Asymptotic Instability Research Articles

Linear asymptotic instability of a stationary flow of a polymeric medium in a plane channel in the case of periodic perturbations

We study the linear stability of a stationary flow of an incompressible viscoelastic polymeric fluid in a plane infinite channel in the case of periodic perturbations with respect to a variable related to the channel length.

Dynamical Nonlinear Neural Networks with Perturbations Modeling and Global Robust Stability Analysis

paper is devoted to studying both the global and local stability of dynamical neural networks. In particular, it has focused on nonlinear neural networks with perturbation. Properties relating to asymptotic and exponential stability and instability have been detailed. This paper also looks at the robustness of neural networks to perturbations and examines if the related properties have been preserved. Circumstances for global and local exponential stability of nonlinear neural network dynamics have been studied. We mentioned that the local exponential stability of any equilibrium point of dynamical neural networks is equivalent to the stability of the linearized system around that equilibrium point. From this, some well-known and new sufficient conditions for local exponential stability of neural networks have been obtained. The Lyapunov's procedure has been used to analyze the stability property of nonlinear dynamical systems and many outcomes have been combined and generalized. A kind of Lyapunov's stability of the stable points of Hopfield neural network (HNN) have been proven, which means that if the initial state of the network is close enough to a stable point, then the network state will remain in a small neighborhood of the stable point. These stability results indicate the convergence of the memory process of HNN. The theoretical results are illustrated through a few problem cases for a nonlinear dynamical system with perturbation behavior. Keywordserturbed nonlinear systems, Hopfield neural network,

Stability Analysis of a Class of Higher Order Difference Equations

We consider the sufficient conditions for asymptotic stability and instability of certain higher order nonlinear difference equations with infinite delays in finite-dimensional spaces. With the aid of the general comparison condition on the right-hand side functionfk(·), we generalize the stability and instability result.

The motion of the thread with a variable length

This paper is devoted to the peculiarities of a process of inertial tether system deployment of the great length (tens of kilometers). And one of the main practical issues of this article is the behavior of maximum oscillation amplitude.First of all the exact analytical solution of the original problem is received. Furthermore it is shown that ratio of transverse wave velocity of the tether to the velocity of deployment is one of the main parameters of the problem. In limiting case, when initial tether length equals zero, the solution is received using another method – the method of progressing waves. Also it is demonstrated that the limiting case agrees with the previously received results.Analysis of the solution has revealed that the maximum oscillation amplitude of the tether almost always rises with the time. This effect corresponds to asymptotic instability of the solution (in this article instability is regarded as rising of oscillation amplitude with the time). Only in specific degenerate conditions the solution is stable. The received results of amplitude increase have to be considered in simulation of the tether system behavior as far as small perturbative vibrations at the reel system always take place.

Two-phase flow in rotating Hele-Shaw cells with Coriolis effects

The free boundary problem of a two phase flow in a rotating Hele-Shaw cell with Coriolis effects is studied. Existence and uniqueness of solutions near spheres is established, and the asymptotic stability and instability of the trivial solution is characterized in dependence on the fluid densities.

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.

On the Kolmogorov-Petrovskii-Piskunov equation

We prove existence and uniqueness of the solutions of Kolmogorov Petrovskii-Piskunov (KPP) equation. We study asymptotic stability and instability of the equilibrium solution u(x; t) = 0 of KPP equation with subject to the traveling wave solutions. We show that KPP equation has not got any periodic traveling wave solution. Also, we obtain some exact traveling wave solutions of KPP equation by the first integral method.

Stabilization of a Kind of Nonlinear Discrete Singular Large-Scale Control Systems

This paper studies the state feedback stabilization of a kind of nonlinear discrete singular large-scale control systems by using Lyapunov matrix equation, generalized Lyapunov function method and matrix theory. There gives some sufficient conditions for determining the asymptotical stability and instability of the corresponding singular closed-loop large-scale systems while the subsystems are regular, causal and R-controllable. At last, an example is given to show the application of main result. Key words: Nonlinear discrete singular large-scale system; Control system; Asymptotical stability; Stabilization; State feedback

Stability and instability of controlled motions of a two-mass pendulum of variable length

The problem of parametric control of plane motions of a two-mass pendulum (swing) is considered. The swing model is a weightless rod with two lumped masses one of which is fixed on the rod and the other slides along it within bounded limits. The control is the distance from the suspension point to the moving point. The proposed control law of swing excitation and damping consists in continuously varying the pendulumsuspension length depending on the phase state. The stability of various controlled motions, including the motions near the upper and lower equilibria, is studied. The Lyapunov functions that prove the asymptotic stability and instability of the pendulum lower position in the respective cases of the pendulum damping and excitation are constructed for the proposed control law. The influence of the viscous friction forces on the pendulum stable motions and the onset of stagnation regions in the case of its excitation is analyzed. The theoretical results are confirmed by graphical representation of the numerical results.

