Pullback exponential attractors for evolution processes in Banach spaces: properties and applications

This article is a continuation of our previous work [5], where we formulated general existence theorems for pullback exponential attractors for asymptotically compact evolution processes in Banach spaces and discussed its implications in the autonomous case. We now study properties of the attractors and use our theoretical results to prove the existence of pullback exponential attractors in two examples, where previous results do not apply.

An extension of the concept of gradient semigroups which is stable under perturbation

In this article we introduce the concept of a gradient-like nonlinear semigroup as an intermediate concept between a gradient nonlinear semigroup (those possessing a Lyapunov function, see [J.K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., vol. 25, Amer. Math. Soc., 1989]) and a nonlinear semigroup possessing a gradient-like attractor. We prove that a perturbation of a gradient-like nonlinear semigroup remains a gradient-like nonlinear semigroup. Moreover, for non-autonomous dynamical systems we introduce the concept of a gradient-like evolution process and prove that a non-autonomous perturbation of a gradient-like nonlinear semigroup is a gradient-like evolution process. For gradient-like nonlinear semigroups and evolution processes, we prove continuity, characterization and (pullback and forwards) exponential attraction of their attractors under perturbation extending the results of [A.N. Carvalho, J.A. Langa, J.C. Robinson, A. Suarez, Characterization of non-autonomous attractors of a perturbed gradient system, J. Differential Equations 236 (2007) 570-603] on characterization and of [A.V. Babin, M.I. Vishik, Attractors in Evolutionary Equations, Stud. Math. Appl.. vol. 25, North-Holland, Amsterdam, 1992] on exponential attraction. (C) 2009 Elsevier Inc. All rights reserved.

On backwards and forwards reachable sets bounding for perturbed time-delay systems

Linear systems with interval time-varying delay and unknown-but-bounded disturbances are considered in this paper. We study the problem of finding outer bound of forwards reachable sets and inter bound of backwards reachable sets of the system. Firstly, two definitions on forwards and backwards reachable sets, where initial state vectors are not necessary to be equal to zero, are introduced. Then, by using the Lyapunov-Krasovskii method, two sufficient conditions for the existence of: (i) the smallest possible outer bound of forwards reachable sets; and (ii) the largest possible inter bound of backwards reachable sets, are derived. These conditions are presented in terms of linear matrix inequalities with two parameters need to tuned, which therefore can be efficiently solved by combining existing convex optimization algorithms with a two-dimensional search method to obtain optimal bounds. Lastly, the obtained results are illustrated by four numerical examples.

Invariability, orbits and fuzzy attractors

In this paper, we present a generalization of a new systemic approach to abstract fuzzy systems. Using a fuzzy relations structure will retain the information provided by degrees of membership. In addition, to better suit the situation to be modelled, it is advisable to use T-norm or T-conorm distinct from the minimum and maximum, respectively. This gain in generality is due to the completeness of the work on a higher level of abstraction. You cannot always reproduce the results obtained previously, and also sometimes different definitions with different views are obtained. In any case this approach proves to be much more effective when modelling reality.

Effect of light-sticks and electralunie attractors on surface-longline catches of swordfish (Xiphias gladius, Linnaeus, 1959) in the southwest equatorial Atlantic

Two experimental fishing trials were carried out off the coast of Pernambuco, Brazil, in 1999 and 2001, using a small artisanal longliner. In experiment 1, six-hook baskets with three chemical light-sticks on alternating hooks had significantly higher catch rates than those with zero or with a light-stick on every hook, with most swordfish accounted for by hooks with light-sticks. Analysis of the data from experiment 2 showed no significant difference between electralume attractors, consisting of AA lithium batteries protected by a solid cover and light-sticks that produce a fluorescent light when two chemical products are mixed. Significant differences were detected in mean CPUE by size class, with most swordfish belonging to class 'b' (125-170 cm lower jaw to fork length (LJFL)). No differences, however, were found for swordfish catches in classes 'a' (< 125 cm LJFL) and V, and no evidence was found of interaction between the two factors (attractor and size class). Although there was no significant difference between the total length-frequency distributions of swordfish caught with light-sticks and electralume attractors, signibcant differences were found for fish smaller than 125 cm LJFL, with electralume catches consisting of smaller swordfish than those of gear using light-sticks. (c) 2004 Elsevier B.V. All rights reserved.

