The stability of non-linear pulse system in the first approximation

A pulse system, described by a non-linear functional-differential equation, and an “equivalent” continuous non-linear system, obtained from the initial system by replacing the pulse modulator by its static characteristic is considered. It is shown that, for a sufficiently high pulse frequency, asymptotic stability of a state of equilibrium of the pulse system arises from the stability of the equivalent system in the first approximation.     

Non-linear stability characterization of the thin micropolar liquid film flowing down the inner surface of a rotating vertical cylinder

Linear and non-linear stability analysis for characterization of micropolar film flowing down the inner surface of a rotating infinite vertical cylinder is given. A generalized non-linear kinematic model is derived to represent the physical system and is solved by the long wave perturbation method in the following procedure. First, the normal mode method is used to characterize the linear behaviors. Then, an elaborated non-linear film flow model is solved by using the method of multiple scales to characterize flow behaviors at various states of sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion. The modeling results indicate that by increasing the rotation speed, Ω, and the radius of cylinder, R, the film flow will generally stabilize the flow system.     

The stability and stabilization of the motion of non-conservative mechanical systems

Two stability problems are solved. In the first, the stability of mechanical systems, on which dissipative, gyroscopic, potential and positional non-conservative forces (systems of general form) act, is investigated. The stability is considered in the case when the potential energy has a maximum at equilibrium. The condition for asymptotic stability is obtained by constructing Lyapunov's function. In the second problem, the possibility of stabilizing a gyroscopic system with two degrees of freedom up to asymptotic stability using non-linear dissipative and positional non-conservative forces is investigated. Stability of the gyroscopic system is achieved by gyroscopic stabilization. The stability conditions are obtained in terms of the system parameters. Cases when the gyroscopic stabilization is disrupted by these non-linear forces are indicated.     

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

A non-linear controlled dynamical system that describes the dynamics of a broad class of non-linear mechanical and electromechanical systems (in particular, electromechanical robot manipulators) is considered. It is proposed that the real parameter vector of a non-linear controlled dynamical system belongs to an assigned (admissible) constrained closed set and is assumed to be unknown. The programmed motion of the non-linear controlled dynamical system and the programmed control that produces it are assigned (constructed) by using an estimate, that is, the nominal value of the parameter vector of the non-linear controlled dynamical system, which differs from its actual value. A procedure for synthesizing stabilizing control laws with linear feedback with respect to the state that ensure stabilization of the programmed motions of the non-linear controlled dynamical system under parametric perturbations is proposed. A non-singular linear transformation of the coordinates of the state space that transforms the original non-linear controlled dynamical system in deviations (from the programmed motion and programmed control) into a certain non-linear controlled dynamical system of special form, which is convenient for analysing and synthesizing laws for controlling the motion of the system, is constructed. A certain non-linear controlled dynamical system of canonical form is derived in the original non-linear controlled dynamical system in deviations. The transformation of the coordinates of the state space constructed and the Lyapunov function methodology are used to synthesize stabilizing control laws with linear feedback with respect to the state, which ensure asymptotic stability as a whole of the equilibrium position of the non-linear controlled dynamical system of canonical form and dissipativity “in the large” of the non-linear controlled dynamical system of special form and of the original non-linear controlled dynamical system in deviations. In the control laws synthesized, the formulae for the elements of their matrices of the feedback loop gains do not depend on the real parameter vector of the non-linear controlled dynamical system, and they depend solely on the constants from certain estimates that hold for all of its possible values from an assigned set. Estimates of the region of dissipativity “in the large” of the non-linear controlled dynamical system of special form and the original non-linear controlled dynamical system in deviations closed by the stabilizing control laws synthesized are given, and estimates for their limit sets and regions of attraction are presented.     

