Robust invariant stabilization of nonlinear continuous and discrete systems

Two classes of continuous systems described by differential equations in which coefficients are functions of the state vector are considered. The systems are subject to two scalar controls and a constantly acting scalar perturbation.An analytical synthesis of a control is performed under which the system is invariant in the sense that the scalar output of the system approaches zero as time tends to infinity and does not depend on the perturbation; moreover, the limit norm of the state vector is bounded above by the least upper bound for the norm of perturbation. The case where the coefficients of the system are subject to an uncontrolled additive perturbation is considered. In this case, the limit of the output norm is bounded above by a known function of the perturbation value. The method of synthesis is based on constructing the Lyapunov function as a positive definite quadratic form with Jacobian matrix.     

Golden mean relevance for chaos inhibition in a system of two coupled modified van der Pol oscillators

In this work, we present a novel evidence of the importance of the golden mean criticality of a system of oscillators in agreement with El Naschie’s E-infinity theory. We focus on chaos inhibition in a system of two coupled modified van der Pol oscillators. Depending on the coupling between the two oscillators, the system shows chaotic behavior for different ranges of the coupling parameter. Chaos suppression, as a transition from irregular behavior to a periodical one, is induced by perturbing the system with a harmonic signal with amplitude considerably lower than the value which causes entrainment. The frequency of the perturbation is related to the main frequencies in the spectrum of the freely running system (without perturbation) by the golden mean. We demonstrate that this effect is also obtained for a perturbation with frequency such that the ratio of half the frequency of the first main component in the freely running chaotic spectrum over the frequency of the perturbation is very close (five digits coincidence) to the golden mean. This result is shown to hold for arbitrary values of the coupling parameter in the various ranges of chaotic dynamics of the free running system.     

Analysis of effect of random perturbation on dynamic response of gear transmission system

In order to investigate the effects of random perturbation of a low-frequency excitation caused by torque fluctuations, gear damping ratio, gear backlash, meshing frequency and meshing stiffness, the random dynamic model of a single pair of three-degree-of-freedom spur gear transmission system is established. With gear meshing frequency changing, the dynamic characteristics of the gear transmission system were analyzed by bifurcation diagram, phase diagram, time course diagram and Poincaré map of the system. The effects of random perturbation caused by a low-frequency excitation caused by torque fluctuations, gear damping ratio, gear backlash, meshing frequency and meshing stiffness were comparative analyzed. Numerical simulation shows that the gear transmission system with nonlinear clearance exists rich period-doubling bifurcation phenomenon. With the increasing of the gear meshing frequency, gear transmission system will be from the chaotic motion to periodic motion by inverse period-doubling bifurcation. The effect of the meshing frequency random perturbation on the gear transmission system movement is largest. On the contrary, the effect of the meshing stiffness random perturbation on the system is minimum.     

Effective stability for slow time-dependent near-integrable Hamiltonians and application

The aim of this note is to prove a result of effective stability for a non-autonomous perturbation of an integrable Hamiltonian system, provided that the perturbation depends slowly on time. Then we use this result to clarify and extend a stability result of Giorgilli and Zehnder for a mechanical system with an arbitrary time-dependent potential.     

On Lyapunov equivalence of linear differential systems with unbounded coefficients

We show that any linear homogeneous differential system can be reduced by some linear piecewise differentiable transformation whose matrix, together with its inverse, is bounded on the half-line to a system with piecewise constant coefficients of the same growth order, and any system with a uniformly small perturbation can be reduced by this linear transformation to the same system with a piecewise constant perturbation of the same smallness.     

