Sense Of Liapunov Research Articles

Stability of steady states in ferromagnetic rings

In this paper, we consider a one-dimensional model of ferromagnetic rings, taking into account curvature and anisotropy effects. We describe relevant stationary configurations of the magnetization and we investigate their stability in the Liapunov sense.

Differential systems stability analysis based on matrix multiplicative criteria

This article proposes a method of stability analysis in the sense of Liapunov for systems of ordinary differential equations. The method is based on matrix multiplicative stability criteria obtained from transformations of numerical integration difference schemes. Criteria are formed in the form of necessary and sufficient conditions. A distinctive feature of the criteria is that they do not employ the methods of the qualitative theory of differential equations. Specifically, for the case of linear systems, the evaluation of characteristic numbers and characteristic indicators is not necessary. When analyzing the stability of nonlinear systems, the construction of Liapunov functions is not required. Matrix multiplicative form of the criteria allows computerization of stability analysis. The software implementation of the criteria is performed in real time and, based on its results, allows to make an unambiguous conclusion about the nature of stability of the system under study.

STABILITY IN THE EQUILIBRIUM POSITION OF AN EXTENSIBLE CABLE-CONNECTED TWO SATELLITE SYSTEM UNDER PERTURBATIVE FORCE IN CIRCULAR ORBIT

One stable position of equilibrium must exist for a system of two satellites connected by extensible string whether perturbative forcesact on it or not. In this paper we have obtained an equilibrium point for the system when perturbative force like air resistance acts on it and has been shown to be stable in the sense of Liapunov.

Establishment of infinite dimensional Hamiltonian system of multilayer quasi-geostrophic flow & study on its linear stability☆☆Project supported by the National Natural Science Foundation of China (Grant Nos. 41575026, 41275113, and 41475021).

A multilayer flow is a stratified fluid composed of a finite number of layers with densities homogeneous within one layer but different from each other. It is an intermediate system between the single-layer barotropic model and the continuously stratified baroclinic model. Since this system can simulate the baroclinic effect simply, it is widely used to study the large-scale dynamic process in atmosphere and ocean. The present paper is concerned with the linear stability of the multilayer quasi-geostrophic flow, and the associated linear stability criteria are established. Firstly, the nonlinear model is turned into the form of a Hamiltonian system, and a basic flow is defined. But it cannot be an extreme point of the Hamiltonian function since the system is an infinite-dimensional one. Therefore, it is necessary to reconstruct a new Hamiltonian function so that the basic flow becomes an extreme point of it. Secondly, the linearized equations of disturbances in the multilayer quasi-geostrophic flow are derived by introducing infinitesimal disturbances superposed on the basic flows. Finally, the properties of the linearized system are discussed, and the linear stability criteria in the sense of Liapunov are derived under two different conditions with respect to certain norms.

The linearization of periodic Hamiltonian systems with one degree of freedom under the Diophantine condition

In this paper we are concerned with the periodic Hamiltonian system with one degree of freedom, where the origin is a trivial solution. We assume that the corresponding linearized system at the origin is elliptic, and the characteristic exponents of the linearized system are ±iω with ω be a Diophantine number, moreover if the system is formally linearizable, then it is analytically linearizable. As a result, the origin is always stable in the sense of Liapunov in this case.

On Stability Criterion of 4-th Order Quasilinear Differential Equations with Quasiderivatives

Abstract This paper deals with stability in the Liapunov sense of the 4-th order quasilinear differential equations with quasiderivatives. An asymptotic stability criterion is derived. An illustrative example is added.

Nonlinear Differential Equations and Feedback Control Design for the Urban–Rural Resident Pension Insurance in China

Facing many problems of the urban–rural resident pension insurance system in China, one should firstly make sure that this system can be optimized. This paper, based on the modern control theory, sets up differential equations as models to describe the urban–rural resident pension insurance system, and discusses the globally asymptotic stability in the sense of Liapunov for the urban–rural resident pension insurance system in the new equilibrium point. This research sets the stage for our further discussion, and it is theoretically important and convenient for optimizing the urban–rural resident pension insurance system.

About the stability of nonlocal viscoelastic struts lying on an elastic foundation

The present paper is devoted to the investigation of the stability of an infinitely long strut, lying on a continuous elastic foundation. The material of the strut is characterized by nonlocal viscoelastic properties. The strut is compressed by a longitudinal force, constant or changing periodically with time. The stability is understood in Liapunov sense. The effect of parameters, characterizing nonlocalness of viscoelastic properties of the material and of the force on the stability of the strut, is explored. This analysis is performed with help of the top Liapunov exponent.

