Liapunov's Direct Method Research Articles

Hydrodynamic regime and cold plasmas hit by short laser pulses

AbstractWe briefly report and elaborate on some conditions allowing a hydrodynamic description of the impact of a very short and arbitrarily intense laser pulse onto a cold plasma, as well as the localization of the first wave-breaking due to the plasma inhomogeneity. We use a recently developed fully relativistic plane model whereby we reduce the system of the Lorentz–Maxwell and continuity PDEs into a 1-parameter family of decoupled systems of non-autonomous Hamilton equations in dimension 1, with the light-like coordinate $$\xi =ct\!-\!z$$ ξ = c t - z replacing time t as an independent variable. Apriori estimates on the Jacobian $$\hat{J}$$ J ^ of the change from Lagrangian to Eulerian coordinates in terms of the input data (initial density and pulse profile) are obtained applying Liapunov direct method to an associated family of pairs of ODEs; wave-breaking is pinpointed by the inequality $$\hat{J}\le 0$$ J ^ ≤ 0 . These results may help in drastically simplifying the study of extreme acceleration mechanisms of electrons, which have very important applications.

A note on the stability analysis of nonlinear fractional difference equations: Comparative approach

In this study, we focus on nonlinear forward fractional difference equations with order ν∈(0,1] and construct stability analysis regarding h-stability and Mittag-Leffler stability notions. The main results of the paper are obtained by equiparating the equation in the spotlight with an auxiliary fractional difference equation. The outcomes of the manuscript provide an alternative approach to the ongoing theory of discrete fractional equations since the method used in the main results deviates from the fundamental tools of stability theory, namely fixed point theory and Liapunov's direct method.

Nonlinear stability for stationary helical vortices

We present a stability result for stationary smooth helical ideal fluid flows by using the direct method of Liapunov. It is enunciated in terms of the norm $$L^{2}$$ of the velocity and vorticity. A stability for monotonic helical vortices in the norm $$L^{1}$$ is also obtained.

Dynamic stability of Timoshenko beams on Pasternak viscoelastic foundation

The dynamic stability problem of a Timoshenko beam supported by a generalized Pasternak-type viscoelastic foundation subjected to compressive axial loading, where rotary inertia is neglected, is investigated. Each axial force consists of a constant part and a time-dependent stochastic function. By using the direct Liapunov method, bounds of the almost sure asymptotic stability of a beam as a function of viscous damping coefficient, variance of the stochastic force, shear correction factor, parameters of Pasternak foundation, and intensity of the deterministic component of axial loading are obtained. With the aim of justifying the use of the direct Liapunov method analytical results are firstly compared with numerically obtained results using Monte Carlo simulation method. Numerical calculations are further performed for the Gaussian process with a zero mean as well as a harmonic process with random phase. The main purpose of the paper is to point at significance damping parameter of foundation on dynamic stability of the structure.

Design of parallel cyclones based on stability analysis

The uniform distribution of gas solids flow across parallel cyclones is required for high efficiency. In this study, we introduced mass flow rate ratio between solids and gas (CT) to present multi‐phase interaction. And the direct Liapunov method is used to detect the instability of uniformity. Due to the special symmetry in this system, the criterion can be simplified into identifying the concavity (concave or convex) of pressure drop across a single cyclone with respect to CT. Then, based on the stability analysis of uniformity, a novel design principle is provided to prevent non‐uniform distribution at dense phase. The effect of geometrical factor, i.e. dimensionless vortex finder diameter dr, on the stability of uniformity has been further investigated. The phase diagram, illustrating the effects of both operational parameter (CT) and geometrical parameter (dr) on stability of uniformity is calculated to give a clue of designing a robust parallel cyclones system. © 2016 American Institute of Chemical Engineers AIChE J, 62: 4251–4258, 2016

