Multidimensional bony attractors

In this paper we study attractors of skew products, for which the following dichotomy is ascertained. These attractors either are not asymptotically stable or possess the following two surprising properties. The intersection of the attractor with some invariant submanifold does not coincide with the attractor of the restriction of the skew product to this submanifold but contains this restriction as a proper subset. Moreover, this intersection is thick on the submanifold, that is, both the intersection and its complement have positive relative measure. Such an intersection is called a bone, and the attractor itself is said to be bony. These attractors are studied in the space of skew products. They have the important property that, on some open subset of the space of skew products, the set of maps with such attractors is, in a certain sense, prevalent, i.e., ??big.?? It seems plausible that attractors with such properties also form a prevalent subset in an open subset of the space of diffeomorphisms.     

Comparison of various concepts of a random attractor:¶ A case study

Several non-equivalent definitions of an attractor of a random dynamical system have been proposed in the literature. We consider a rather simple special case: random dynamical systems with state space [0, ￥) [0, \infty) which fix 0. We examine conditions under which the set {0} is an attractor for three different notions of an attractor. It turns out that even in this simple case the various concepts are quite different. The purpose of this note is to highlight these differences and thus provide a basis for discussion about the "correct" concept of a random attractor.     

Non-autonomous dynamics of a semi-Kolmogorov population model with periodic forcing

In this paper we study a semi-Kolmogorov type of population model, arising from a predator–prey system with indirect effects. In particular we are interested in investigating the population dynamics when the indirect effects are time dependent and periodic. We first prove the existence of a global pullback attractor. We then estimate the fractal dimension of the attractor, which is done for a subclass by using Leonov’s theorem and constructing a proper Lyapunov function. To have more insights about the dynamical behavior of the system we also study the coexistence of the three species. Numerical examples are provided to illustrate all the theoretical results.     

Extended Fisher-Kolmogorov系统的渐近吸引子

考虑了ExtendedFisher-Kolmogorov系统的解的长时间行为,构造了一个有限维解序列即该系统的渐近吸引子,证明了它在长时间后无限趋于方程的整体吸引子,并给出了渐近吸引子的维数估计.     

On the “genetic memory” of chaotic attractor of the barotropic ocean model

The structure of attractor of barotropic ocean model is studied in this paper. Theorems of the existence of the attractor for the finite dimensional approximation of this model are proved as well as its convergence to the attractor of the model itself. Some properties of stationary solutions of this model and their stability are discussed.The structure of the attractor is partially explained by the sequence of bifurcations the system is subjected to by variations of leading parameters. The principal feature of the studied system is the existence of two “almost invariant” basins of chaotic attractor with very rare transitions between them. This is related to the rise of a couple of non-symmetric stable stationary solutions in the model with symmetric forcing.The “memory” of chaos appears also in the presence of maxima in the spectrum of energy. These maxima correspond either to the principal frequency of the limit cycle which arose in the Hopf bifurcation, or to the frequencies of the Feigenbaum phenomenon.     

Sine—Gordon方程的全局吸引子的维数估计

本文得到了阻尼Ｓｉｎｅ－Ｇｏｒｄｏｎ方程的狄氏问题的全局吸引子的Ｈａｕｓｄｏｒｆｆ维数以偶数上界的参数条件，特别地，当阻尼与Ｌａｐｌａｅ算子的第一个特征值适当大时，全局吸引子是零维的，零维吸引子恰是系统的唯一平衡解并且指数吸引相空间的有界集。     

吸引子的逼近

本文考虑耦合振子系统及对其按时间离散化(时间长为h)得到的离散动力系统.证明在一定条件下,这两个系统都有一维整体吸引子l和l_h.进一步证明当时间步长h→0时,l_h→l.     

