Notion Of Attractor Research Articles

On some results related to the Karlsson–Nussbaum conjecture in geodesic spaces

We show a Wolff–Denjoy type theorem in the case of a one-parameter continuous semigroup of nonexpansive mappings in which there is a compact mapping. Using the notion of attractor, we are also able to prove some specific properties directly related to the Karlsson–Nussbaum conjecture.

Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system

<p style='text-indent:20px;'>At the very beginning of the theory of finite dynamical systems, it was discovered that some relatively simple systems, even of ordinary differential equations, can generate very complicated (chaotic) behaviors. Furthermore these systems are extremely sensitive to perturbations, in the sense that trajectories with close but different initial data may diverge exponentially. Very often, the trajectories of such chaotic systems are localized, up to some transient process, in some subset of the phase space, the so-called strange attractors. Such subset have a very complicated geometric structure. They accumulate the nontrivial dynamics of the system.</p><p style='text-indent:20px;'>For a distributed system, whose time evolution is usually governed by partial differential equations (PDEs), the phase space X is (a subset of) an infinite dimensional function space. We will thus speak of infinite dimensional dynamical systems. Since the global existence and uniqueness of solutions has been proven for a large class of PDEs arising from different domains as Mechanics and Physics, it is therefore natural to investigate whether the features, in particular the notion of attractor, obtained for dynamical systems generated by systems of ODEs generalizes to systems of PDEs.</p><p style='text-indent:20px;'>In this paper we give a positive aftermath by proving the existence of pullback <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{D} $\end{document}</tex-math></inline-formula>-attractor. The key point is to find a bounded family of pullback <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{D} $\end{document}</tex-math></inline-formula>-absorbing sets then we apply the decomposition techniques and a method used in previous works to verify the pullback <inline-formula><tex-math id="M4">\begin{document}$ w $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M5">\begin{document}$ \mathcal{D} $\end{document}</tex-math></inline-formula>-limit compactness. It is based on the concept of the Kuratowski measure of noncompactness of a bounded set as well as some new estimates of the equicontinuity of the solutions.</p>

Toward the abstractors: modes of care and lineages of becoming

ABSTRACTThis paper explores an ontological version of ‘abstraction’ as it manifests in the commonalities and differences across social scientific research events. Drawing on a range of writings that focus on the potentiality of the event, on Whitehead’s concept of the ‘eternal object’, and on the notion of attractor as discussed by DeLanda, the notion of abstractor’ is tentatively proposed. The aim is to show how abstractors introduce particular potentialities, or ‘lineages of becoming’. However, in the specific context of the research event, these abstractors are subject to modes of care (with their own linages of becoming) that inform how analysts might engage with the potentialities of a research event (understood as an inventive problem space). This broad schema is initially illustrated through a particular abstractor, that of the ‘blackest black’ as partially actualized in the nanotechnology VANTAblack. Subsequently, the case of VANTAblack is used to prompt a number of heuristic questions with regard to how we might practically and carefully explore the commonalities and differences across research events. The paper closes with reflections on the broader status of the approach sketched here.

On three types of dynamics and the notion of attractor

We propose a theoretical framework for explaining the numerically discovered phenomenon of the attractor–repeller merger. We identify regimes observed in dynamical systems with attractors as defined in a paper by Ruelle and show that these attractors can be of three different types. The first two types correspond to the well-known types of chaotic behavior, conservative and dissipative, while the attractors of the third type, reversible cores, provide a new type of chaos, the so-called mixed dynamics, characterized by the inseparability of dissipative and conservative regimes. We prove that every elliptic orbit of a generic non-conservative time-reversible system is a reversible core. We also prove that a generic reversible system with an elliptic orbit is universal; i.e., it displays dynamics of maximum possible richness and complexity.

Attractors for impulsive non-autonomous dynamical systems and their relations

In this work, we deal with several different notions of attractors that may appear in the impulsive non-autonomous framework and we explore their relationships to obtain properties regarding the different scenarios of asymptotic dynamics, such as the cocycle attractor, the uniform attractor and the global attractor for the impulsive skew-product semiflow. Lastly, we illustrate our theory by exhibiting an example of a non-classical non-autonomous parabolic equation with subcritical nonlinearity and impulses.

