Periodic non-autonomous second-order dynamical systems

We study the existence of periodic solutions for a second-order non-autonomous dynamical system. We give three sets of hypotheses which guarantee the existence of non-constant solutions. We were able to weaken the hypotheses considerably from those used previously for such systems. We employ a saddle point theorem using linking methods.     

Invariant normalization of non-autonomous Hamiltonian systems

A new algorithm is proposed for reducing non-autonomous Hamiltonian systems to normal Birkhoff form. The criterion for the normal form is the condition that the vector fields of the perturbed and unperturbed parts of the system should commute. The invariant character of the criterion enables the system to be normalized in a unified way, without first simplifying the unperturbed part and without distinguishing between resonance and non-resonance, or autonomous and non-autonomous, cases. The whole algorithm reduces to a one-dimensional recurrence formula. The result is obtained by using the Campbell-Hausdorff formula for the ring of asymptotic forms, as well as the solution of a homologicla equation in the form of a quadrature. Three examples are considered to illustrate the various special features of the new algorithm. One of the examples is of interest for nuclear magnetic resonance theory.     

Sensitivity of non-autonomous discrete dynamical systems

In this paper, the sensitivity for non-autonomous discrete systems is investigated. First of all, two sufficient conditions of sensitivity for general non-autonomous dynamical systems are presented. At the same time, one stronger form of sensitivity, that is, cofinite sensitivity, is introduced for non-autonomous systems. Two sufficient conditions of cofinite sensitivity for general non-autonomous dynamical systems are presented. We generalized the result of sensitivity and strong sensitivity for autonomous discrete systems to general non-autonomous discrete systems, and the conditions in this paper are weaker than the correlated conditions of autonomous discrete systems.     

Skew-product semi-flows and non-autonomous control systems

In this paper we give a necessary and sufficient conditions for exact and approximate controllability of a wide class of linear infinite-dimensional non-autonomous control systems. This is done by employing skew-product semi-flow technique. Finally, we apply these results to prove the controllability of a broad class of non-autonomous reaction diffusion equations in Hilbert spaces.     

Representation of infinite-dimensional neutral non-autonomous control systems

This paper studies the well-posedness of a class of non-autonomous neutral control systems in Banach spaces. We prove that such systems are represented by absolutely regular non-autonomous linear systems in the sense of Schnaubelt [R. Schnaubelt, Feedback for non-autonomous regular linear systems, SIAM J. Control Optim. 41 (2002) 1141-1165]. This paper can be considered as the non-autonomous version of the work presented in [H. Bounit, S. Hadd, Regular linear systems governed by neutral FDEs, J. Math. Anal. Appl. 320 (2006) 836-858].     

On the asymptotic properties of non-autonomous systems

By giving a new method, we study asymptotic behavior of weakly almost nonexpansive sequences and curves introduced by Djafari Rouhani (J Differ Equ 229:412–425, 2006) in a reflexive Banach space X. Subsequently, we apply our results to study the asymptotic properties of unbounded trajectories for the quasi-autonomous dissipative system ${du/dt +Au\ni f}By giving a new method, we study asymptotic behavior of weakly almost nonexpansive sequences and curves introduced by Djafari Rouhani (J Differ Equ 229:412–425, 2006) in a reflexive Banach space X. Subsequently, we apply our results to study the asymptotic properties of unbounded trajectories for the quasi-autonomous dissipative system du/dt +Au ' f{du/dt +Au\ni f}, where A is an accretive (possibly multivalued) operator in X × X, and for some f_￥ ? X{f_{\infty}\in X} and 1 ≤ p < ∞ we have g ? L^p((1,+￥);X){g\in L^p((1,+\infty);X)}, so that g(θ) = (f(θ) − f _∞)/θ, ${(\forall \theta > 1)}${(\forall \theta > 1)}. Our results extend and improve many previously known results. Moreover, we answer an open question raised by B. Djafari Rouhani.     

Periodic solutions of the higher-dimensional non-autonomous systems

In this paper, the new existence theorem, unique theorem of periodic solution of the periodic system

and new stationary oscillation theorem of the periodic system

are derived by using functional analysis method, algebraic method and the new estimated formulas of solution of the homogenous linear system. Our results extend and improve some main results related to references. In addition, these criteria are of great interest in many applications such as computation.     

