Energy Method for Computing Periodic Solutions of Strongly Nonlinear Autonomous Systems with Multi-Degree-of-Freedom

In this paper with the use of conservation of average energy, a newmethod for computing the periodic solutions of strongly nonlinearautonomous systems with multi-degree-of-freedom is suggested. Thismethod cannot only decide the existence, but also give the approximateexpressions of the periodic solutions.     

Construction of Periodic Solutions of Linear Systems of Partial Differential Equations with Linear Deviations of Arguments

We prove a theorem on the existence of a solution of a system of partial differential equations with linearly transformed arguments that is continuously differentiable and bounded on ².__________Translated from Neliniini Kolyvannya, Vol. 7, No. 4, pp. 462–467, October–December, 2004.     

Periodic Orbits in Hamiltonian Systems with Involutory Symmetries

We study the existence of families of periodic solutions in a neighbourhood of a symmetric equilibrium point in two classes of Hamiltonian systems with involutory symmetry. In both classes, the involution reverses the sign of the Hamiltonian function, and the system is in 1:1 resonance. In the first class we study a Hamiltonian system with a reversing involution R acting symplectically. We first recover a result of Buzzi and Lamb showing that the equilibrium point is contained in a three dimensional conical subspace which consists of a two parameter family of periodic solutions with symmetry R, and furthermore that there may or may not exist two families of non-symmetric periodic solutions, depending on the coefficients of the Hamiltonian (correcting a minor error in their paper). In the second problem we study an equivariant Hamiltonian system with a symmetry S that acts anti-symplectically. Generically, there is no S-symmetric solution in a neighbourhood of the equilibrium point. Moreover, we prove the existence of at least 2 and at most 12 families of non-symmetric periodic solutions. We conclude with a brief study of systems with both forms of symmetry, showing they have very similar structure to the system with symmetry R.     

Stability of Equilibrium Solutions of Autonomous and Periodic Hamiltonian Systems with <Emphasis Type="Italic">n</Emphasis>-Degrees of Freedom in the Case of Single Resonance

This paper concerns with the study of the stability of an equilibrium solution of an analytic Hamiltonian system in a neighborhood of the equilibrium point with n-degrees of freedom, in the autonomous and periodic case under the presence of a single resonance. Our Main Theorem generalizes several results existing in the literature and we also give a geometrical interpretation of the hypotheses involved there. In particular, our Main Theorem provides necessary and sufficient conditions for the stability of the equilibrium solutions under the existence of a single resonance, depending on the coefficients of the Hamiltonian function.     

Exponential Dichotomy for Asymptotically Hyperbolic Two-Dimensional Linear Systems

We consider the problem of determining the existence of exponential dichotomy for a class of linear nonautonomous ODEs. An approach is explored that combines numerical techniques with rigorous perturbation theory. It is applicable to a given problem within the class we consider. Numerical results illustrate the utility of the approach.     

Periodic Solutions of Linear Impulsive Differential Inclusions

For linear differential inclusions with pulse action at fixed times, we establish conditions for the existence of periodic ordinary solutions and R-solutions.__________Translated from Neliniini Kolyvannya, Vol. 7, No. 4, pp. 495–515, October–December, 2004.     

On Reliable Stabilization of Linear Periodic Systems

An algorithm is proposed to synthesize a reliable controller with a given stability margin for linear and periodic systems optimized with respect to a quadratic performance criterion. A reliable controller synthesized by the algorithm guarantees the stability margin and is close to the linear-quadratic requlator. The importance of ensuring the stability margin is demonstrated. The proposed algorithm is based on methods of linear matrix inequalities and can be implemented using standard MATLAB routines. As an example, a reliable controller that stabilizes the program motion of a hopping machine is synthesized __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 10, pp. 114–126, October 2005.     

线性哈密顿系统的本征摄动法

利用哈密顿算子辛自共轭的特点讨论了保守哈密顿体系的摄动问题,给出了哈密顿矩阵的本征值与本征向量的二阶摄动分析方法。即当系统在哈密顿框架下进行较小修改时,不重复求解大型哈密顿矩阵的本征问题,只需在原系统的模态参数基础上进行模态分析即可,这种矩阵摄动法给出了修改后矩阵的二阶本征值和本征向量,为一般线性保守体系的本征摄动求解提出了一个新方法。     

Dynamical Methods for Linear Hamiltonian Systems with Applications to Control Processes

The nonautonomous version of the Yakubovich Frequency Theorem characterizes the solvability of an infinite horizon optimization problem in terms of the validity of the Frequency and Nonoscillation Conditions for a linear Hamiltonian system, which is defined from the coefficients of the quadratic functional to be minimized. This paper describes those nonautonomous linear Hamiltonian systems satisfying the required properties. Two groups appear, depending on whether they are uniformly weakly disconjugate or not. It also contains a previous analysis of the long-term behavior of the Grassmannian and Lagrangian flows under the presence of exponential dichotomy, which is required for the classification and has interest by itself.     

