The nonlinear global exponential pointwise stability of a vertical steady flow driven by a horizontal periodic temperature gradient in a porous layer is performed. It is shown that the stability threshold depends on the supremum of a quadratic functional, having non constant coefficients, and new in the literature on the convection problem. In solving the variational problem, a suitable functional transformation is used.Received: 27 January 2003, Accepted: 10 March 2003, Published online: 12 September 2003 Correspondence toF. Capone     

磁浮列车的动力稳定性分析与Liapunov指数

针对弹性轨道上的磁悬浮列车线动力控制系统,给出了采用Ｌｉａｐｕｎｏｖ特性指数送别动力系统稳定性的判据：当动力系统的全部Ｌｉａｐｕｎｏｖ特性指数小于零时,动力控制系统是稳定的;而当动力系统有一Ｌｉａｐｕｎｏｖ特性指数大于零时,动力系统就成为不稳定的,由于Ｌｉａｐｕｎｏｖ特性指数可以由数值积分方式方便地给出,这一判断方法对于不便搜索参数稳定区域的高维动力系统,与寻求周期变系数线性常微分方法动力系统的Ｆ     

Nonlinear Stability of Convection in a Porous Layer with Solid Partitions

We show that for many classes of convection problem involving a porous layer, or layers, interleaved with finite but non-deformable solid layers, the global nonlinear stability threshold is exactly the same as the linear instability one. The layer(s) of porous material may be of Darcy type, Brinkman type, possess an anisotropic permeability, or even be such that they are of local thermal non-equilibrium type where the fluid and solid matrix constituting the porous material may have different temperatures. The key to the global stability result lies in proving the linear operator attached to the convection problem is a symmetric operator while the nonlinear terms must satisfy appropriate conditions.     

Technical stability of nonlinear time-varying systems with small parameters

ＩｎｔｒｏｄｕｃｔｉｏｎＳｔａｂｉｌｉｔｙｐｒｏｂｌｅｍｓａｒｉｓｉｎｇｆｒｏｍｅｎｇｉｎｅｅｒｉｎｇａｐｐｌｉｃａｔｉｏｎｓａｒｅｕｓｕａｌｌｙｒｅｌａｔｅｄｔｏｃｅｒｔａｉｎｑｕａｎｔｉｔｉｅｓｔｈａｔｓｐｅｃｉｆｙｔｈｅｓｔｒｅｎｇｔｈｏｆａｄｍｉｓｓｉｂｌｅｄｉｓｔｕｒｂａｎｃｅｓａｎｄｔｈｅｌｉｍｉｔｓｏｎｄｅｖｉａｔｉｏｎｓｏｆｍｏｔｉｏｎｏｆｔｈｅｄｉｓｔｕｒｂｅｄｓｙｓｔｅｍ .Ｉｎｔｈｉｓｒｅｇａｒｄ ,ｔｈｅｃｏｎｖｅｎｔｉｏｎａｌＬｉａｐｕｎｏｖｓｔａｂｉｌｉｔｙｃｏｎｃｅｐｔ…     

