The effect of inversely unstable solutions on the attractor of the forced pendulum equation with friction

Consider the attractor A of a periodically forced equation of pendulum type with linear friction, in the cylinder. Levi and independently Min, Xian and Jinyan show that if the friction coefficient is larger than a certain bound then A is homeomorphic to the circle. We shall give a topological version of the definition of inversely unstable solution of N. Levinson and show that the appearance of such solutions imply that A is not homeomorphic to the circle. As an application we shall show that the bounds on the friction coefficient obtained before are optimal.     

The attractor on viscosity Degasperis–Procesi equation

In this paper the dynamical behaviors of a dispersive shallow water equation with viscosity, viscosity Degasperis–Procesi equation, are investigated. The existence of global solution to viscosity Degasperis–Procesi equation in L² under the periodical boundary condition is studied and the existence of the global attractor of semi-group to solution on viscosity Degasperis–Procesi equation in H² is obtained.     

The global attractor of the viscous Fornberg–Whitham equation

This paper aims to present a proof of the existence of the attractor for the one-dimensional viscous Fornberg–Whitham equation. In this paper, the global existence of solution to the viscous Fornberg–Whitham equation in L² under the periodic boundary conditions is studied. By using the time estimate of the Fornberg–Whitham equation, we get the compact and bounded absorbing set and the existence of the global attractor for the viscous Fornberg–Whitham equation.     

Homoclinic and heteroclinic flows in global attractor for Davey-stewartson II equation

By the variable transformation and generalized Hirota method, exact homoclinic and heteroclinic solutions for Davey-Stewartson II （DSII） equation are obtained. For perturbed DSII equation, the existence of a global attractor is proved. The persistence of homoclinic and heteroclinic flows is investigated, and the special homoclinic and heteroclinic structure in attractors is shown.     

The attractor of the stochastic generalized Ginzburg-Landau equation

The stochastic generalized Ginzburg-Landau equation with additive noise can be solved pathwise and the unique solution generates a random system.Then we prove the random system possesses a global random attractor in H_0~1.     

The Rovella attractor is a homoclinic class

Rovella proved the existence of measure-persistent attractors for flows exhibiting a unique singularity with three real eigenvalues satisfying λ₂ < λ₃ < 0 < λ₁ < −λ₃ ([Ro]). In this paper we prove that most of them are in fact homoclinic classes. *Partially supported by IMPA and CNPq. **Partially supported by CNPq, FAPERJ and PRONEX/DYN-SYS. from Brazil.     

Globally and locally attractive solutions for quasi-periodically forced systems

We consider a class of differential equations, , with ω∈Rd, describing one-dimensional dissipative systems subject to a periodic or quasi-periodic (Diophantine) forcing. We study existence and properties of trajectories with the same quasi-periodicity as the forcing. For g(x)=x2p+1, p∈N, we show that, when the dissipation coefficient is large enough, there is only one such trajectory and that it describes a global attractor. In the case of more general nonlinearities, including g(x)=x² (describing the varactor equation), we find that there is at least one trajectory which describes a local attractor.     

KdV-Burgers-Kuramoto系统的渐近吸引子

研究了 KdV-Burgers-Kuramoto 方程的渐近吸引子,即利用正交分解法构造一个有限维解序列。首先用数学归纳法证明了该解序列不会远离方程的整体吸引子,接着证明解序列在长时间后无限趋于方程的整体吸引子,最后给出渐近吸引子的维数估计。     

Global attractor for Hirota equation

The long time behavior of solution for Hirota equation with zero order dissipation is studied. The global weak attractor for this system in Hper^k is constructed. And then by exact analysis of the energy equation, it is shown that the global weak attractor is actually the global strong attractor in Hper^k.     

Kolmogorov-Spieqel-Sivashinsky方程的渐近吸引子及维数估计

研究了周期边界条件下Kolmogorov-Spieqel-Sivashinsky方程的渐近吸引子,并给出了它的维数估计.首先利用正交分解法构造了一个有限维解序列,然后分两步证明该解序列收敛于方程的真实解.     

Global attractor for the Kirchhoff type equation with a strong dissipation

The paper studies the longtime behavior of the Kirchhoff type equation with a strong dissipation utt−M(‖∇u^‖2)Δu−Δut+h(ut)+g(u)=f(x). It proves that the related continuous semigroup S(t) possesses in the phase space with low regularity a global attractor which is connected. And an example is shown.     

