Concept Of Attractor Research Articles

Morphodynamical Abduction. Causation by Attractors Dynamics of Explanatory Hypotheses in Science

Philosophers of science today by and large reject the cataclysmic and irrational interpretation of the scientific enterprise claimed by Kuhn. Many computational models have been implemented to rationally study the conceptual change in science. In this recent tradition a key role is played by the concept of abduction as a mechanism by which new explanatory hypotheses are introduced. Nevertheless some problems in describing the most interesting abductive issues rise from the classical computational approach. It describes a cognitive process (and so abduction) by the manipulation of internal symbolic representations of external world. This view assumes a discrete set of representations fixed in discrete time jumps, and cannot adequately account for the issue of anticipation and causation of a new hypothesis. An integration of the traditional computational view with some ideas developed inside the so-called dynamical approach can suggest some important insights. The concept of attractor is very significant. It permits a description of the abductive generation of new hypotheses in terms of a catastrophic rearrangement of the parameters responsible for the behavior of the system.

FINITE DIMENSIONALITY OF ATTRACTORS FOR NON-AUTONOMOUS DYNAMICAL SYSTEMS GIVEN BY PARTIAL DIFFERENTIAL EQUATIONS

In the last decade, the concept of pullback attractor has become one of the usual tools to describe some qualitative properties of non-autonomous partial differential equations. A pullback attractor is a family of compact sets, invariant for the corresponding process related to the equation, and attracting from the past, and it assumes a natural generalization of the now classical concept of global attractor for autonomous partial differential equations. In this work we give sufficient conditions in order to prove the finite Hausdorff and fractal dimensionality of pullback attractors for non-autonomous infinite dimensional dynamical systems, and we apply our results to a generalized non-autonomous partial differential equation of Navier–Stokes type.

Complexity analysis, uncertainty management and fuzzy dynamical systems

In this paper, we present a brief study on various paradigms to tackle complexity or in other words manage uncertainty in the context of understanding science, society and nature. Fuzzy real numbers, fuzzy logic, possibility theory, probability theory, Dempster‐Shafer theory, artificial neural nets, neuro‐fuzzy, fractals and multifractals, etc. are some of the paradigms to help us to understand complex systems. We present a very detailed discussion on the mathematical theory of fuzzy dynamical system (FDS), which is the most fundamental theory from the point of view of evolution of any fuzzy system. We have made considerable extension of FDS in this paper, which has great practical value in studying some of the very complex systems in society and nature. The theories of fuzzy controllers, fuzzy pattern recognition and fuzzy computer vision are but some of the most prominent subclasses of FDS. We enunciate the concept of fuzzy differential inclusion (not equation) and fuzzy attractor. We attempt to present this theoretical framework to give an interpretation of cyclogenesis in atmospheric cybernetics as a case study. We also have presented a Dempster‐Shafer's evidence theoretic analysis and a classical probability theoretic analysis (from general system theoretic outlook) of carcinogenesis as other interesting case studies of bio‐cybernetics.

The Survival Attractor in the Sensory Functions: The Example of Hearing

High noise levels may have an adverse effect on the normal cochlea function and lead to significant hearing loss. Clinically, exposure to high intensity impulse noise produces a wide range of audiometric effects which may result in long term or even irreversible symptoms. Nevertheless, there is sometimes a spontaneous rebound recovery of the auditory function. This phenomenon was previously studied in the vision, another sensory function. It was called the visual survival attractor. In view of the importance that the sensory organs have for the brain, and in particular in its function of recognising and dealing with its environment, it was interesting to know whether this survival attractor concept already described for vision occurs more generally in all the sensory functions. With this in mind we present here the results of a new study, this time on hearing. This study was carried out on guinea pigs subjected to a pulsed acoustic trauma simulating the sound of a gun going off. Auditory function was explored using electrocochleography and two types of investigation were carried out. At first the change in hearing loss was studied in relation to frequencies varying between 2 and 24 kHz by calculating the difference, at each frequency, between the compound action potential thresholds measured before and then 20 mn, 24 h and 7 days after the acoustic trauma. On the other hand, the change in compound action potential amplitude was determined by varying the stimulating sound level from the auditory threshold up to 90 dB. This change was also recorded at the same time intervals as the investigation of hearing loss. In both the analysis of changes to hearing loss and in the investigation of variations in compound action potential, the results of the electrocochleographic investigations after acoustic trauma confirm the presence of a butterfly catastrophe type reaction process, that is positive evidence of a hearing survival attractor.

On the concept of attractor for community-dynamical processes I: the case of unstructured populations.

We introduce a notion of attractor adapted to dynamical processes as they are studied in community-ecological models and their computer simulations. This attractor concept is modeled after that of Ruelle as presented in [11] and [12]. It incorporates the fact that in an immigration-free community populations can go extinct at low values of their densities.