On the motions of a double pendulum with vibrating suspension point

We consider the motions of a system consisting of two pivotally connected physical pendulums rotating about horizontal axes. We assume that the system suspension point, which coincides with the suspension point of one of the pendulums, performs harmonic vibrations of high frequency and small amplitude along the vertical. We also assume that the system has four relative equilibrium positions in which the suspension points and the pendulum centers of mass lie on one vertical line. We study the stability of these relative equilibria. For arbitrary physical pendulums, we obtain stability conditions in the linear approximation. For a system consisting of two identical rods, we solve the stability problem the in nonlinear setting. For the same system, we study the existence, bifurcations, and stability of high-frequency periodic motions of small amplitude other than the relative equilibria on the vertical line. The studies of dynamic stability augmentation in mechanical systems under the action of high-frequency perturbations was initiated in the paper [1], where it was shown that the unstable inverted equilibrium of a pendulum may become stable if the suspension point vibrates rapidly. This idea was developed in [2–10] and other papers, where several aspects of motion of a mathematical pendulum in the case of rapid small-amplitude vibrations of the suspension point were studied in the linear setting and also (without full mathematical rigor) in the nonlinear setting. The motions of the suspension point along an arbitrary oblique straight line [2, 4, 7, 8], along the vertical [3, 5, 6], along the horizontal [9], and in the case of damping [8] were considered. The monograph [10] deals with the stabilization of a pendulum or a system of pendulums under periodic and conditionally periodic vibrations of the suspension point along the vertical, along an oblique straight line, and along an ellipse. A rigorous nonlinear analysis of the existence and stability of periodic motions of the mathematical pendulum under horizontal and oblique vibrations of the suspension point at arbitrary frequencies and amplitudes can be found in [11, 12]. For the case of vertical vibrations of the suspension point at an arbitrary frequency and amplitude, a rigorous stability analysis of the relative equilibria of the pendulum on the vertical was carried out in [13].     

Stabilization of Parametric Vibrations of a Nonlinear Continuous System

The purpose of the present paper is to solve an active control problem of nonlinear continuous system parametric vibrations excited by the fluctuating force. The problem is solved using the concept of distributed piezoelectric sensors and actuators with a sufficiently large value of velocity feedback. The direct Liapunov method is proposed to establish criteria for the almost sure stochastic stability of the unperturbed (trivial) solution of the shell with closed-loop control. The distributed control is realized by the piezoelectric sensor and actuator, with the changing widths, glued to the upper and lower shell surface. The relation between the stabilization of nonlinear problem and a linearized one is examined. The fluctuating axial force is modeled by the physically realizable ergodic process. The rate velocity feedback is applied to stabilize the shell parametric vibrations.     

On stability of relative equilibria of a double pendulum with vibrating suspension point

We consider the motions of a double pendulum consisting of two hinged identical rods. The pendulum suspension point is assumed to perform harmonic vibrations of arbitrary frequency and arbitrary amplitude in the vertical direction. We carry out a complete nonlinear analysis of the stability of the four pendulum relative equilibria on the vertical. The problem on the stability of the relative equilibria of the mathematical pendulum in the case where the suspension point performs vertical harmonic vibrations of arbitrary frequency and arbitrary amplitude was considered in a linear setting [1–3] and a nonlinear setting [4, 5]. In the case of small-amplitude rapid vertical vibrations of the suspension point, linear and (mathematically not fully rigorous) nonlinear stability analysis of the relative equilibria was carried out for an ordinary pendulum [6–9] and a double pendulum [10, 11]. In [12], for the same case of rapid vibrations, stability conditions in the linear approximation were obtained for the four relative equilibria of a system consisting of two physical pendulums. In the special case of a system consisting of two identical rods, the problem was solved in the nonlinear setting.     

Thermal system identification using fractional models for high temperature levels around different operating points

This work aims at the preparation of an experiment for the thermal modeling of an ARMCO iron sample (iron of the American Rolling Mill COmpany) for small temperature variations around different operating points. Fractional models have proven their efficacy for modeling thermal diffusion around the ambient temperature and for small variations. Due to their compactness, as compared to rational models and to finite element models, they are suitable for modeling such diffusive phenomena. However, for large temperature variations, thermal characteristics such as thermal conductivity and specific heat vary along with the temperature. In this context, the thermal diffusion obeys a nonlinear partial differential equation and cannot be modeled by a single linear model. In this paper, thermal diffusion of the iron sample is modeled around different operating points for temperatures ranging from 400 to 1070?K, which is above the Curie point (In physics and materials science, the Curie temperature (T _C), or Curie point, is the temperature at which a ferromagnetic or a ferrimagnetic material becomes paramagnetic.) showing that for a large range of temperature variations, a nonlinear model is required. Identification and validation data are generated by finite element methods using COMSOL Software.     