A TOPOLOGICAL CHARACTERIZATION OF ­Ω-LIMIT SETS ON DYNAMICAL SYSTEMS

Abstract. In this article, we deal with the notion of ›-limit setsin dynamical systems. We show that the ›-limit set of a compactsubset of a phase space is quasi-attracting. 1. IntroductionThe theory for the notion of attractors is important for the classi-cal theory of dynamical systems. Conley [7] introduced a topologicaldeflnition of attractors for a °ow on a compact metric space. Hurley[9, 10] obtained results which is related to the correspondence betweenattractors and Lyapunov functions on noncompact spaces. Akin [1] andMcGehee [12] obtained many properties of attractors in set-valued dy-namics.The concept of omega-limit set, arising from their ubiquitous applica-tions in dynamical systems, is also an extremely used tool in the abstracttheory of dynamical systems. Especially, the notion of omega-limit setsis much related to the notion of attractors. These notions are used todescribe eventually the positive time behavior for dynamical systems.In recent years, Choy and Chu [4] described the characterizations ofomega-limit sets for analytic °ows on R

Continuous or dicrete? Attractor dynamics and spatial representations in a model of the hippocampal network

There is now strong evidence of the presence of a vast integrated neural system, which involves the hippocampus and several parahippocampal regions, dedicated to spatial navigation and memory. 'Place cells' and 'Grid cells' have been identified as neural correlates of the ability to remember an environment and to successfully move through it. The storage of a spatial map in the hippocampal circuits poses important computational issues. It asks for an extension of the classical notion of attractor, successfully used to model the storage of discrete memory items, like episodic memories, to a new entity that captures the continuous nature of space. In a continuous attractor activity smoothly changes with the position of the animal in the environment, so that each point in space is uniquely represented in the hippocampus. This situation corresponds to the presence of an infinite number of distinct stable configurations of the system, each for a different point of the environment. It is not known how much this idealized notion corresponds to the actual structure of the maps stored in the hippocampus, and how well attractor dynamics can approximate a continuous representation. We address these questions, within a simplified mathematical network model. The model network simulates the storage in CA3 of a spatial representation, its retrieval and its transfer to CA1. Through both analytical calculations and computer simulations we evaluate the ability of the network to represent different points in space and we study the properties of the attractor landscape when different kinds of disorder (connections dilution, heterogeneous place field distribution, multiple maps) are introduced in the system. Moreover we investigate the possible role of CA1 in the realization of a continuous attractor. We find that even networks of considerable size can only approximate the idealized notion of a 2D quasi-continuous dynamical attractor, but that the presence of CA1 generates a 'smoother' representation of space.

Fatal Attraction? Why Sperber’s Attractors do not Prevent Cumulative Cultural Evolution

In order to explain why cultural traits remain stable despite the error-proneness of social learning, Dan Sperber has proposed that human psychology and ecology lead to cultural traits being transformed in the direction of attractors. This means that simple-minded Darwinian models of cultural evolution are not appropriate. Some scientists and philosophers have been concerned that Sperber’s notion of attractors might show more than this, that attractors destroy subtle cultural variation and prevent adaptive cultural evolutionary processes from occurring. I show that Sperber’s view does not have this consequence—that even if there are attractors, cumulative cultural evolution can still occur. 1 Introduction2 Why Attractors are Supposed to Pose a Problem for Cumulative Cultural Evolution3 Why Attractors do not Prevent Cumulative Cultural Evolution 3.1 Reconstruction probably retains adaptively important features of cultural traits 3.2 Cultural selection may counteract the effects of attractors when cultural variation is adaptively important 3.3 Attractors may be numerous 3.4 Factors of attraction may often be working on the components of cultural traits 3.5 Attractors are highly variable, even within local populations4 Conclusion

On statistical attractors and the convergence of time averages

AbstractThere are various notions of attractor in the literature, including measure (Milnor) attractors and statistical (Ilyashenko) attractors. In this paper we relate the notion of statistical attractor to that of the essential ω-limit set and prove some elementary results about these. In addition, we consider the convergence of time averages along trajectories. Ergodicity implies the convergence of time averages along almost all trajectories for all continuous observables. For non-ergodic systems, time averages may not exist even for almost all trajectories. However, averages of some observables may converge; we characterize conditions on observables that ensure convergence of time averages even in non-ergodic systems.

Game as a career metaphor: a chaos theory career counselling application

ABSTRACT The potential of game as a career metaphor for use in counselling is explored and it is argued that it has been largely overlooked in the literature to date. This metaphor is then explicitly linked with the Chaos Theory of Careers (CTC), by showing how the notion of attractors within the CTC can be illustrated effectively using games metaphors. Games simultaneously illustrate the closed and open systems aspects of human endeavours and therefore can be a useful way of encouraging clients to appreciate the contingent and uncertain nature of their career development. It is argued, therefore, that metaphors provide an example of analogical reasoning that is useful for dealing with the modern counselling realities of complexity, connectedness, systems, changeability and chance.