一维不定参数结构系统振动特征问题的摄动传递矩阵法

基于Riccati传递矩阵法，给出了一维不确定参数结构系统振动特征问题的二阶摄动计算方法，该方法适用于一般的一维结构系统的实数和复数特征问题的分析，并给出了结构振动特征的灵敏度计算公式．算例对转子的陀螺特征值问题进行了摄动分析，摄动结果和精确计算结果吻合良好．     

Exponential stabilization of a class of 1-D hyperbolic PDEs

The problem of exponential stabilization of 1-D hyperbolic system with spatially varying coefficients is investigated. The main strategy reposes on mapping the original system into a target one by an invertible Volterra transformation with a kernel satisfying an appropriate PDE. This enables to convert a multiplicative perturbation exerted from the whole domain to a boundary perturbation in the target system. The problem is reformulated in the context of semigroups theory and solved via a quadratic Lyapunov functional. The stabilizer is explicitly constructed by means of a collocated-type controller of the auxiliary system combined with a term containing the solution of the kernel PDE. The technics of the feedback law construction also offer information about the stabilization mechanism which makes the proposed controller realizable in concrete situations.     

Impulsive perturbations in a predator–prey model with dormancy of predators

In this paper, a predator–prey model consisting of active and dormant states of predators with impulsive control strategy is established. Using Floquet theories, the small amplitude perturbation technique and the piecewise Lyapunov function method, the conditions of local and global asymptotical orbital stability of the prey-eradication periodic solution are obtained. The boundness and permanence of the impulsive system are proved by the comparison principle. Through numerical simulations, the effects of the impulsive perturbation on the inherent oscillation are investigated, which implies that the impulsive perturbation can lead to period-doubling bifurcation, chaos, and period-halving bifurcation. Moreover, the effects of the impulsive perturbation and hatching rate on the chaos of the system are comparatively studied by numerical simulation. These obtained results can be useful for ecosystem management and for explaining complex phenomena of ecosystems.     

Stochastic properties of autowave turbulence elimination

The complex spatio-temporal behavior due to spiral breakup and wave interaction, in a 2D reaction diffusion system is eliminated by spatially uniform perturbation. This autowave turbulence is modeled by a stochastic process of defect production and annihilation. The constructed solution agrees well with probabilities – in particular probability to have no defects – estimated from the reaction diffusion system. The time to extinction, related to the probability of eliminating all defects, depends on the medium size and the applied perturbation amplitude.     

Traveling wave solutions for Schrödinger equation with distributed delay

The paper is devoted to study of traveling waves of nonlinear Schrödinger equation with distributed delay by applying geometric singular perturbation theory, differential manifold theory and the regular perturbation analysis for a Hamiltonian system. Under the assumptions that the distributed delay kernel is strong general delay kernel and the average delay is small, we first investigate the existence of solitary wave solutions by differential manifold theory. Then by utilizing the regular perturbation analysis for a Hamiltonian system, we explore the periodic traveling wave solutions.     

Dissipation of one-species VMB system with non-uniform ionic background

The Vlasov–Maxwell–Boltzmann system is a system that describes the dynamics of charged particles when there is electromagnetic field. There have been extensive studies on the energy estimates and global existence of solutions to the one- and two-species Vlasov–Maxwell–Boltzmann system when the background density is constant. In this work, we first deal with the energy estimates of the one-species Vlasov–Maxwell–Boltzmann system when the background density is a small perturbation from the stationary state. Then, we handle the dissipation rate for the linearized one-species Vlasov–Maxwell–Boltzmann system with the background density having a small perturbation around the stationary state.     

一类时变非线性系统的一致有界性的注记

主要研究带干扰的广义齐次系统的一致有界和一致最终有界性 ,证明了当干扰项满足一致有界及Lp 可积时系统的一致有界性及一致最终有界性 ,本质推广了最近相关文献中的有关系统一致有界性的结果 .     

B-Spline Collocation Method for Nonlinear Singularly-Perturbed Two-Point Boundary-Value Problems

A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary conditions. The quasilinearization technique is used to linearize the original nonlinear singular perturbation problem into a sequence of linear singular perturbation problems. The B-spline collocation method on piecewise uniform mesh is derived for the linear case and is used to solve each linear singular perturbation problem obtained through quasilinearization. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighborhood of the boundary layers. The convergence analysis is given and the method is shown to have second-order uniform convergence. The stability of the B-spline collocation system is discussed. Numerical experiments are conducted to demonstrate the efficiency of the method.     

