Stability Conditions for Linear Impulsive Systems with Delay

Sufficient asymptotic-stability conditions for a linear impulsive system with delay are established on the basis of Lyapunov—Razumikhin functions. A scalar equation is considered as an example __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 130–138, June 2005.     

Stability for Nonautonomous Linear Differential Systems with Infinite Delay

Journal of Dynamics and Differential Equations - We study the stability of general n-dimensional nonautonomous linear differential equations with infinite delays. Delay independent criteria, as...     

Averaging of Nonlinear Oscillation Systems of Higher Approximation with Delay

For oscillation resonance systems with delay on a finite segment or a semiaxis, we justify the averaging method with respect to fast variables. We also obtain error estimates for this method, which explicitly depend on a small parameter.     

Dimensional Reduction for Nonlinear Time-Delayed Systems Composed of Stiff and Soft Substructures

This paper presents a new approach, based on the center manifoldtheorem, to reducing the dimension of nonlinear time-delay systemscomposed of both stiff and soft substructures. To complete the reductionprocess, the dynamic equation of a delayed system is first formulated asa set of singular perturbed equations that exhibit dynamic behaviorevolving in two different time scales. In terms of the fast time scale,the dynamic equation of system can be converted into the standard formof a functional differential equation in critical cases, namely, to aform that can be treated by means of the center manifold theorem. Then,the approximated center manifold is determined by solving a series ofboundary-value problems. The center manifold theorem ensures that thedominant dynamics of the system is described by a set of ordinarydifferential equations of low order, the dimension of which is identicalto that of the phase space of slowly variable states. As an applicationof the proposed approach, a detailed stability analysis is made for aquarter car model equipped with an active suspension with a time delaycaused by a hydraulic actuator. The analysis shows that the dimensionalreduction is surprisingly effective within a wide range of the systemparameters.     

Iteration Method for Systems of Nonlinear Differential Equations with Delay and Restrictions

We establish a consistency condition for systems of nonlinear differential equations with delay and restrictions and justify the applicability of the iteration method to these problems.     

Periodic Response and Chaos in Nonlinear Systems with Parametric Excitation and Time Delay

In this paper, the periodic motions of a nonlinear system with quadratic,cubic, and parametrically excited stiffness terms and with time-delayterms are obtained by the incremental harmonic balance (IHB) method. Theelements of the Jacobian matrix and residue vector arising in the IHBformulation are derived in closed form. A mechanism model representingthe one-mode oscillation of beams and plates is considered as anexample. A path-following algorithm with an arc-length parametriccontinuation procedure is used to obtain the response diagrams. Thesystem also exhibits chaotic motion through a cascade of period-doublingbifurcations, which is characterized by phase planes, Poincaré sectionsand Lyapunov exponents. The interpolated cell mapping (ICM) procedure isused to obtain the initial condition map corresponding to multiplesteady-state solutions.     

NGPG-Stability of Linear Multistep Methods for Systems of Generalized Neutral Delay Differential Equations

The stability analysis of linear multistep methods for the numerical solutions of the systems of generalized neutral delay differential equations is discussed. The stability behaviour of linear multistep methods was analysed for the solution of the generalized system of linear neutral test equations. After the establishment of a sufficient condition for asymptotic stability of the solutions of the generalized system, it is shown that a linear multistep method is NGP _G -stable if and only if it is A-stable.     

Linear and Nonlinear Structural Mechanics

Book Review

Linear and Nonlinear Structural Mechanics     

Application of Special Nonsmooth Temporal Transformations to Linear and Nonlinear Systems under Discontinuous and Impulsive Excitation

Linear and nonlinear mechanical systems under periodic impulsive excitation are considered. Solutions of the differential equations of motion are represented in a special form which contains a standard pair of nonsmooth periodic functions and possesses a convenient structure. This form is also suitable in the case of excitation with a periodic series of discontinuities of the first kind (a stepwise excitation). The transformations are illustrated in a series of examples. An explicit form of analytical solutions has been obtained for periodic regimes. In the case of parametric impulsive excitation, it is shown that a nonequidistant distribution of the impulses with dipole-like temporal shifts may significantly effect the qualitative characteristics of the response. For example, the sequence of instability zones loses its different subsequences depending on the parameter of the shifts. It is shown that the method's applicability can be extended for nonperiodic regimes by involving the idea of averaging.     

On the Absolute Exponential Stability of Solutions of Systems of Linear Parabolic Differential Equations with One Delay

We investigate necessary conditions for the absolute exponential stability of a system of linear parabolic differential equations with one delay.     

Finite Precision Representation of the Conley Decomposition

We present a theoretical basis for a novel way of studying and representing the long-time behavior of finite-dimensional maps. It is based on a finite representation of -pseudo orbits of a map by the sample paths of a suitable Markov chain based on a finite partition of phase space. The use of stationary states of the chain and the corresponding partition elements in approximating the attractors of maps and differential equations was demonstrated in Refs. 7 and 3 and proved for a class of stable attracting sets in Ref. 8. Here we explore the relationship between the communication classes of the Markov chain and basic sets of the Conley decomposition of a dynamical system. We give sufficient conditions for the existence of a chain transitive set and show that basic sets are isolated from each other by neighborhoods associated with closed communication classes of the chain. A partition element approximation of an isolating block is introduced that is easy to express in terms of sample paths. Finally, we show that when the map supports an SBR measure there is a unique closed communication class and the Markov chain restricted to those states is irreducible.     