Horizontal thermal convection in a porous medium

Linearised instability and nonlinear stability bounds for thermal convection in a fluid-filled porous finite box are derived. A nonuniform temperature field in the steady state is generated by maintaining the vertical walls at different temperatures. The linearised instability threshold is shown to be well above the global stability boundary, which is strongly suggested by the lack of symmetry in the perturbed system. The numerical results are evaluated utilising a newly developed Legendre-polynomial-based spectral method.     

有约束阻尼的非保守力学系统的稳定性

利用Ｌｙａｐｕｎｏｖ直接法，研究了有势力、陀螺力、Ｒａｙｌｅｉｇｈ阻尼和约束阻尼同时作用的非线性非保守力学系统的稳定性．假设陀螺力依赖某参数ｈ，得到系统渐近稳定的两个定理．     

A stability investigation for an incompressible simple fluid with fading memory

The nonlinear equations of motion for an incompressible simple fluid, occupying a fixed bounded container, are formulated on the basis of the finitelinear viscoelasticity theory for materials with fading memory; this formal boundary-initial value problem is then viewed as a nonlinear abstract evolution equation on a certain Hilbert space. It is shown that a linearized version of this evolution equation is associated with a linear dynamical system on this Hilbert space, and several results for stability and asymptotic behavior for this linearized problem are proved through the use of Liapunov stability methods. On the assumption that the original nonlinear evolution equation also is associated with some dynamical system on the same space, it is shown that the rest condition of the fluid is stable and all motions are bounded. The Liapunov function employed for this purpose can be interpreted as a mechanical energy function for the fluid.E. F. Infante's work was supported in part by the U.S. Office of Naval Research (grant N0014-76-C-0278P002), the U.S. National Science Foundation (grant MCS-76-07247 A03), and the U.S. Army Research Office (grant AROD 31-124-73-G-130); that of J. A. WALKER was supported in part by the U.S. National Science Foundation (grant ENG76-81570) and the U.S. Air Force (grant AFOSR-76-3063A).     

RESEARCH OF THE PERIODIC MOTION AND STABILITY OF TWO-DEGREE-OF-FREEDOM NONLINEAR OSCILLATING SYSTEMS

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｓｙｓｔｅｍｏｆｎｏｎｌｉｎｅａｒｏｓｃｉｌｌａｔｉｎｇ ,ｐｅｒｉｏｄｉｃｍｏｔｉｏｎｉｓｏｆｐｒｉｍｅｉｍｐｏｒｔａｎｃｅ .Ｂｕｔｅｘｉｓｔｅｎｃｅｏｆｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｓｉｓａｖｅｒｙｄｉｆｆｉｃｕｌｔｑｕｅｓｔｉｏｎ .Ｌｕｃｋｉｌｙｔｈｅｒｅｅｘｉｓｔｓｏｍｅｋｉｎｄｓｏｆｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｉｎａｃｔｕａｌｐｈｙｓｉｃａｌｓｙｓｔｅｍｓ .Ｔｈｅｒｅｆｏｒｅ ,ｗｅｕｓｕａｌｌｙｃｏｎｃｅｎｔｒａｔｅｄｏｕｒａｔｔｅｎｔｉｏｎｏｎｔ…     

Traveling Waves for a Biological Reaction-Diffusion Model

We investigate the existence of traveling wave solutions for a system of reaction–diffusion equations that has been used as a model for microbial growth in a flow reactor and for the diffusive epidemic population. The existence of traveling waves was conjectured early but only has been proved recently for sufficiently small diffusion coefficient by the singular perturbation technique. In this paper we show the existence of traveling waves for an arbitrary diffusion coefficient. Our approach is a shooting method with the aid of an appropriately constructed Liapunov function.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.Wenzhang Huang-Research was supported in part by NSF Grant DMS-0204676.     