Almost sure asymptotic stability analysis of the θ-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations

AbstractWe perform an almost sure linear stability analysis of the θ-Maruyama method, selecting as our test equation a two-dimensional system of Itô differential equations with diagonal drift coefficient and two independent stochastic perturbations which capture the stabilising and destabilising roles of feedback geometry in the almost sure asymptotic stability of the equilibrium solution. For small values of the constant step-size parameter, we derive close-to-sharp conditions for the almost sure asymptotic stability and instability of the equilibrium solution of the discretisation that match those of the original test system. Our investigation demonstrates the use of a discrete form of the Itô formula in the context of an almost sure linear stability analysis.

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.

Linearized flavor-stability analysis of dense neutrino streams

Neutrino-neutrino interactions in dense neutrino streams, like those emitted by a core-collapse supernova, can lead to self-induced neutrino flavor conversions. While this is a nonlinear phenomenon, the onset of these conversions can be examined through a standard stability analysis of the linearized equations of motion. The problem is reduced to a linear eigenvalue equation that involves the neutrino density, energy spectrum, angular distribution, and matter density. In the single-angle case, we reproduce previous results and use them to identify two generic instabilities: The system is stable above a cutoff density ("cutoff mode"), or can approach an asymptotic instability for increasing density ("saturation mode"). We analyze multi-angle effects on these generic types of instabilities and find that even the saturation mode is suppressed at large densities. For both types of modes, a given multi-angle spectrum typically is unstable when the neutrino and electron densities are comparable, but stable when the neutrino density is much smaller or much larger than the electron density. The role of an instability in the SN context depends on the available growth time and on the range of affected modes. At large matter density, most modes are off-resonance even when the system is unstable.

Asymptotic stability and instability criteria for nonautonomous systems

The classical criterion of asymptotic stability of the zero solution of equations x′ = f(t, x) is that there exists a function V (t, x), a(∥x∥) ≤ V (t, x) ≤ b(∥x∥) for some a, b ∈ K such that $$ \dot{V} $$ (t, x) ≤ −c(∥x∥) for some c ∈ K. In this paper, we prove that if $$ \mathop {V}\limits^{(m + {1})} $$ (t, x) is bounded on some set [tk − T, tk + T] × BH(tk → +∞ as k → ∞), then the condition that $$ \dot{V} $$ (t, x) ≤ −c(∥x∥) can be weakened and replaced by that $$ \dot{V} $$ (t, x) ≤ 0 and − (− $$ \dot{V} $$ (tk, x)| + − $$ \ddot{V} $$ (tk, x)| + ⋯ + − $$ \mathop {V}\limits^{(m)} $$ (tk, x)|) ≤ −c′(∥x∥) for some c′ ∈ K. Moreover, the author also presents a corresponding instability criterion. [1–10]

An Experimental Paradigm to Assess Postural Stabilization: No More Movement and Not Yet Posture

A ground reaction based method is proposed to evaluate the hypothesis that a stabilization phase occurs in transitions towards erect posture, following the macroscopic movement and preceding the quiet final erect posture, whose aim is to control and dissipate the residual inertial unbalancing forces occurring at the transition end. The experimental protocol considers three tasks leading to the final erect posture: taking a step forward (F), sit-to-stand (S), and bending the trunk forward (B), The method mainly consists of the fitting of a negative exponential function on the instability time profile following the end of the transition movement. The model parameters Y(0), T, and Y(inf), respectively, quantify the initial instability rate, a time duration related to the stabilization, and the final asymptotic instability rate. Results from a sample of 40 adult able bodied subjects demonstrated that a postural stabilization phase actually occurs: Y(inf) is smaller (0.010, 0.010, and 0.008 m/s(2) for, respectively, F, S, and B tasks) than Y(0) (0.081, 0.137, and 0.057 m/s(2)). Tis in the order of seconds (0.95, 0.51, and 1.00 s). No trial with large values of both Y(0) and T was observed, evidencing that large initial instability rates are quickly controlled and reduced. The Y(0) and T parameters distribution are discussed according to the possible underlying active and/or passive stabilization mechanisms. The test-retest reliability overall figure (mean ICC 0.45 for 12 indexes) increased, when dropping the indexes related to the less reliable B task, to values (mean ICC 0.56 for eight indexes) comparable to published posturographic data.