Crisis transitions in excitable cell models

It is believed that sudden changes both in the size of chaotic attractor and in the number of unstable periodic orbits on chaotic attractor are sufficient for interior crisis. In this paper, some interior crisis phenomena were discovered in a class of physically realizable dissipative dynamical systems. These systems represent the oscillatory activity of membrane potentials observed in excitable cells such as neuronal cells, pancreatic β-cells, and cardiac cells. We examined the occurrence of interior crises in these systems by two means: (i) constructing bifurcation diagrams and (ii) calculating the number of unstable periodic orbits on chaotic attractor. Bifurcation diagrams were obtained by numerically integrating the simultaneous differential equations which simulate the activity of excitable membranes. These bifurcation diagrams have shown an apparent crisis activity. We also demonstrate in terms of the associated Poincaré maps that the number of unstable periodic orbits embedded in a chaotic attractor suddenly increases or decreases at the crisis.     

Invariant set and attractor of nonautonomous functional differential systems

The paper is devoted to the invariant set and periodic attractor for nonautonomous functional differential systems. Some sufficient criteria of the invariant set and periodic attractor are derived. Particularly, we have provided an estimate on existence range of periodic attractor by using the properties of nonnegative matrices and differential inequality technique.     

（E0，E）型渐近光滑映射和耗散波方程

本文在无穷维空间引入（Ｅ_０，Ｅ）型渐近光滑映射的概念，研究了其基本性质和变为Ｅ中渐近光滑映射的条件，我们证明了（Ｅ_０，Ｅ）型吸引子存在性定理和（Ｅ_０，Ｅ）型吸引子转化为Ｅ中吸引子的条件定理，所有结果都应用于一类耗散波方程渐近性态的研究。映射，吸引子，耗散波     

Structure of the global attractor for a second order strongly damped lattice system

In this paper, we consider the dynamical behavior of a second order strongly damped lattice system where the coupled operator is nonnegative definite symmetric. Firstly, we prove the existence of a global attractor, and give an upper bound of Hausdorff dimension of the global attractor, which keeps bounded for large strongly damping. Then we show that when the damping term is linear and the damping is suitable large, the system has an unbounded one-dimensional global attractor, which is a restricted horizontal curve.     

Collapse of attractors for ODEs under small random perturbations

This paper concerns comparisons between attractors for random dynamical systems and their corresponding noiseless systems. It is shown that if a random dynamical system has negative time trajectories that are transient or explode with probability one, then the random attractor cannot contain any open set. The result applies to any Polish space and when applied to autonomous stochastic differential equations with additive noise requires only a mild dissipation of the drift. Additionally, following observations from numerical simulations in a previous paper, analytical results are presented proving that the random global attractors for a class of gradient-like stochastic differential equations consist of a single random point. Comparison with the noiseless system reveals that arbitrarily small non-degenerate additive white noise causes the deterministic global attractor, which may have non-zero dimension, to ‘collapse’. Unlike existing results of this type, no order preserving property is necessary.      

Model based method for estimating an attractor dimension from uni/multivariate chaotic time series with application to Bremen climatic dynamics

In this paper, a method for estimating an attractor embedding dimension based on polynomial models and its application in investigating the dimension of Bremen climatic dynamics are presented. The attractor embedding dimension provides the primary knowledge for analyzing the invariant characteristics of the attractor and determines the number of necessary variables to model the dynamics. Therefore, the optimality of this dimension has an important role in computational efforts, analysis of the Lyapunov exponents, and efficiency of modeling and prediction. The smoothness property of the reconstructed map implies that, there is no self-intersection in the reconstructed attractor. The method of this paper relies on testing this property by locally fitting a general polynomial autoregressive model to the given data and evaluating the normalized one step ahead prediction error. The corresponding algorithms are developed in uni/multivariate form and some probable advantages of using information from other time series are discussed. The effectiveness of the proposed method is shown by simulation results of its application to some well-known chaotic benchmark systems. Finally, the proposed methodology is applied to two major dynamic components of the climate data of the Bremen city to estimate the related minimum attractor embedding dimension.     