Convection in a rotating medium above a thermally inhomogeneous horizontal surface

The linear stationary problem of convection in a medium rotating about a vertical axis above a thermally inhomogeneous horizontal surface is theoretically investigated. Attention is mainly focused on the case of a homogeneous medium, but certain stratification effects and especially the convection characteristics in binary mixtures (for example, in saline sea water) are also considered. When the rotation is rapid (large Taylor numbers) the convective cells are strongly elongated in the vertical direction, though they also contain a thin Ekman boundary layer. The importance of the boundary conditions on the horizontal surface (in parallel with the no-slip conditions, more general conditions that may follow from the quadratic turbulent friction model are considered) is shown. In the case of binary mixtures, the differential diffusion and rotation effects may together result in the appearance of “induced salt fingers”, the deep penetration of convection into an arbitrarily stably stratified medium. The convective motions may then have a considerable effect on the background vertical temperature and admixture distributions. Attention is drawn to an original manifestation of the analogy between the rotation and stratification effects: in a non-rotating, stably stratified medium, near a thermally inhomogeneous vertical surface, the convection also penetrates deep into the medium, but in the horizontal direction, so that, when the coordinate system is rotated through 90°, the solution coincides with the case of a rotating non-stratified fluid considered here.     

Crack front shape effects on propagation stability in thermosetting polyesters

Fracture mechanics testing of the resistance of a polymer to slow crack growth often reveal it to be a unique function of crack speed. However, several thermosetting polyesters, tested using the Double Torsion (DT) and Tapered Double Cantilever Beam (TDCB) techniques, seem unable to sustain stable propagation over particular, isolated ranges of crack velocity with specimen-dependent limits. The meaning of ‘propagation stability’ in this context is discussed, distinguished from the static stability concept normally used, and applied using Liapunov criteria. The corresponding hypothesis that isolated unstable regimes symptomise locally-falling sectors of the resistance versus crack speed crack speed characteristic is supported by an observed crack shape influence, due to which DT tests are inherently more stable than TDCB ones.     

时滞系统的实用稳定性和Liapunov稳定性

本文主要研究非线性时滞系统在两种度量下的实用稳定性问题．首先引入一类Ｒａｚｕｍｉｋｈｉｎ型微分比较原理和单调性准则，在此基础上提出一种Ｌｉａｐｕｎｏｖ－Ｒａｚｕｍｉｋｈｉｎ型直接方法，建立一般形式的实用稳定性直接判据．这些判据将问题约化为一组有限维的微分或积分不等式，可以直接根据系统方程进行检验，便于实际应用．然后利用这些结果研究时滞系统的Ｌｉａｐｕｎｏｖ稳定性．最后示例说明本文主要结果．     

Equilibrium characteristics of the secondary convection in a vertical fluid layer between two flat plates 1

The nonlinear stability of the natural convection in a vertical fluid layer between two flat plates with different temperatures is investigated by a direct method to find the equilibrium states of the secondary convection. We confine ourselves to two-dimensional flows and assume that the aspect ratio of the fluid layer is very large. Since the Prantl number is assumed to be very small, the buoyancy effect caused by temperature disturbances is negligible. As a result we obtained a neutral surface of the energy of the fundamental mode of the secondary convection. It is concluded that there is no finite amplitude instability below the critical Grashof number derived from linear stability theory, and that both the unstable equilibrium solution (threshold amplitude solution) and the stable equilibrium solution (finite amplitude solution) are found outside the neutral curve of the linear stability. Our results are almost consistent with those of Nagata and Busse (1983), but are more accurate and more thorough.     

Linear and nonlinear double diffusive convection in a rotating sparsely packed porous layer using a thermal non-equilibrium model

Double diffusive convection in a fluid-saturated rotating porous layer is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and nonlinear stability analyses. The Brinkman model that includes the Coriolis term is employed as the momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for the energy equation. The onset criterion for stationary, oscillatory, and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal diffusion, solute diffusion, and rotation that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz number, and Taylor number on the stability of the system is investigated. The nonlinear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out. Some of the convection systems previously reported in the literature is shown to be special cases of the system presented in this study.