Generation of Accurate Finite Element Models of Nonlinear Systems – Application to an Aeroplane-Like Structure

Model updating and validation is currently a central issue in the fields of computational structural mechanics and dynamics. The vast majority of applications however concerns linear structures. On the other hand, updating nonlinear models is something the structural dynamicist prefers to avoid mainly because tools such as modal analysis are no longer available. The objective of the present study is to propose a two-step methodology for dealing with nonlinear systems. Its most appealing feature is that it decouples the estimation of the linear and nonlinear parameters. A numerical application consisting of an aeroplane-like structure is used to assess the efficiency of the procedure.     

Effects of Random Noise and Internal Delay in Nonlinear Psychophysics

The effects of introducing second-order random noise on to parameters, so that they are unstable over time, and the effects of internal delay in a deterministic and therefore noise-free recursion are compared. This is done by examining changes in the shape of the escarpment region which corresponds to the traditional psychometric function in sensation intensity or threshold experiments. Some of the grosser psychophysical response surface features are preserved, but only over a limited region of the parameter space. The system is robust against low noise and very brief internal delays, but will lose information and stability outside the region corresponding to low inputs and medium stability. This finding is compatible with what is reported on nonlinear cellular neural networks, for which a few analytical results on stability have been derived.     

Linear Equations of Motion in Finite Elasticity

In this note we show that it may be possible and useful to construct valid strain-energy functions that lead directly to linear equilibrium equations for problems in isotropic homogeneous unconstrained nonlinear elasticity. While it is possible to make some general progress the final outcome will depend on the geometry and kinematics of the problem under consideration. Specific examples are given to show how exact solutions, via the linear equations of motion, can be found to non-trivial problems for physically meaningful constitutive models.      

非线性随机过程的有限元分析

本文根据实际复杂工程结构独立随机变量多，计算工作量大的特点，提出了用广义随机变量替代一般的独立随机变量以减少基本随机变量的数目，用响应曲面法替代敏度法或摄动法以提高计算效率。算例证明，以上措施取得了良好的效果，有助于非线性随机有限元法在实际工程结构可靠性分析中的应用。     

Traveling Wave Fronts of Reaction-Diffusion Systems with Delay

This paper deals with the existence of traveling wave front solutions of reaction-diffusion systems with delay. A monotone iteration scheme is established for the corresponding wave system. If the reaction term satisfies the so-called quasimonotonicity condition, it is shown that the iteration converges to a solution of the wave system, provided that the initial function for the iteration is chosen to be an upper solution and is from the profile set. For systems with certain nonquasimonotone reaction terms, a convergence result is also obtained by further restricting the initial functions of the iteration and using a non-standard ordering of the profile set. Applications are made to the delayed Fishery–KPP equation with a nonmonotone delayed reaction term and to the delayed system of the Belousov–Zhabotinskii reaction model. An erratum to this article is available at .     

A Finite Dimensional Property of a Parabolic Partial Differential Equation

In this note, we prove that there exists an inertial manifold for a parabolic system which is related to the transformed Navier-Stokes equations. For the proof we find negatively bounded solutions. 1991 Mathematics Subject Classification: 35A05; 35B40; 35B42; 35K57.     

Nonlinear System Identification of Multi-Degree-of-Freedom Systems

A nonlinear system identification methodology based on theprinciple of harmonic balance is extended tomulti-degree-of-freedom systems. The methodology, called HarmonicBalance Nonlinearity IDentification (HBNID), is then used toidentify two theoretical two-degree-of-freedom models and anexperimental single-degree-of freedom system. The three modelsand experiments deal with self-excited motions of afluid-structure system with a subcritical Hopf bifurcation. Theperformance of HBNID in capturing the stable and unstable limitcycles in the global bifurcation behavior of these systems is alsostudied. It is found that if the model structure is well known,HBNID performs well in capturing the unknown parameters. If themodel structure is not well known, however, HBNID captures thestable limit cycle but not the unstable limit cycle.     

Nonlinear Vibrations of Fractionally Damped Systems

This paper deals with the harmonic oscillations of periodically excited nonlinear systems where hysteresis is simulated via fractional operator representations. Employing a diophantine version of the fractional operational powers, the energy constrained Lindstedt–Poincaré perturbation procedure is utilized to establish the harmonic solution. The constrained perturbation procedure was employed since it allows for the handling of strong damping and exciting forces over the full span of the driving frequency range. Based on the approach taken, the long time behavior of the fractionally damped Duffing's equation is studied in detail. Of special interest is the determination of the influence of fractional order on the frequency amplitude response behavior.