Persistence of Diophantine flows for quadratic nearly integrable Hamiltonians under slowly decaying aperiodic time dependence

The aim of this paper is to prove a Kolmogorov type result for a nearly integrable Hamiltonian, quadratic in the actions, with an aperiodic time dependence. The existence of a torus with a prefixed Diophantine frequency is shown in the forced system, provided that the perturbation is real-analytic and (exponentially) decaying with time. The advantage consists in the possibility to choose an arbitrarily small decaying coefficient consistently with the perturbation size. The proof, based on the Lie series formalism, is a generalization of a work by A. Giorgilli.     

Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions

In a bounded domain, we consider an Euler–Bernoulli-type thermoelastic plate equation with perturbed boundary conditions. The boundary conditions are such that when the perturbation parameter goes to infinity, we recover the hinged boundary conditions, while one recovers the clamped boundary conditions when the perturbation parameter goes to zero. Relying on resolvent estimates, we show that the underlying semigroup is uniformly, with respect to the perturbation parameter, analytic and exponentially stable. The main features of our proof are appropriate decompositions of the components of the system and the use of Lions? interpolation inequalities.     

On the qualitative behaviour of symplectic integrators Part I: Perturbed linear systems

Summary. We consider a dissipative perturbation of non–resonant harmonic oscillators. Under the perturbation the system admits a weakly attractive invariant torus. We apply a Runge-Kutta method to the system. If the integration method is symplectic then it also admits an attractive invariant torus, the step-size being independent of the perturbation parameter. For non–symplectic methods the discrete system only admits an attractive invariant torus if the step-size is so small such that the discretisation error is smaller than the perturbation. Received May 17, 1996     

具拟周期摄动的近可积Hamiltonian系统的KAM型定理

In this paper we consider the persistence of invariant tori of an integrable Hamiltonian system with a quasiperiodic perturbation. It is proved that if the unperturbed system satisfies the Rtissmann non-degenerate condition and the perturbed system satisfies the co-linked non-resonant condition, then the majority of invariant tori is persistent under the perturbation.     

A fast algorithm for solving Toeplitz penta-diagonal systems

In this paper a fast algorithm for solving a large system with a symmetric Toeplitz penta-diagonal coefficient matrix is presented. This efficient method is based on the idea of a system perturbation followed by corrections and is competitive with standard methods. The error analysis is also given.     

A new method of performance sensitivity analysis for non-Markovian queueing networks

The paper develops a new method of calculating and estimating the sensitivities of a class of performance measures with respect to a parameter of the service or interarrival time distributions in queueing networks. The distribution functions may be of a general form. The study is based on perturbation analysis of queueing networks. A new concept, the realization factor of a perturbation, is introduced for the network studied. The properties of realization factors are discussed, and a set of linear differential equations specifying the realization factors are derived. The sensitivity of the steady-state performance with respect to a parameter can be expressed in a simple form using realization factors. Based on this, the sensitivity can be estimated by applying a perturbation analysis algorithm to a sample path of the system. We show that the derivative of the performance measure with respect to a parameter based on a single sample path converges with probability one to the derivative of the steady-state performance as the length of the sample path goes to infinity. The results provide a new analytical method of calculating performance sensitivities and justifies the application of perturbation analysis algorithms to non-Markovian queueing networks.     

静力分析的一般随机摄动法

本文对向量值和矩阵值函数的不确定结构的静力响应和可靠性进行了研究。基于Ｋｒｏｎｅｃｋｅｒ代数和摄动理论导出了随机结构的有限元分析方法，随机变量和系统导数很方便地排列到２Ｄ矩阵中，给出了一般的数学表达式．