Attractors and recurrence for dendrite-critical polynomials

We call a rational map f dendrite-critical if all its recurrent critical points either belong to an invariant dendrite D or have minimal limit sets. We prove that if f is a dendrite-critical polynomial, then for any conformal measure μ either for almost every point its limit set coincides with the Julia set of f, or for almost every point its limit set coincides with the limit set of a critical point c of f. Moreover, if μ is non-atomic, then c can be chosen to be recurrent. A corollary is that for a dendrite-critical polynomial and a non-atomic conformal measure the limit set of almost every point contains a critical point.     

Recurrent critical points and typical limit sets of rational maps

We consider a rational map of the Riemann sphere with normalized Lebesgue measure and show that if there is a subset of the Julia set of positive -measure whose points have limit sets not contained in the union of the limit sets of recurrent critical points, then for -a.e. point and is conservative, ergodic and exact.

     

Backward stability for polynomial maps with locally connected Julia sets

We study topological dynamics on unshielded planar continua with weak expanding properties at cycles for which we prove that the absence of wandering continua implies backward stability. Then we deduce from this that a polynomial with a locally connected Julia set is backward stable outside any neighborhood of its attracting and neutral cycles. For a conformal measure this easily implies that one of the following holds: 1. for -a.e. , ; 2. for -a.e. , for a critical point depending on .

     

Attractors synthesis for a Lotka-Volterra like system

The aim of this paper is to prove numerically, via computer graphic simulations, that the synthesis algorithmprovided by Danca et al. [1] can be utilized to synthesize any attractor of a dynamical system modeling a two-predator, one prey Lotka-Volterra like system. The algorithm switches in a periodic deterministic or a random way the control parameter inside a set of a chosen values. The obtained attractor is the same with the attractor obtained for parameter value taken as averaged value of the switched control values. This simple and effective algorithm relies on a convex property induced in the set of the attractors corresponding to the chosen switching parameters. The algorithm was tested successfully on systems depending linearly on the control parameter like Lorenz, Chen, Rossler, networks and other systems.     

Rigidity for rational maps with Cantor Julia sets

In this paper, we prove that two rational maps with the Cantor Julia sets are quasicon- formally conjugate if they are topologically conjugate.     

Attractors and Dimensions for Discretizations of a Generalized Ginzburg-Landau Equation

In this paper, we discretize the generalized Ginzburg-Landau equations with the periodic boundary condition by the finite difference method in spatial direction. It is proved that for each mesh size, there exist at tractors for the discretized systems. The bounds for the Hausdorff dimensions of the discrete attractors are obtained, and the various bounds are independtmt of tho mesh sizes.     

一类非线性非自治可伸缩板方程的一致吸引子

讨论了一类带有初值条件和固定边界条件的非线性非自治可伸缩板方程u_(tt)+△~2u+αu-△u_(tt)-M(||▽u||~2)△u-γ△u_t+f(u)=σ(x,t)证明了该系统的整体适定性.进一步研究了该系统的长时间动力学,利用建立整体弱解所生成的半过程一致渐近紧性,得到了有界区域Ω■R~n(n≥1)或无界开集中一致吸引子的存在性.     

Attractors for the multilayer quasi-geostrophic equations of the ocean with delays

In this article, we study the multilayer quasi-geostrophic equations of the ocean with delays. We prove the existence of an attractor using the theory of pullback attractors.     

齐次Neumann边界条件下反应扩散方程的指数吸引子

利用构造挤压性的方法,讨论了齐次Neumann边界条件下反应扩散方程u_t-△u+λu=f(u)+β在H_(0¹)(Ω)中的指数吸引子的存在性.     

非线性梁方程强全局吸引子的存在性

非线性梁方程描述了桥面在竖直平面内的振动.作者利用文献[3]中给出的一种新的验证紧性的方法讨论了这类方程强解的全局吸引子.     

Conformal measures and Hausdorff dimension for infinitely renormalizable quadratic polynomials

We show in this paper that iff is a quadratic infinitely many times renormalizable polynomial of sufficient high combinatorial type, then: HD (J(f))= inf{: -conformal measure for f} We use Lyubich's construction of the principal nest ([Lyu97]) in order to prove this result.Partially supported by CNP q-Brazil grant # 300534/96-5     

On the dynamics of the singularly perturbed rational maps

In this paper, we study the dynamics of the family of rational maps fλ,（z） = zn - λ/zm, n ≥2, m ≥ 1,λ ∈ C. We construct an example of buried Sierpinski curve Julia set in this family. We also give an estimate of the location of bifurcation locus of fλ.     

一类二维非线性Schrodinger方程的吸引子及其维数

本文研究了一类二维非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程解的有限维行为，我们得到了此方程存在吸引子，并得到了此吸引子维数的上界估计     

Attractors for the strongly damped wave equation with p‐Laplacian

This paper is concerned with the initial‐boundary value problem for one‐dimensional strongly damped wave equation involving p‐Laplacian. For p > 2 , we establish the existence of weak local attractors for this problem in . Under restriction 2 < p < 4, we prove that the semigroup, generated by the considered problem, possesses a strong global attractor in , and this attractor is a bounded subset of . Copyright © 2017 John Wiley & Sons, Ltd.     

Specifying attracting cycles for Newton maps of polynomials

We show that for any set of n distinct points in the complex plane, there exists a polynomial p of degree at most n+1 so that the corresponding Newton map, or even the relaxed Newton map, for p has the given points as a super-attracting cycle. This improves the result in Plaza and Romero [6], which shows how to find such a polynomial of degree 2n. Moreover, we show that in general one cannot improve upon degree n+1. Our methods allow us to give a simple, constructive proof of the known result that for each cycle length n ≥ 2 and degree d ≥ 3, there exists a polynomial of degree d whose Newton map has a super-attracting cycle of length n.     

记忆型梁方程强解的全局吸引子

记忆型梁方程出现于—般的Kirchhoff粘弹性梁模型中．本文在记忆核满足指数衰退的条件下证明了系统的能量也是指数衰退的．进一步,通过对条件(C)的验证获得了系统强解的全局吸引子．     

巴拿赫空间中半群指数吸引子的存在性

We construct an exponential attractor for semigroup in Banach space by using ω-limit compactness method and provide a new method to prove the existence of an exponential attractor in uniformly convex Banach space. As a simple application, we prove the existence of an exponential attractor for reaction diffusion equations.