Robust design of rule-based fuzzy controllers

This paper deals with the design of stable and robust rule-based fuzzy control systems. New expressions to compute indices which provide a measure of the stability and robustness of fuzzy control systems are presented. The relation between the modification of the rules and the stability is studied through the so-called sensitivity indices. The paper presents procedures that make use of these indices to improve the design of fuzzy control systems, including the modification of the rules to obtain the global stability of an unstable system with multiple attractors, and to improve the dynamic behavior or the robustness of a non-linear plant. An example with a fuzzy controller for a system with non-linear damping and saturation in the actuation is presented to illustrate the design procedure.     

A reduction principle for asymptotically triangular differential systems

Problems of the stability of non-linear non-autonomous systems of differential equations with a special class of asymptotically vanishing perturbations are considered. The problem of reducing a problem on the stability of the equilibrium of a perturbed system to a problem on stability with respect to a non-linear approximation system which has a triangular form is solved. Applications of the results of the investigations to mechanical systems with a variable mass and time-varying equations of the constraints are presented.     

广义大系统的Lyapunov稳定性分析

广义大系统的稳定性是一个非常重要的问题 ,由于广义大系统的复杂性 ,对其稳定性的研究也是一件相当困难的事情 .本文利用 Lyapunov方程 ,应用 Lyapunov函数法 ,研究了广义线性大系统和广义非线性大系统的稳定性和不稳定性 ,得到了系统的关联稳定参数域和不稳定域 .给出例子说明该方法的可行性 .     

Non-linear Interaction of Morphological and Convective Instabilities During Solidification of a Dilute Binary Alloy

We consider the stability of a horizontal, planar solid-meltinterface during the solidification of a dilute binary alloy.Linear theory has shown that instability may occur due to eithersolutal convection in the melt or due to morphological effects.We investigate a system in which two modes of different horizontalwavenumber become unstable simultaneously. One mode representsthe morphological instability and the other represents the convectiveinstability. We carry out a weakly non-linear analysis, in orderto determine the non-linear evolution equations for the amplitudesof these two modes. The coefficients of these equations determinethe non-linear interaction which occurs near the onset of instability.We present an example of the application of the theory to thelead-tin alloy system, and show that the interaction is weak.     

Problems of the partial stability and detectability of dynamical systems

The conditions under which uniform stability (uniform asymptotic stability) with respect to a part of the variables of the zero equilibrium position of a non-linear non-stationary system of ordinary differential equations signifies uniform stability (uniform asymptotic stability) of this equilibrium position with respect the other, larger part of the variables, which include an additional group of coordinates of the phase vector, are established. These conditions include the condition for uniform asymptotic stability of the zero equilibrium position of the “reduced” subsystem of the original system with respect to the additional group of variables. Since within the conditions obtained the stability with respect to the remaining unmeasured coordinates of the phase vector remains undetermined or is investigated additionally, partial zero-detectability of the original system occurs in this case, and the conditions obtained supplement the series of known results from partial stability theory. The application of the results obtained to problems of the partial stabilization of non-linear controlled systems, particularly to the problem of stabilizing an asymmetric rigid body relative to an assigned direction in an inertial space, is considered. The partial detectability of linear systems with constant coefficients is also investigated.     

Non-linear stability for the Vlasov–Poisson system—the energy-Casimir method

The problem of stability of stationary solutions of the Vlasov–Poisson system has received a lot of attention in the physics literature, both in the stellar dynamics and the plasma physics cases. The energy-Casimir method has been used to prove non-linear stability for various conservative systems, but no rigorous application to the Vlasov–Poisson system has been given yet. We employ this method to prove non-linear stability of stationary solutions for the plasma physics case in three geometrically different settings.     