关于非自治动力系统中的h-极小覆盖

设（X,d1,f1∞）与（Y ,d2,g1,∞）为两个非自治动力系统,h是从（X,d1,f．∞）到（Y,d2,g1∞）的拓扑半共轭．通过对自治动力系统中的h一极小覆盖的研究,本文得到了以下结论：1）对于任意的Y∈Y及X∈h-1（y）,orb（x,f1∞）被h映射为orb（y,g1∞）,w（x,f1∞）被h映射为w（y,g1∞）;2）在（X,d1,f1∞）中引入关于拓扑半共轭的h-极小覆盖的定义,证明了h一极小覆盖的存在性;3）对于任意的XEX和Y∈Y,在（w（z,f1∞）,f1∞。（x,f1,∞）与（w（y,g∞）,g1,∞（y,g1∞））均构成原系统的子系统的前提下,R（f1∞）被h映射为R（g1∞）．这些结论丰富了非自治动力系统的内容．     

Convergence for non-autonomous semidynamical systems with impulses

The present paper deals with impulsive non-autonomous systems with convergence. We show that the structure of the Levinson center of a compact dissipative system is preserved under homomorphism in impulsive convergent systems. Also, we present some criteria of convergence using Lyapunov functions.     

Poincaré type theorems for non-autonomous systems

In this paper we establish analytic equivalence theorems of Poincaré and Poincaré-Dulac type for analytic non-autonomous differential systems based on the dichotomy spectrum of their linear part. As applications of the theorem, normal forms linearize for two illustrative examples.     

Stabilization of the motions of non-autonomous mechanical systems

The problem of stabilizing the motions of mechanical systems that can be described by non-autonomous systems of ordinary differential equations is considered. The sufficient conditions for stabilizing of the motions of mechanical systems with assigned forces due to forces of another structure are obtained by constructing a vector Lyapunov function and a reference system. Examples of the solution of the problems of stabilizing the rotational motion of an axisymmetric satellite in an elliptic orbit, a non-tumbling gyro horizon, etc. are considered ©2009     

Periodic solutions of quasilinear non-autonomous systems with impulses

The paper considers a system of differential equations with impulse perturbations at fixed moments in time of the form where x ? R ⁿ, ε is a small parameter, Sufficient conditions have been found for existence of the periodic solution of the given system in the critical and non-critical cases.     

Estimations of topological entropy for non-autonomous discrete systems

This paper is concerned with estimations of topological entropy for non-autonomous discrete systems. An estimation of lower bound of topological entropy for coupled-expanding systems associated with transition matrices in compact Hausdorff spaces is given. Estimations of upper and lower bounds of topological entropy for systems in compact metric spaces are obtained by their topological equi-semiconjugacy to subshifts of finite type under certain conditions. One example is provided for illustration.     

Global exponential stability of non-autonomous Nicholson-type delay systems

In this paper, we present some sufficient conditions for the global exponential stability of the non-autonomous Nicholson-type delay systems. Moreover, we give an example to illustrate our main results.     

非自治二阶Hamilton系统的周期解

本文推广了Willem的一个结果,Willem研究的受迫周期振动要求受迫势能关于空间变量是周期的,而本文只要求受迫势能对时间变量积分后关于空间变量是周期的,该结果包括了单摆的受迫振动.本文将用直接变分最小方法和Rabinowtz的鞍点定理来研究当势函数对时间变量积分后是周期时受迫摆方程的周期解.     

Homoclinic solutions of some second order non-autonomous systems

In this paper we investigate the existence of homoclinic solutions for the following second order non-autonomous system

where A is an antisymmetric constant matrix, is a symmetric and positive definite matrix for all , W(t,q)=a(t)V(q) such that is a continuous function and . Assuming that V(q) is subquadratic as q→+∞ and some technical assumptions on A and L, we establish two existence criteria to guarantee that (DS) has at least one nontrivial homoclinic solution by using a standard minimizing argument. Besides that, in some particular case, for the first time the uniqueness of homoclinic solutions of (DS) is also obtained. Recent results in the literature are generalized and significantly improved.     

Periodic solutions of non-autonomous second order Hamiltonian systems

The purpose of this paper is to study the existence of periodic solutions for a class of non-autonomous second order Hamiltonian system. Some new existence theorems are obtained by the least action principle.