On Stability of Certain Class of 3-Rd Order Nonlinear Control Systems

Abstract The paper deals with a certain class of nonautonomous ordinary third-order nonlinear differential equations L3y=f(t,L0y,L1y,L2y) with quasi-derivatives. A criterion of asymptotic stability in Liapunov sense as well as a criterion of instability in Liapunov sense is derived. The results are illustrated by two examples.

On the Dynamical Systems Stability Control and Applications

The stability control analyzed by us, in this show, is based on our results in the domain of dynamical systems that depend of parameters. Any dynamical system can be considered as dynamical system that depends of parameters, without numerical particularization of them. All concrete dynamical systems, meted in the specialized literature, underline the property of separation between the stable and unstable zones, in sense of Liapunov, for two free parameters. This property can be also seen for one or more free parameters. Some mathematical conditions of separation between stable and unstable zones for linear dynamical systems are identified by us. For nonlinear systems, the conditions of separation may be identified using the linear system of first approximation attached to nonlinear system. A necessary condition of separation between stable and unstable zones, identified by us, is the sufficient order of differentiability or conditions of continuity for the functions that define the dynamical system. The property of stability zones separation can be used in defining the strategy of stability assurance and optimizing of the parameters, in the manner developed in the paper. The cases of dynamical systems that assure the separations of the stable and unstable zones, in your evolution, and permit the stability control, are analyzed in the paper.

Hybrid Control for Seismological Nonlinear Structures on Liapunov’s Theory

In this paper, the hybrid control method of earthquake excited high-raised buildings is put forword. The building is modeled as a shear-wall type structure with non-linear hysteretic restoring forces after the structure enters the period of nonlinear and plasticity. A passive base-isolation is combined with actuators applied at the basement of the structure. A candidate for Liapunov function is found out based on the theory of energy. A non-linear control law is designed following the theory of Liapunov, since small residual deformations have to be tolerated due to inelastic energy dissipation, asymptotic stability will not be required, but only stability in the sense of Liapunov has to be guaranteed. Computer simulations demonstrate the efficiency of the proposed control algorithm.

A feedback neural network with weights of sinusoidal functions

A new feedback neural network model is proposed. The network has the sinusoidal basis functions as its weights. Neuronal activation function is a linear function. The network connection form is feedback structure. An energy function is defined for the feedback neural network. And then, the network stability issue in operation is analyzed. In the Liapunov sense, the proposed feedback network stability is proved. During the operation of the network, the network states are changed ceaselessly but network weights vary according to time-dependent sinusoidal law. As the network state changes continuously, its energy will be reduced. Finally, when network comes to a stable state, its energy arrivs at a minimum value. The network is particularly suited for the adaptive approximation and the detection for periodic signals because of its sinusoidal basis function weights. It is, in practice, a new and effective way for periodic signal detection and processing. The very good detection results are obtained in the detection of power system voltage sag characteristics. Simulation examples show that the dynamic response speed of the network is very high.

A note on periodic solutions of a forced Liénard-type equation

Criteria for guaranteeing the existence, uniqueness and asymptotic stability (in the sense of Liapunov) of periodic solutions of a forced Liénard-type equation under certain assumptions are presented. These criteria are obtained by application of the Manásevich–Mawhin continuation theorem, Floquet theory, Liapunov stability theory and some analysis techniques. An example is provided to demonstrate the applicability of our results. doi:10.1017/S1446181110000805

Stability of the Rydberg atom in the crossed magnetic and electric fields

AbstractThe stability of equilibrium positions of the Rydberg atom exposed to the uniform crossed electric and magnetic fields is analyzed. The dynamics of the system is described by an autonomous Hamiltonian depending on parameters a and f. By the normalization of the quadratic part of the Hamiltonian expansion in the neighborhood of the equilibrium position it is proved that for any f < 0 and ${1 \over 2} < a < {1 \over 2} + {{( - f)^{3/2} } \over {3\sqrt 3 }}$, the equilibrium solution of the equations of motion is stable in Liapunov sense, while for f > 0 and a < 1/2, there is a domain of instability in the plain of parameters Ofa bounded by the curve d3 = 0. In the domain of linear stability, it is proved that there are two curves in the plane Ofa, where the resonance conditions of third (ω1 = 2ω2) and fourth (ω1 = 3ω2) order are fulfilled. Moreover, by the normalization of the third‐ and fourth‐order terms in the Hamiltonian expansion it is proved that in the case of the third‐order resonance, the equilibrium position is unstable for all f > 0 different from f = 0.111572 and f = 0.281144, for which the stability takes place. In the case of the fourth‐order resonance, there are two intervals of parameters for which the equilibrium position is unstable. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Fake superpotential for large and small extremal black holes