Fractional-order attractors synthesis via parameter switchings

In this paper we provide numerical evidence, via graphics generated with the help of computer simulations, that switching the control parameter of a dynamical system belonging to a class of fractional-order systems in a deterministic way, one obtains an attractor which belongs to the class of all admissible attractors of the considered system. For this purpose, while a multistep numerical method for fractional-order differential equations approximates the solution to the mathematical model, the control parameter is switched periodically every few integration steps. The switch is made inside of a considered set of admissible parameter values. Moreover, the synthesized attractor matches the attractor obtained with the control parameter replaced with the averaged switched parameter values. The results are verified in this paper on a representative system, the fractional-order Lü system. In this way we were able to extend the applicability of the algorithm presented in earlier papers using a numerical method for fractional differential equations.     

One-dimensional attractor for a strongly damped lattice system

In this paper, we consider an attractor of a second order strongly damped lattice system with the non-negative definite symmetric coupled operator. We obtain the existence of a global attractor and prove this attractor is homeomorphic to the circle.     

Dynamical bifurcation of the damped Kuramoto–Sivashinsky equation

In this paper, we study the primary instability of the damped Kuramoto–Sivashinsky equation under a periodic boundary condition. We prove that it bifurcates from the trivial solution to an attractor which determines the long time dynamics of the system. Using the attractor bifurcation theorem and the center manifold theory, we describe the bifurcated attractor in detail.     

具有线性乘积白噪声的随机非自治吊桥方程的随机指数吸引子

本文考虑具有线性乘积白噪声的随机非自治吊桥方程长时间行为.首先,建立了所研究共圈系统的适定性;第二步,研究了该系统随机吸引子的存在性;第三步,当随机系数趋于0时,得到了随机吸引子的上半连续性;第四步,通过``迭代''法证明了随机吸引子在高正则空间中的正则性;最后,给出了该系统随机指数吸引子的存在性,同时得到了吸引子的有限分形维数.     

Attractors of generalized IFSs that are not attractors of IFSs

Mihail and Miculescu introduced the notion of a generalized iterated function system (GIFS in short), and proved that every GIFS generates an attractor. (In our previous paper we gave this notion a more general setting.) In this paper we show that for any m≥2$m\ge 2$, there exists a Cantor subset of the plane which is an attractor of some GIFS of order m  , but is not an attractor of a GIFS of order m−1$m-1$. In particular, this result shows that there is a subset of the plane which is an attractor of some GIFS, but is not an attractor of an IFS. We also give an example of a Cantor set which is not an attractor of a GIFS.     

Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system

In this paper we construct a dynamical process (in general, multivalued) generated by the set of solutions of an optimal control problem for the three-dimensional Navier-Stokes system. We prove the existence of a pullback attractor for such multivalued process. Also, we establish the existence of a uniform global attractor containing the pullback attractor. Moreover, under the unproved assumption that strong globally defined solutions of the three-dimensional Navier-Stokes system exist, which guaranties the existence of a global attractor for the corresponding multivalued semiflow, we show that the pullback attractor of the process coincides with the global attractor of the semiflow.     

Forward attraction in nonautonomous difference equations

Two types of attractors consisting of families of sets that are mapped into each other under the dynamics have been defined for nonautonomous difference equations, one using pullback convergence with information about the system in the past and the other using forward convergence with information about the system in the future. In both cases, the component sets are constructed using a pullback argument within a positively invariant family of sets. The forward attractor so constructed also uses information about the past, which is very restrictive and not essential for determining future behaviour. Here an alternative is investigated, essentially the omega-limit set of the system, which Chepyzhov and Vishik called the uniform attractor. It is shown here that this set is asymptotically positively invariant, thus providing it with an hitherto missing form of invariance, if in somewhat weaker than usual, that one expects an attractor to possess. As a consequence this set provides useful information about the behaviour in current time during the approach to the limit.