Asymptotically finite dimensional pullback behaviour of non-autonomous PDEs

The asymptotic behaviour of general non-autonomous partial differential equations can be described using the concept of pullback attractor. This is, under suitable hypotheses, a time-dependent family of finite-dimensional compact sets. In this work we investigate how this finite-dimensional dynamics on the attractor determines the asymptotic behaviour of non-autonomous PDEs.

On the concept of attractor for community-dynamical processes II: the case of structured populations.

In Part I of this paper Jacobs and Metz (2003) extended the concept of the Conley-Ruelle, or chain, attractor in a way relevant to unstructured community ecological models. Their modified theory incorporated the facts that certain parts of the boundary of the state space correspond to the situation of at least one species being extinct and that an extinct species can not be rescued by noise. In this part we extend the theory to communities of physiologically structured populations. One difference between the structured and unstructured cases is that a structured population may be doomed to extinction and not rescuable by any biologically relevant noise before actual extinction has taken place. Another difference is that in the structured case we have to use different topologies to define continuity of orbits and to measure noise. Biologically meaningful noise is furthermore related to the linear structure of the community state space. The construction of extinction preserving chain attractors developed in this paper takes all these points into account.

Attracting sets in a deterministic discrete traffic model

The fundamental diagram of the Nagel-Schreckenberg traffic model is derived analytically for the deterministic case using methods and concepts from nonlinear dynamics. It is shown that the possible states of the long-term behaviour form a globally attractive subset which can be well characterized. This attractive set of states is composed of coexisting attractors. The attractor concept is applied to a slow-to-start extension of the model. For this example it is shown that the attractive set consists of coexisting attractors with different macroscopic properties, that can be determined analytically.

Finite-dimensional dynamics of distributed optical systems with delayed feedback

A mathematical approach for evaluation the complexity of spatio-temporal dynamics for delayed feedback optical systems is suggested. It is based on the concept of global attractor for discrete semigroups. Both the retarded differential-difference and retarded diffusion models of feedback optical systems are similarly investigated. Upper estimates for the number of determining modes and Hausdorff dimension of attractors are obtained.

Hausdorff dimension of a random invariant set

The notion of random attractor for a dissipative stochastic dynamical system has recently been introduced. It generalizes the concept of global attractor in the deterministic theory. It has been shown that many stochastic dynamical systems associated to a dissipative partial differential equation perturbed by noise do possess a random attractor. In this paper, we prove that, as in the case of the deterministic attractor, the Hausdorff dimension of the random attractor can be estimated by using global Lyapunov exponents. The result is obtained under very natural assumptions. As an application, we consider a stochastic reaction-diffusion equation and show that its random attractor has finite Hausdorff dimension.

Determination of Fatigue Crack Paths

The complete solution of the problem of a fatigue crack growing from an initial crack includes determination of the crack path. Macroscopic aspects of the main features of the path taken by a growing fatigue crack in a metallic material are reasonably well understood in a qualitative sense. However, the governing equations are highly nonlinear, and are only partially understood. Microstructural irregularities have a significant effect on fatigue crack path behaviour. The nonlinear dynamics concept of attractor (in a physical rather than a state space sense) as a line or surface to which a fatigue crack path trends, appears to be helpful in providing a unified view of crack path behaviour. If the attractor for a particular configuration is known, then the crack path can be estimated. At present attractors have to be obtained intuitively on the basis of experience.

Nonautonomous systems, cocycle attractors and variable time-step discretization

The concept of an attractor for autonomous systems is generally too restrictive in the nonautonomous context. An appropriate generalization is the cocycle attractor which consists of a family of equivariant sets. Here the cocycle description of a nonautonomous system and the concept of a cocycle attractor are reviewed in the context of nonautonomous ordinary differential equations and variable time-step numerical schemes for autonomous ordinary differential equations. In the latter case, theorems are stated for the existence and convergence of numerical cocycle attractors to an assumed attractor of an autonomous ordinary differential equations.

On fatigue crack paths

The main features of the path taken by a propagating fatigue crack are discussed. It is assumed that a material is homogeneous and isotropic on a macroscopic scale, that behaviour is essentially elastic, and that the initial crack is ‘long’. Under these conditions the path taken by a propagating fatigue crack is reasonably well understood in a qualitative sense. There is a strong tendency for crack propagation to be in Mode I; the main exception is the slant crack propagation sometimes observed in thin sheets. A crack path is influenced by the precise characteristics of the crack tip plastic zone, by microstructural irregularities, and by the line tension of the crack front. The application of chaos theory to crack paths is explored briefly, and it is found that the concept of attractor helps in qualitative descriptions.