State Feedback Stabilization of Discrete Singular Large-scale Control Systems

Abstract: This paper studies the state feedback stabilization of discrete singular large-scale control systems by using Lyapunov matrix equation, generalized Lyapunov function method and matrix theory. There gives some sufficient conditions for determining the asymptotical stability and instability of the corresponding singular closed-loop large-scale systems while the subsystems are regular, causal and R-controllable. At last, an example is given to show the application of main result. Key words: Discrete Singular Large-Scale Control System; Asymptotical Stability; Stabilization; Generalized Lyapunov Function; State Feedback

Weak continuity of dynamical systems for the KdV and mKdV equations

In this paper we study weak continuity of the dynamical systems for the KdV equation in $H^{-3/4}(\mathbb{R})$ and the modified KdV equation in $H^{1/4}(\mathbb{R})$. This topic should have significant applications in the study of other properties of these equations such as finite time blow-up and asymptotic stability and instability of solitary waves. The spaces considered here are borderline Sobolev spaces for the corresponding equations from the viewpoint of the local well-posedness theory. We first use a variant of the method of [5] to prove weak continuity for the mKdV, and next use a similar result for an mKdV system and the generalized Miura transform to get weak continuity for the KdV equation.

Stochastic instability via nonlocal continuum mechanics

The dynamic stability problem is solved for one-dimensional structures subjected to time-dependent deterministic or stochastic axial forces. The stability analysis of structures under time-dependent forces strongly depends on dissipation energy. The paper is concerned with the stochastic parametric vibrations of micro- and nano-rods based on Eringen’s nonlocal elasticity theory and Euler–Bernoulli beam theory. The asymptotic instability, and 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 asymptotic instability, and the almost sure asymptotic instability of undeflected form of beam are derived. The nonlocal Euler–Bernoulli beam accounts for the scale effect, which becomes significant when dealing with short micro- and nano- rods. From 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.

Sufficient and necessary conditions for Lyapunov stability of Lorenz system and their application

The sufficient and necessary conditions for Lyapunov stability of the zero equilibrium point of Lorenz system are discussed, and some brief criteria are presented for globally exponential stability, globally asymptotical stability and instability. Furthermore, the behavior of the non-zero equilibrium point of Lorenz system is also investigated, and several sufficient and necessary conditions are provided for locally exponential stability and instability. The established theorems in this paper develop and extend the existing achievements on Lyapunov stability of Lorenz system. In conclusion, by applying the obtained results, some less conservative feedback-control laws are derived to guarantee the globally exponential stability of the chaos control of Chen system, Lu system, Yang-Chen system and Yu-Xia Li system.

Role of long waves in the stability of the plane wake

This work is directed toward investigating the fate of three-dimensional long perturbation waves in a plane incompressible wake. The analysis is posed as an initial-value problem in space. More specifically, input is made at an initial location in the downstream direction and then tracing the resulting behavior further downstream subject to the restriction of finite kinetic energy. This presentation follows the outline given by Criminale and Drazin [W. O. Criminale and P. G. Drazin, Stud. Appl. Math. 83, 123 (1990)] that describes the system in terms of perturbation vorticity and velocity. The analysis is based on large scale waves and expansions using multiscales and multitimes for the partial differential equations. The multiscaling is based on an approach where the small parameter is linked to the perturbation property independently from the flow control parameter. Solutions of the perturbative equations are determined numerically after the introduction of a regular perturbation scheme analytically deduced up to the second order. Numerically, the complete linear system is also integrated. Since the results relevant to the complete problem are in very good agreement with the results of the first-order analysis, the numerical solution at the second order was deemed not necessary. The use for an arbitrary initial-value problem will be shown to contain a wealth of information for the different transient behaviors associated to the symmetry, angle of obliquity, and spatial decay of the long waves. The amplification factor of transversal perturbations never presents the trend--a growth followed by a long damping--usually seen in waves with wave number of order one or less. Asymptotical instability is always observed.

The effect of infinitesimal damping on nonconservative divergence instability systems

The present work discuss the local dynamic asymptotic stability of 2-DOF weakly damped nonconser-vative systems under follower compressive loading in regions of divergence, using the Lienard–Chipartstability criterion. Individual and coupling effects of the mass and stiffness distributions on the localdynamic asymptotic stability in the case of infinitesimal damping are examined. These autonomoussystems may either be subjected to compressive loading of constant magnitude and varying direction(follower) with infinite duration or be completely unloaded. Attention is focused on regions of diver-gence (static) instability of systems with positive definite damping matrices. The aforementioned massand stiffness parameters combined with the algebraic structure of positive definite damping matrices mayhave under certain conditions a tremendous effect on the Jacobian eigenvalues and thereafter on the localdynamic asymptotic stability of these autonomous systems. It is also found that contrary to conservativesystems local dynamic asymptotic instability may occur, strangely enough, for positive definite dampingmatrices before divergence instability, even in the case of infinitesimal damping (failure of Ziegler’skinetic criterion).