We consider the fist order, gradient-flow, description of the scalar fields coupled to spherically symmetric, asymptotically flat black holes in extended supergravities. Using the identification of the fake superpotential with Hamilton’s characteristic function we clarify some of its general properties, showing in particular (besides reviewing the issue of its duality invariance) that W has the properties of a Liapunov’s function, which implies that its extrema (associated with the horizon of extremal black holes) are asymptotically stable equilibrium points of the corresponding first order dynamical system (in the sense of Liapunov). Moreover, we show that the fake superpotential W has, along the entire radial flow, the same flat directions which exist at the attractor point. This allows to study properties of the ADM mass also for small black holes where in fact W has no critical points at finite distance in moduli space. In particular the W function for small non-BPS black holes can always be computed analytically, unlike for the large black-hole case.

Modal Interpretation for the Ekman Destabilization of Inviscidly Stable Baroclinic Flow in the Phillips Model

Abstract Ekman boundary layers can lead to the destabilization of baroclinic flow in the Phillips model that, in the absence of dissipation, is nonlinearly stable in the sense of Liapunov. It is shown that the Ekman-induced instability of inviscidly stable baroclinic flow in the Phillips model occurs if and only if the kinematic phase velocity associated with the dissipation lies outside the interval bounded by the greatest and least neutrally stable Rossby wave phase velocities. Thus, Ekman-induced destabilization does not correspond to a coalescence of the barotropic and baroclinic Rossby modes as in classical inviscid baroclinic instability. The differing modal mechanisms between the two instability processes is the reason why subcritical baroclinic shears in the classical theory can be destabilized by an Ekman layer, even in the zero dissipation limit of the theory.

A NOTE ON PERIODIC SOLUTIONS OF A FORCED LIÉNARD-TYPE EQUATION

AbstractCriteria for guaranteeing the existence, uniqueness and asymptotic stability (in the sense of Liapunov) of periodic solutions of a forced Liénard-type equation under certain assumptions are presented. These criteria are obtained by application of the Manásevich–Mawhin continuation theorem, Floquet theory, Liapunov stability theory and some analysis techniques. An example is provided to demonstrate the applicability of our results.

Zonal shear flows with a free surface: Hamiltonian formulation and linear and nonlinear stability

General theoretical results via a Hamiltonian formulation are developed for zonal shear flows with the inclusion of the vortex stretching effect of the deformed free surface (equivalent barotropic model). These results include a generalization of the Flierl–Stern–Whitehead zero angular momentum theorem for localized nonlinear structures (whether or not on a β-plane), and sufficient conditions for linear and nonlinear stability in the Liapunov sense–the latter are given as estimates in terms of an L 2-type perturbation norm which are global in time and are derived via bounds on the equilibrium potential vorticity gradient.

Ekman destabilization of inertially stable baroclinic abyssal flow on a sloping bottom

Baroclinic abyssal currents on a sloping bottom, which are nonlinearly stable in the sense of Liapunov in the absence of dissipation, are shown to be destabilized by the presence of a bottom Ekman boundary layer for any positive value of the Ekman number. When the abyssal flow is baroclinically unstable, the dissipation acts to reduce the inviscid growth rates except near the marginal stability boundary where it acts to increase the inviscid growth rates. It is shown that when the abyssal flow is baroclinically stable, the Ekman destabilization corresponds to the kinematic wave phase velocity lying outside the range of the inertial topographic Rossby phase velocities. The transition mechanism described here might provide a dynamical bridge between the nonrotational roll-wave instability that can occur in supercritical abyssal overflows and frictionally induced destabilization in subinertial geostrophically balanced baroclinic abyssal currents. In addition, the theory presented here suggests a dissipation-induced destabilization mechanism for coastal downwelling fronts whose cross-slope potential vorticity gradient does not satisfy the necessary condition for baroclinic instability.

New Results on Existence of Asymptotically Stable Periodic Solutions of a Forced Liénard Type Equation

In this work, we will be concerned with the following forced Lienard type equation: $$x^{\prime \prime} (t) + f(x(t))x^{\prime} (t) + g(t, x(t)) = e(t)$$ . Under certain assumptions, some new criteria for guaranteeing the existence of asymptotically stable (in the sense of Liapunov) periodic solutions of this equation are presented by applying Manasevich–Mawhin continuation theorem, Floquet theory, Liapunov stability theory and some analysis techniques. Moreover, an example is provided to demonstrate the applications of our results.