Dynamic instability of coupled nanobeam systems

The dynamic instability problem of coupled nanobeam system subjected to compressive axial loading is investigated. The paper is concerned with the stochastic parametric vibrations of nanobeams based on Eringen’s nonlocal elasticity theory of Helmholtz and bi-Helmholtz type of kernel and Euler–Bernoulli beam theory. Each pair of axial forces consists of a constant part and a time-dependent stochastic function. By using the direct Liapunov method, bounds of the almost sure asymptotic instability of a coupled nanobeam system as a function of viscous damping coefficient, stiffness of the coupling elastic medium, variances of the stochastic forces, scale coefficients, and intensity of the deterministic components of axial loading are obtained. Analytical results are verified with the numerical results obtained from the Monte Carlo simulation. Numerical calculations are performed for the Gaussian process with a zero mean as well as a harmonic process with random phase.

Volterra integral equations on time scales: stability under constant perturbations via Liapunov direct method

In this paper we consider Volterra integral equations on time scales and describe our study about the long time behavior of their solutions. We provide sufficient conditions for the stability under constant perturbations by using the direct Lyapunov method and we present some examples of application.

Dynamic stability of nonlocal Voigt–Kelvin viscoelastic Rayleigh beams

The dynamic stability problem of a viscoelastic nanobeam subjected to compressive axial loading, where rotary inertia is taken into account, is investigated. The paper is concerned with the stochastic parametric vibrations of a Voigt–Kelvin nanobeam based on Eringen’s nonlocal elasticity theory of the Helmholtz and bi-Helmholtz type of kernel. The axial force consists of a constant part and a time-dependent stochastic function. By using the direct Liapunov method, bounds of the almost sure asymptotic stability of a viscoelastic nanobeam are obtained as a function of retardation time, variance of the stochastic force, geometric ratio, scale coefficients, and intensity of the deterministic component of axial loading. Numerical calculations were done for the Gaussian and harmonic process. When the excitation is a real noise process, the advanced numerical simulation based on the Monte Carlo method is presented for moment Liapunov exponents numerical determination.

Control of Trajectory Kinematics for Flexible Manipulators Applying Integral Variable Structure Strategy

Tracking control of manipulators with joint flexibility is considered to achieve a higher control technique performance. An integral variable structure control (IVSC) approach for robot manipulators is presented for accurate servo-tracking in the presence of load variation, parameter uncertainty and nonlinear dynamic interactions. A procedure is proposed for choosing the control function so that it guarantees the existence of the sliding mode and for determining the coefficients of the switching plane and integral control gain. Furthermore, a modified proper continuous function is introduced to overcome the chattering problem. The proposed (IVSC) approach has been simulated for a two-degree of freedom flexible joint robot, each joint modeled by two-equations of second order. Stability is insured by the use of Liapunov's direct method. The simulation results demonstrate the potential of the proposed scheme. Keywords: Flexible joint robot, Integral variable structure system, Sliding mode, Switching plane, Servo- tracking, Stability criterion, Dynamic uncertainty, Chatter.

Higher Performance Adaptive Control of a Flexible Joint Robot Manipulators

Joint flexibility limits the performance of an industrial robot by introducing resonant frequencies in the same range of the control bandwidth. If joint flexibility is considered in the modelling, advanced control techniques can be designed to achieve a higher performance. This paper presents an adaptive control scheme for a twodegree of freedom flexible joint robot. The adaptive approach is applied to design a control signal for each joint (modeled by twoequations of second order), to withstand all uncertainties in parameters and load disturbances. Asymptotic stability is insured regardless of the joint flexibility value, by the use of Liapunov's direct method, i.e., the results are not restricted to weak joint elasticity. A simulation study was carried out to evaluate the performance of the proposed control law in case of flexible joint robot model. It was found that the adaptation technique is robust to parameters and load changes.