Reaction of the organism to stress: The survival attractor concept

This paper outlines a phenomenological approach for describing physiological reactions occurring immediately after vital threats. This exemplified by data taken from previous studies relative to chemical intoxications of rats by a neurotoxical drug. The survival rate of the animals and the variations of their cerebral acetylcholinesterase activity are both reported as a function of the drug concentration, and with respect to their age. The collecting of the results may be described as the cusp, a bifurcation set of Thom's Catastrophe Theory. The young animals react by a vital burst which modifies the shape of the cusp. A new fold or pocket takes place, changing it to the butterfly bifurcation set. This pocket supports the survival attractor concept, which appears reactional, transitory and variable in its intensity. In the last part of the paper, an extension of this concept is proposed to the visual domain. Some of patients who have suddenly and partially lost their visual acuity can recover some visual capacity by developing a new ability to see in weak contrasts. This reaction may be described as a visual survival attractor.

Physical models of cognition

This paper presents and discusses physical models for simulating some aspects of neural intelligence, and, in particular, the process of cognition. The main departure from the classical approach here is in utilization of a terminal version of classical dynamics introduced by the author earlier. Based upon violations of the Lipschitz condition at equilibrium points, terminal dynamics attains two new fundamental properties: it is spontaneous and nondeterministic. Special attention is focused on terminal neurodynamics as a particular architecture of terminal dynamics which is suitable for modeling of information flows. Terminal neurodynamics possesses a well-organized probabilistic structure which can be analytically predicted, prescribed, and controlled, and therefore which presents a powerful tool for modeling real-life uncertainties. Two basic phenomena associated with random behavior of neurodynamic solutions are exploited. The first one is a stochastic attractor—a stable stationary stochastic process to which random solutions of a closed system converge. As a model of the cognition process, a stochastic attractor can be viewed as a universal tool for generalization and formation of classes of patterns. The concept of stochastic attractor is applied to model a collective brain paradigm explaining coordination between simple units of intelligence which perform a collective task without direct exchange of information. The second fundamental phenomenon discussed is terminal chaos which occurs in open systems. Applications of terminal chaos to information fusion as well as to explanation and modeling of coordination among neurons in biological systems are discussed. It should be emphasized that all the models of terminal neurodynamics are implementable in analog devices, which means that all the cognition processes discussed in the paper are reducible to the laws of Newtonian mechanics.

Exponential attractors for non‐autonomous systems: Long‐time behaviour of vibrating beams

AbstractWe introduce the concept of exponential attractor for non‐autonomous systems. Then we prove the existence and finite dimensionality of the attractor for the model equation magnified image where K and f are quasiperiodic in time.

Chaos, Topology, and Social Organization

This paper attempts to apply chaos theory to social organization. It begins with a mathematical definition of chaos. Specifically, the geometric concept of attractor is explored; phase space and Poincaré maps are discussed and applied to the concept of attractor; and nonlinearity is conceptually defined. We then apply the mathematics of chaos to social systems by showing how a nonlinear equation might be used to describe organization and how data derived from a simple univariate equation can be converted into a multivariate Poincaré map. The conclusion section identifies three approaches to analyzing chaos in social organization: metaphorical analysis, mathematical modeling, and data collection. Finally, possible uses of chaos are explored, as are the shortcomings of such analysis.

The concept of minimal attractor and maximal attractors of partial differential equations of the Kuramoto-Sivashinsky type.

The first part of the paper is devoted to definition of the smallest set that may be regarded as the attractor of the dissipative system. Simple properties of this set are discussed. In the second part some geometrical facts concerning attractors of the partial differential equations are revealed. They include well-known results for the two-dimensional Navier-Stokes equation and new ones for the Kuramoto-Sivashinsky and similar equations.

Real-time attractors

The global properties of attractors of a class of dynamical systems are studied in the state space. The concept of real-time attractor is introduced with the view on practical applications. The abstract properties of attractors and real-time attractors are illustrated on classical examples in mechanics by computing the domains of attraction of asymptotically stable equilibria and periodic solutions using the method of point mapping. The properties of transient attractors are also studied. One of the possible applications here is to use them in generating the map of a differential game in the state space

Cortex, spikes and waves: the silent areas revisited

The concept of attractor neural networks (ANN) has led to biologically plausible models for neocortical working-associative memory, depending on excitatory feedback within modules of 103-105 neurons, asynchronous updating by units within discrete intervals approximating the spike refractory period and the stabilisation of attractors as different modular patterns of correlated burst firing which would persist for many seconds (of Hebbian reverberatory circuits). Recent studies of up to 11 units simultaneously, in nonspecific cortex of behaving monkeys, confirm traditional impressions of relatively low firing rates and apparently little bursting correlation in these conventionally ‘silent areas’ (in contrast to sensorimotor-specific neocortex) during a bimodal-sensory, complex localisation task. Considering the supposed cognitive functions of nonspecific cortex, the intracortical pathways involved, and not least the data from population-vector and oscillatory slow-wave studies in sensory-specific areas of t...