Complexity for Extended Dynamical Systems

We consider dynamical systems for which the spatial extension plays an important role. For these systems, the notions of attractor, epsilon-entropy and topological entropy per unit time and volume have been introduced previously. In this paper we use the notion of Kolmogorov complexity to introduce, for extended dynamical systems, a notion of complexity per unit time and volume which plays the same role as the metric entropy for classical dynamical systems. We introduce this notion as an almost sure limit on orbits of the system. Moreover we prove a kind of variational principle for this complexity.

Attractors for Relative Semi-Dynamical Systems

In this paper a model for the concept of bornology from the point of view of an observer is presented. By the use of relative bornological spaces, the notion of attractor for relative semi-dynamical systems is considered.

Morse decompositions of nonautonomous dynamical systems

The global asymptotic behavior of dynamical systems on compact metric spaces can be described via Morse decompositions. Their components, the so-called Morse sets, are obtained as intersections of attractors and repellers of the system. In this paper, new notions of attractor and repeller for nonautonomous dynamical systems are introduced which are designed to establish nonautonomous generalizations of the Morse decomposition. The dynamical properties of these decompositions are discussed, and nonautonomous Lyapunov functions which are constant on the Morse sets are constructed explicitly. Moreover, Morse decompositions of one-dimensional and linear systems are studied.

Approximation des ensembles ω-limites des difféomorphismes par des orbites périodiques

Let M be a compact manifold, D the set of its C 1-diffeomorphisms (possibly symplectic or volume preserving). We prove that there exists a dense G δ G of D such that if f∈ G , every ω-limit set of f is the limit (for the Hausdorff topology) of a sequence of periodic orbits. This has certain interesting consequences concerning the structure of the ω-limit sets. Moreover, we define a new notion of attractors and describe them precisely in different cases.

A NEURAL NETWORK PRIMER

Neural networks are composed of basic units somewhat analogous to neurons. These units are linked to each other by connections whose strength is modifiable as a result of a learning process or algorithm. Each of these units integrates independently (in paral lel) the information provided by its synapses in order to evaluate its state of activation. The unit response is then a linear or nonlinear function of its activation. Linear algebra concepts are used, in general, to analyze linear units, with eigenvectors and eigenvalues being the core concepts involved. This analysis makes clear the strong similarity between linear neural networks and the general linear model developed by statisticians. The linear models presented here are the perceptron and the linear associator. The behavior of nonlinear networks can be described within the framework of optimization and approximation techniques with dynamical systems (e.g., like those used to model spin glasses). One of the main notions used with nonlinear unit networks is the notion of attractor. When the task of the network is to associate a response with some specific input patterns, the most popular nonlinear technique consists of using hidden layers of neurons trained with back-propagation of error. The nonlinear models presented are the Hopfield network, the Boltzmann machine, the back-propagation network and the radial basis function network.

Attractors and confiners in demography

Various dynamical models have been suggested by demographers to understand or to better forecast fertility fluctuations. Usual linear techniques have proved poor, but nonlinear techniques may help us to find out some temporal qualitative structure, and to take into account breakdown phenomena. For instance, the 1931–1985 fluctuations may be better described by drawing first return functions, which show a precise qualitative behaviour: two attractors exchanging the trajectory in the phase space under the perturbing environment of the historical context. A second example is drawn from a 17th century population, at a time when people were submitted to high and very irregular mortality. Because of the stochastic nature of the process, attractors are difficult to determine, and we resort to tools which are different from usual asymptotic attractors. These tools, called confiners, are the stochastic analogue of the notion of attractor for stochastic dynamical processes. They have been defined by Cosnard and Demongeot on the basis of the set of accumulation points of the trajectory, and rely on the essential intuitive property of invariance through the double operation consisting first in searching the confining basin of a set, then in taking the limit of this basin. The notion of confiner enables a rigourous description of the equilibrium states of the process: For fertility fluctuations during the 17th century, it reveals two confining zones exchanging the trajectory. Moreover, a simulation system shows how the existence of these two zones are directly linked to the frequencyv of mortality crisis occurring, that is to say, according to the valuev, the population dynamics will more or less balance towards a resistance to the crisis or towards a usual renewal of the population.