Almost sure stochastic stability of a viscoelastic double-beam system

This paper investigates the dynamic stability of a viscoelastic double-beam system under parametric excitations. It is assumed that the two beams, made from Voigt–Kelvin material, are simply supported and continuously joined by a Winkler elastic layer. Each pair of axial forces consists of a constant part and a time-dependent stochastic function. In the case of “non-white” excitations, by using the direct Liapunov method, bounds of the almost sure stability of the double-beam system as a function of retardation time, bending stiffness, stiffness modulus of the Winkler layer, variances of the stochastic forces and the intensity of the deterministic components of axial loading are obtained. Numerical calculations are performed for the Gaussian process with a zero mean, as well as a harmonic process with a random phase. When the excitations are wideband noises, almost sure stability is obtained within the concept of the Liapunov exponent. White noise and Ornstein–Uhlenbeck processes are considered as models of wideband noises.

Three Fixed Point Theorems: Periodic Solutions of a Volterra Type Integral Equation with Infinite Heredity

AbstractIn this paper we study the existence of periodic solutions of a Volterra type integral equation with infinite heredity. Banach fixed point theorem, Krasnosel'skii's fixed point theorem, and a combination of Krasnosel'skii's and Schaefer's fixed point theorems are employed in the analysis. The combination theorem of Krasnosel'skii and Schaefer requires an a priori bound on all solutions. We employ Liapunov's direct method to obtain such an a priori bound. In the process, we compare these theorems in terms of assumptions and outcomes.

On the stability of non-autonomous perturbed Lotka–Volterra models

The paper is devoted to an extended Lotka–Volterra system of differential equations of predator–prey model. The extension is proposed with perturbation terms, which are null for the positive equilibrium state. In the original Lotka–Volterra system, the equilibrium state is not asymptotically stable due to the fact that perturbations are periodic in time. The aim of the paper is to characterize a form of perturbation terms guaranteeing the asymptotic stability or instability of equilibrium state. The reason of the proposed model is that for large time scale, the Lotka–Volterra model is too simple to be realistic. In the paper, the non-autonomous perturbations do not change the equilibrium state but introduce functions of time as well as for additional perturbed terms as for the main part of the equations modified from Lotka–Volterra model. Theorems are proposed in a renormalized form of the differential equations for time and the two variables. The key point of the paper comes from the use of a Liapunov function introduced in Section 2 which allows to obtain conditions for the asymptotic stability (Section 3) and instability (Section 4) by using a Cetaiev instability theorem following conditions on the renormalized coefficients in time of System (6). An appendix recalls the main results of the Liapunov Direct Method for non-autonomous binary systems of ordinary differential equations.

Stability by Krasnoselskii’s theorem in totally nonlinear neutral differential equations

In this paper we use fixed point methods to prove asymptotic stability results of the zero solution of a class of totally nonlinear neutral differential equations with functional delay. The study concerns \[x'(t)=a(t)x^3(t)+c(t)x'(t-r(t))+b(t)x^3(t-r(t)).\] The equation has proved very challenging in the theory of Liapunov's direct method. The stability results are obtained by means of Krasnoselskii-Burton's theorem and they improve on the work of T.A. Burton (see Theorem 4 in [Liapunov functionals, fixed points, and stability by Krasnoselskii's theorem, Nonlinear Studies 9 (2001), 181-190]) in which he takes \(c=0\) in the above equation.

Influence of rotatory inertia on dynamic stability of the viscoelastic symmetric cross-ply laminated plates

The dynamic stability problem of the symmetrically laminated cross-ply plates made of the viscoelastic Voigt–Kelvin material, compressed by time-dependent stochastic membrane forces, is investigated. The effect of rotatory inertia is included in the present formulation. It is assumed that all elastic moduli have the same retardation times. By using the direct Liapunov method, bounds of the almost sure stability of cross-ply plates as a function of viscous damping coefficient, retardation time, variances of the stochastic forces, ratio of the principal lamina stiffnesses, number of layers, plate aspect ratio, cross-ply ratio and intensity of the deterministic components of axial loading are obtained. Numerical calculations are performed for the Gaussian process with a zero mean, as well as an harmonic process with random phase.