Time-dependent global attractor for extensible Berger equation

In this paper, we investigate the asymptotic behavior of the nonautonomous Berger equation

$\epsilon (t)u_{tt}+{\mathrm{\Delta }}^{2}u?(Q+\underset{\mathrm{\Omega }}{\int }|\mathrm{?}u{|}^{2}dx)\mathrm{\Delta }u+g(u_t)+\phi (u)=f,t>\tau ,$

on a bounded smooth domain $\mathrm{\Omega }?{\mathbb{R}}^N$ with hinged boundary condition, where $\epsilon (t)$ is a decreasing function vanishing at infinity. Under suitable assumptions, we establish an invariant time-dependent global attractor within the theory of process on time-dependent space.     

ZERO-DIMENSIONAL GLOBAL ATTRACTOR OF DAMPED SINE-GORDON EQUATION

1.IntroductionandMainResultsInthispaper,weconsiderthedampedsine-Gordonequation,withhomogeneousDirichletboundarycondition:whereu=u(x,t)ER,xEn,fiisaboundeddomaininRe(m=1,2,3)withsmoothboundaryoff,thedampingcoefficientor>0,thediffusingconstantd>0.Inthesequel,insteadofconsideringsystem(1.1),weinvestigatethefollowingsysteminHilbertspaceE=Ha(fi)xL'(fl):inwhichu(t)CHI(fl),v(t)6L'(fl)foranyt>0,A=--dA,G(u)=(--sine f),fEHa(~~),noEV.=Ha(~~),itoEH.=L'(fl).Let11'Onlayrwriteequatioll(l.2)asillwhi…     

The Cauchy problem for the dissipative Boussinesq equation

This work is devoted to the solvability and finite time blow-up of solutions of the Cauchy problem for the dissipative Boussinesq equation in all space dimension. We prove the existence and uniqueness of local mild solutions in the phase space by means of the contraction mapping principle. By establishing the time-space estimates of the corresponding Green operators, we obtain the continuation principle. Under some restriction on the initial data, we further study the results on existence and uniqueness of global solutions and finite time blow-up of solutions with the initial energy at three different level. Moreover, the sufficient and necessary conditions of finite time blow-up of solutions are given.     

The existence of global attractor for a fourth-order parabolic equation

This article is concerned with a fourth-order parabolic equation. Based on the regularity estimates for the semigroups and the classical existence theorem of global attractors, we prove that the fourth-order parabolic equation possesses a global attractor in H ^k (0?≤?k?H ^k (Ω) in the H ^k -norm.     

Uniform attractor for non‐autonomous Boussinesq‐type equation with critical nonlinearity

In this paper, we study the longtime dynamics of the non‐autonomous Boussinesq‐type equation with critical nonlinearity, and time‐dependent external forcing, which is translation bounded but not translation compact. We prove the existence of a uniform attractor in . Copyright © 2015 John Wiley & Sons, Ltd.     

Attractor for the viscous two-component Camassa-Holm equation

This paper is concerned with the attractor for a viscous two-component generalization of the Camassa-Holm equation subject to an external force, where the viscosity term is given by a second order differential operator. The global existence of solution to the viscous two-component Camassa-Holm equation with the periodic boundary condition is studied. We obtain the compact and bounded absorbing set and the existence of the global attractor in H²×H² for the viscous two-component Camassa-Holm equation by uniform prior estimate and many inequalities.     

Dynamical bifurcation for the Kuramoto-Sivashinsky equation

In this paper, by using the center manifold reduction method, together with the eigenvalue analysis, we made bifurcation analysis for the Kuramoto-Sivashinsky equation, and proved that the Kuramoto-Sivashinsky equation with constraint condition bifurcates an attractor A_λ as λ crossed the first critical value λ₀=1 under the two cases. Our analysis was based on a new and mature attractor bifurcation theory developed by Ma and Wang (2005) [17] and [18].     

Synchronization phenomena in the system consisting of m coupled Berger plates

The dynamical system arising in the study of nonlinear oscillations of a number of coupled Berger plates is considered. The dependence of the long-time behavior of the trajectories of the system on the properties of the coupling operator is studied. It is shown that the global attractor of the dynamical system is continuous with respect to the coupling parameter γ expressing the intensity of plate interaction. When γ→∞ it converges upper semicontinuously to the attractor of the system generated by the projection of the vector field of the coupled system on the kernel of the coupling operator. For the particular case of 3-diagonal coupling operator the synchronization phenomenon at the level of attractors is stated for large values of γ as well as the absence of synchronization for γ small. The case of cluster synchronization is also considered.     

The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions

In this paper, we study the global-in-time existence and the pointwise estimates of solutions to the Cauchy problem for the dissipative wave equation in multi-dimensions. Using the fixed point theorem, we obtain the global existence of the solution. In addition, the pointwise estimates of the solution are obtained by the method of the Green function. Furthermore, we obtain the Lp, 1?p?∞, convergence rate of the solution.