非线性系统李雅普诺夫函数构造分析

本文借助常系数线性系统的E.A巴尔巴辛公式,运用“类比法”构造了特殊类型的非线性系统的李雅普诺夫函数,从而推出非线性系统全局稳定性的一些充分条件,特别详细讨论非线性驻定系统(?)+f_1(?)+f_2(?)+f_3=0,其中f_2是二元函数的情形。     

Neutral stability of compression solitons in the bending of a non-linear elastic rod

The spectral stability of compression solitons in non-linear elastic rods with respect to perturbations of the flexural mode of the oscillations of the rod is investigated. The system of equations of the isotropic theory of elasticity, taking account of the non-linear corrections corresponding to the interaction being studied, is used to describe the interaction of longitudinal and flexural waves in the rod. This system of equations describes long longitudinal-flexural waves of small but finite amplitude. It is shown that trapped flexural modes exist, which propagate together with a compression soliton. It is established that these modes, which are the least stable, do not increase with time.     

Exponential Stability in Non-linear Difference Equations

Non-negative definite Lyapunov functionals are employed to obtain sufficient conditions that guarantee exponential stability of the zero solution of a non-linear discrete system. The theory is illustrated with several examples.     

Resonant instability in multidimensional Lyapunov systems

The stability of a point of rest of a Lyapunov system, which describes the perturbed motion of a dynamical system with an arbitrary number of degrees of freedom, is investigated. It is assumed that the characteristic equation of the system of the first approximation has a number of imaginary roots, while the quadratic part of the integral is not sign-definite. The case of third-order resonance is considered, when the stability problem is solved by the first non-linear (quadratic) terms.     

Stability and vibration analysis of a complex flexible rotor bearing system

This paper presents the non-linear dynamic analysis of a flexible rotor having unbalanced and supported by ball bearings. The rolling element bearings are modeled as two degree of freedom elements where the kinematics of the rolling elements are taken into account, as well as the internal clearance and the Hertz contact non-linearity. In order to calculate the periodic response of this non-linear system, the harmonic balance method is used. This method is implemented with an exact condensation strategy to reduce the computational time. Moreover, the stability of the non-linear system is analyzed in the frequency-domain by a method based on a perturbation applied to the known harmonic solution in the time domain.     

A qualitative analysis of the subharmonic oscillations of a parametrically excited flexible rod

A non-autonomous non-linear dynamical system with a small parameter that describes the parametric oscillations of a flexible rod with three static equilibrium positions is obtained. The generating equation of this model is a dynamical system in a plane with a separatrix loop. The qualitative analysis presented includes an investigation of the stability and bifurcation of subharmonic motions at resonance energy levels.     

Dynamic behavior of an AMB supported rotor subject to harmonic excitation

This paper presents a study of the non-linear response of a simple rigid disk-rotor, supported by active magnetic bearings (AMB), without gyroscopic effects. The case of primary resonance is examined under multi-excitation forces. The rotating shaft is described by a coupled second order non-linear ordinary differential equations. Approximate solutions are sought applying the method of multiple scales. Numerical simulations are carried out to illustrate the steady-state response and the stability of the solutions for various parameters using the frequency response function method. It is shown that the system parameters have different effects on the non-linear response of the rotor. For steady-state response, however, multiple-valued solutions and jump phenomenon occur. Results are compared to previously published work.     

具双耦合振子的捕食者——食饵系统的稳定性分析

研究一类具双耦合振子的Lotka-Volterra捕食者—食饵系统的稳定性.通过利用将非线性系统线性化,分析双曲不动点的特征值的实部等方法,获得了系统稳定的一些充分条件.     

On the influence of diffusion on the competition between bacteria and innate immune system of invertebrates

The influence of diffusion on the competition between bacteria and innate immune system of invertebrate animals is described by means of a system of two nonlinear reaction-diffusion equations, with constant coefficients. The existence of absorbing sets, corresponding to the coexistence of the two competing populations, is proven. The linear and non-linear stability properties of the solutions are analysed as well as the stability switches due to the diffusion.     

Non-linear stability of multivalue multiderivative methods

In Burrage and Butcher [2] a stability property suitable for modelling non-linear ordinary differential equation problems was introduced for the family of general linear methods. In the present paper we consider an extension of this non-linear stability property to methods which allow derivative evaluations. The non-linear stability properties of a number of methods are studied and a generalization is made which enables estimates of error growth.