On the long-time dynamics of nonautonomous predator–prey models with mutual interference

The longtime behaviour of a nonautonomous bidimensional Hassell predator–prey model with mutual interference is investigated. The existence of an absorbing set in the phase space is shown, and necessary and sufficient conditions guaranteeing the nonlinear, global, asymptotic stability of the positive solutions have been found by using the Liapunov direct method.

On the nonlinear stability of nonautonomous binary systems

Nonlinear nonautonomous binary systems of O.D.Es are considered. In the framework of the Liapunov Direct Method, stability-instability criteria are obtained via a “sharp” Liapunov function (Cfr. Sect.1). Applications either to the celebrated equation of nonlinear Mechanics x ̈ + p ( t , x , x ̇ ) x ̇ + q ( t , x , x ̇ ) x = 0 (in particular to the Hill, Mathieu–Hill and nonautonomous Duffing equations) or to the nonautonomous Lotka–Volterra system, are furnished.

DYNAMICAL INSTABILITY ANALYSIS OF NANOTUBES USING NONLOCAL SHEAR BEAM THEORY

The dynamics stability of distributed systems (continua) has been an object of considerable attention over the past half century. Numerous papers are available on isotropic and laminated beams, shafts, plates and shells under periodic and random forces. Most papers have applied finite dimensional or modal approximations in the analysis of vibration and stability. This paper focuses on the stochastic parametric vibrations of micro- and nanorods based on Eringens nonlocal elasticity theory and shear beam theory. The almost sure asymptotic instability criteria involving a damping coefficient, structure and loading parameters are derived using Liapunov's direct method. Using the appropriate energy-like Liapunov functional, sufficient conditions for the almost sure asymptotic instability of undeflected form of beam are derived. The nonlocal shear beam accounts for the scale effect, which becomes significant when dealing with short micro- and nanorods. From the obtained analytical formulas it is clearly seen that the small scale effect increases the dynamic instability region. Instability regions are functions of the axial force variance, the constant component of axial force and the damping coefficient.

UNIFORM DISSIPATIVENESS, ROBUST SYNCHRONIZATION AND LOCATION OF THE ATTRACTOR OF PARAMETRIZED NONAUTONOMOUS DISCRETE SYSTEMS

In this series of papers, we study issues related to the synchronization of two coupled chaotic discrete systems arising from secured communication. The first part deals with uniform dissipativeness with respect to parameter variation via the Liapunov direct method. We obtain uniform estimates of the global attractor for a general discrete nonautonomous system, that yields a uniform invariance principle in the autonomous case. The Liapunov function is allowed to have positive derivative along solutions of the system inside a bounded set, and this reduces substantially the difficulty of constructing a Liapunov function for a given system. In particular, we develop an approach that incorporates the classical Lagrange multiplier into the Liapunov function method to naturally extend those Liapunov functions from continuous dynamical system to their discretizations, so that the corresponding uniform dispativeness results are valid when the step size of the discretization is small. Applications to the discretized Lorenz system and the discretization of a time-periodic chaotic system are given to illustrate the general results. We also show how to obtain uniform estimation of attractors for parametrized linear stable systems with nonlinear perturbation.

Influence of transverse shear on stochastic instability of symmetric cross-ply laminated plates

The dynamic instability problem of symmetrically laminated cross-ply plates, compressed by time-dependent stochastic membrane forces, is investigated. The effect of shear deformation is included in the formulation. By using the direct Liapunov method, bounds of the almost sure instability of cross-ply plates are obtained. Furthermore, it is shown how the viscous damping coefficient, variances of the stochastic forces, ratio of the principal lamina stiffnesses, number of layers, plate aspect ratio, and cross-ply ratio influence the instability regions. A special attention is given to the proper choice of two shear correction factors of the laminated plate. Numerical calculations are performed for the Gaussian process with a zero mean and variance σ 2 as well as for the harmonic process with an amplitude A.