Uniform Expansions of Periodic Solutions for the Third Superharmonic Resonance

A new method of uniform expansions of periodic solutions to ordinary differential equations has recently been proposed to study quasi-harmonic processes in non-linear dynamical systems, in particular, when a small parameter of non-linearity is absent. The main idea of the method consists in using the ratio of the amplitudes of higher harmonics to the amplitude of the first harmonic of a periodic solution as a small formal parameter that appears due to descending the amplitudes of harmonics monotonically with increasing their number (this is the condition that the term quasi-harmonic implies). In this paper, the method is generalized for the third superharmonic resonance (when the first and the third harmonics become of the same magnitude) in a harmonically forced oscillator with arbitrary odd polynomial non-linearity.     

Subharmonic Solutions with Prescribed Minimal Period of a Class of Nonautonomous Hamiltonian Systems

In this paper, by using a decomposition technique to estimate the energy (the value of the functional associated with the problem) of a solution in terms of the minimal period of the solution, we give new sufficient conditions for the existence of subharmonic solutions with prescribed minimal period of Hamiltonian systems. Our results improve some known results in the literature. Dedicated to Professor Zhifen Zhang in the occasion of her 80th birthday     

Periodic Solutions for Some Systems of Delay Differential Equations

In this paper, we give sufficient conditions for the existence of periodic orbits of some systems of delay differential equations with a unique delay having 3, 4 or n equations. Moreover, we provide examples of delay systems satisfying the different sets of sufficient conditions.     

强非线性动力系统周期解分析

给出一类强非线性动力系统周期解存在性，唯一性和稳定性的简易差别法以及周期解的摄动法。本差别法把问题归结为干扰力在相应的未扰系统振动周期上的功函数及其导数的讨论，其限制条件比现有结果弱。本摄动法可以认为是经典Ｌｉｎｄｓｔｅｄｔ－Ｐｏｉｎｃａｒｅ（Ｌ－Ｐ）法在强非线性振动系统的推广。它与Ｌ－Ｐ法的主要区别在于假设系统的振动频率为相角的非线性函数。     

The Favard Separation Condition for Almost Periodic Linear Systems

For linear systems which depend almost periodically on time, the Favard separation condition is shown to be equivalent to the following dimensional fact: all the systems in the hull have the same number of independent bounded solutions.     

Minimal Principal Solution at Infinity for Nonoscillatory Linear Hamiltonian Systems

In this paper we open a new direction in the study of principal solutions for nonoscillatory linear Hamiltonian systems. In the absence of the controllability assumption, we introduce the minimal principal solution at infinity, which is a generalization of the classical principal solution (sometimes called the recessive solution) for controllable systems introduced by W. T. Reid, P. Hartman, and/or W. A. Coppel. The term “minimal” refers to the rank of the solution. We show that the minimal principal solution is unique (up to a right nonsingular multiple) and state its basic properties. We also illustrate our new theory by several examples.     

Lyapunov-Like Solutions to Stability Problems for Discrete-Time Systems on Asymptotically Contractive Sets

This paper presents new criteria for stability properties of discrete-time non-stationary systems. The criteria are based on the concept of asymptotically contractive sets. As a result, general necessary conditions are established for asymptotic stability of the zero equilibrium state, the instantaneous asymptotic stability domain of which can be either time-invariant or time-varying and then possibly asymptotically contractive. It is shown that the classical Lyapunov stability conditions including the invariance principle by LaSalle cannot be applied to the stability test as soon as the system instantaneous domain of asymptotic stability is asymptotically contractive. In order to investigate asymptotic stability of the zero state in such a case novel criteria are established. Under the criteria the total first time difference of a system Lyapunov function may be non-positive only and still can guarantee asymptotic stability of the zero state. The results are illustrated by examples.     

Unbounded Solutions to Systems of Differential Equations at Resonance

Journal of Dynamics and Differential Equations - We deal with a weakly coupled system of ODEs of the type $$begin{aligned} x_j'' + n_j^2 ,x_j + h_j(x_1,ldots ,x_d) = p_j(t), qquad...