DCDS-B
Almost periodic and asymptotically almost periodic solutions of Liénard equations
Tomás Caraballo David Cheban
Discrete & Continuous Dynamical Systems - B 2011, 16(3): 703-717 doi: 10.3934/dcdsb.2011.16.703
The aim of this paper is to study the almost periodic and asymptotically almost periodic solutions on $(0,+\infty)$ of the Liénard equation

$ x''+f(x)x'+g(x)=F(t), $

where $F: T\to R$ ($ T= R_+$ or $R$) is an almost periodic or asymptotically almost periodic function and $g:(a,b)\to R$ is a strictly decreasing function. We study also this problem for the vectorial Liénard equation.
   We analyze this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our early papers [3, 7] to prove the existence of almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of Liénard equations (both scalar and vectorial).

keywords: recurrent solutions Non-autonomous dynamical systems asymptotically almost periodic solutions Lienard equation. convergent systems almost automorphic global attractor skew-product systems quasi-periodic almost periodic cocycles
DCDS
Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems
David Cheban Cristiana Mammana
Discrete & Continuous Dynamical Systems - A 2007, 18(2&3): 499-515 doi: 10.3934/dcds.2007.18.499
The paper is dedicated to the study of the problem of continuous dependence of compact global attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems (IIFS). We prove that if a family of non-autonomous dynamical systems $(X,\mathbb T_1,\pi_{\lambda}),(Y,\mathbb T_{2},\sigma),h $depending on parameter $\lambda\in\Lambda$ is uniformly contracting (in the generalized sense), then each system of this family admits a compact global attractor $J^{\lambda}$ and the mapping $\lambda \to J^{\lambda}$ is continuous with respect to the Hausdorff metric. As an application we give a generalization of well known Theorem of Bransley concerning the continuous dependence of fractals on parameters.
keywords: Global attractor; non-autonomous dynamical system; infinite iterated functions systems.
PROC
Global attractors of nonautonomous quasihomogeneous dynamical systems
David Cheban
Conference Publications 2001, 2001(Special): 96-101 doi: 10.3934/proc.2001.2001.96
Please refer to Full Text.
keywords: global attractors. Nonautonomous quasihomogeneous systems
CPAA
On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence
Tomás Caraballo David Cheban
Communications on Pure & Applied Analysis 2013, 12(1): 281-302 doi: 10.3934/cpaa.2013.12.281
The aim of this paper is to describe the structure of global attractors for infinite-dimensional non-autonomous dynamical systems with recurrent coefficients. We consider a special class of this type of systems (the so--called weak convergent systems). We study this problem in the framework of general non-autonomous dynamical systems (cocycles). In particular, we apply the general results obtained in our previous paper [6] to study the almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of different classes of differential equations (functional-differential equations, evolution equation with monotone operator, semi-linear parabolic equations).
keywords: Non-autonomous dynamical systems almost periodic quasi-periodic asymptotically almost periodic solutions convergent systems almost automorphic functional differential equations; evolution equations with monotone operators cocycles global attractor dissipative systems recurrent solutions skew-product systems
DCDS-B
Belitskii--Lyubich conjecture for $C$-analytic dynamical systems
David Cheban
Discrete & Continuous Dynamical Systems - B 2015, 20(3): 945-959 doi: 10.3934/dcdsb.2015.20.945
The aim of this paper is study the problem of global asymptotic stability of solutions for $\mathbb C$-analytical dynamical systems (both with continuous and discrete time). In particular we present some new results for the $C$-analytical version of Belitskii--Lyubich conjecture. Some applications of these results for periodic $\mathbb C$-analytical differential/difference equations and the equations with impulse are given.
keywords: Belitskii--Lyubich conjecture. Global asymptotic stability attractor holomorphic dynamical systems
DCDS
Almost periodic and almost automorphic solutions of linear differential equations
Tomás Caraballo David Cheban
Discrete & Continuous Dynamical Systems - A 2013, 33(5): 1857-1882 doi: 10.3934/dcds.2013.33.1857
We analyze the existence of almost periodic (respectively, almost automorphic, recurrent) solutions of a linear non-homogeneous differential (or difference) equation in a Banach space, with almost periodic (respectively, almost automorphic, recurrent) coefficients. Under some conditions we prove that one of the following alternatives is fulfilled:
  (i) There exists a complete trajectory of the corresponding homogeneous equation with constant positive norm;
  (ii) The trivial solution of the homogeneous equation is uniformly asymptotically stable.
If the second alternative holds, then the non-homogeneous equation with almost periodic (respectively, almost automorphic, recurrent) coefficients possesses a unique almost periodic (respectively, almost automorphic, recurrent) solution. We investigate this problem within the framework of general linear nonautonomous dynamical systems. We apply our general results also to the cases of functional-differential equations and difference equations.
keywords: Almost periodic solution almost automorphic solutions cocycle uniform asymptotic stability linear nonautonomous contractive dynamical systems. nonautonomous dynamical systems
CPAA
On the structure of the global attractor for non-autonomous dynamical systems with weak convergence
Tomás Caraballo David Cheban
Communications on Pure & Applied Analysis 2012, 11(2): 809-828 doi: 10.3934/cpaa.2012.11.809
The aim of this paper is to describe the structure of global attractors for non-autonomous dynamical systems with recurrent coefficients (with both continuous and discrete time). We consider a special class of this type of systems (the so--called weak convergent systems). It is shown that, for weak convergent systems, the answer to Seifert's question (Does an almost periodic dissipative equation possess an almost periodic solution?) is affirmative, although, in general, even for scalar equations, the response is negative. We study this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our paper to the study of almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of different classes of differential equations.
keywords: dissipative systems convergent systems skew-product systems almost periodic global attractor Non-autonomous dynamical systems cocycles almost automorphic quasi-periodic asymptotically almost periodic solutions. recurrent solutions
DCDS-B
Recurrent motions in the nonautonomous Navier-Stokes system
Vena Pearl Bongolan-walsh David Cheban Jinqiao Duan
Discrete & Continuous Dynamical Systems - B 2003, 3(2): 255-262 doi: 10.3934/dcdsb.2003.3.255
We prove the existence of recurrent or Poisson stable motions in the Navier-Stokes fluid system under recurrent or Poisson stable forcing, respectively. We use an approach based on nonautonomous dynamical systems ideas.
keywords: Navier-Stokes equations Nonautonomous dynamical system skew-product flow Poisson stable motion. recurrent motion

Year of publication

Related Authors

Related Keywords

[Back to Top]