Effect of light-sticks and electralume attractors on surface-longline catches of swordfish (Xiphias gladius, Linnaeus, 1959) in the southwest equatorial Atlantic

Two experimental fishing trials were carried out off the coast of Pernambuco, Brazil, in 1999 and 2001, using a small artisanal longliner. In experiment 1, six-hook baskets with three chemical light-sticks on alternating hooks had significantly higher catch rates than those with zero or with a light-stick on every hook, with most swordfish accounted for by hooks with light-sticks. Analysis of the data from experiment 2 showed no significant difference between electralume attractors, consisting of AA lithium batteries protected by a solid cover and light-sticks that produce a fluorescent light when two chemical products are mixed. Significant differences were detected in mean CPUE by size class, with most swordfish belonging to class ‘b’ (125–170 cm lower jaw to fork length (LJFL)). No differences, however, were found for swordfish catches in classes ‘a’ (<125 cm LJFL) and ‘b’, and no evidence was found of interaction between the two factors (attractor and size class). Although there was no significant difference between the total length-frequency distributions of swordfish caught with light-sticks and electralume attractors, significant differences were found for fish smaller than 125 cm LJFL, with electralume catches consisting of smaller swordfish than those of gear using light-sticks.

Symmetry of chaotic attractors and kink-antikink structures in CMLs

In this paper the symmetries of coupled map lattices (CMLs) and their attractors are investigated by group and dynamical system theory, as well as numerical simulation, by means of which the kink-antikink patterns of CMLs in space-amplitude plots are discussed.

A GRADIENT-LIKE NONAUTONOMOUS EVOLUTION PROCESS

In this paper we consider a dissipative damped wave equation with nonautonomous damping of the form u(tt) + beta(t)u(t) - Delta u + f(u) (1) in a bounded smooth domain Omega subset of R(n) with Dirichlet boundary conditions, where f is a dissipative smooth nonlinearity and the damping beta : R -> (0, infinity) is a suitable function. We prove, if (1) has finitely many equilibria, that all global bounded solutions of (1) are backwards and forwards asymptotic to equilibria. Thus, we give a class of examples of nonautonomous evolution processes for which the structure of the pullback attractors is well understood. That complements the results of [Carvalho & Langa, 2009] on characterization of attractors, where it was shown that a small nonautonomous perturbation of an autonomous gradient-like evolution process is also gradient-like. Note that the evolution process associated to (1) is not a small nonautonomous perturbation of any autonomous gradient-like evolution processes. Moreover, we are also able to prove that the pullback attractor for (1) is also a forwards attractor and that the rate of attraction is exponential.

Stability theory and numerical analysis of non-autonomous dynamical systems

The development and use of cocycles for analysis of non-autonomous behaviour is a technique that has been known for several years. Initially developed as an extension to semi-group theory for studying rion-autonornous behaviour, it was extensively used in analysing random dynamical systems [2, 9, 10, 12]. Many of the results regarding asymptotic behaviour developed for random dynamical systems, including the concept of cocycle attractors were successfully transferred and reinterpreted for deterministic non-autonomous systems primarily by P. Kloeden and B. Schmalfuss [20, 21, 28, 29]. The theory concerning cocycle attractors was later developed in various contexts specific to particular classes of dynamical systems [6, 7, 13], although a comprehensive understanding of cocycle attractors (redefined as pullback attractors within this thesis) and their role in the stability of non-autonomous dynamical systems was still at this stage incomplete. It was this purpose that motivated Chapters 1-3 to define and formalise the concept of stability within non-autonomous dynamical systems. The approach taken incorporates the elements of classical asymptotic theory, and refines the notion of pullback attraction with further development towards a study of pull-back stability arid pullback asymptotic stability. In a comprehensive manner, it clearly establishes both pullback and forward (classical) stability theory as fundamentally unique and essential components of non-autonomous stability. Many of the introductory theorems and examples highlight the key properties arid differences between pullback and forward stability. The theory also cohesively retains all the properties of classical asymptotic stability theory in an autonomous environment. These chapters are intended as a fundamental framework from which further research in the various fields of non-autonomous dynamical systems may be extended. A preliminary version of a Lyapunov-like theory that characterises pullback attraction is created as a tool for examining non-autonomous behaviour in Chapter 5. The nature of its usefulness however is at this stage restricted to the converse theorem of asymptotic stability. Chapter 7 introduces the theory of Loci Dynamics. A transformation is made to an alternative dynamical system where forward asymptotic (classical asymptotic) behaviour characterises pullback attraction to a particular point in the original dynamical system. This has the advantage in that certain conventional techniques for a forward analysis may be applied. The remainder of the thesis, Chapters 4, 6 and Section 7.3, investigates the effects of perturbations and discretisations on non-autonomous dynamical systems known to possess structures that exhibit some form of stability or attraction. Chapter 4 investigates autonomous systems with semi-group attractors, that have been non-autonomously perturbed, whilst Chapter 6 observes the effects of discretisation on non-autonomous dynamical systems that exhibit properties of forward asymptotic stability. Chapter 7 explores the same problem of discretisation, but for pullback asymptotically stable systems. The theory of Loci Dynamics is used to analyse the nature of the discretisation, but establishment of results directly analogous to those discovered in Chapter 6 is shown to be unachievable. Instead a case by case analysis is provided for specific classes of dynamical systems, for which the results generate a numerical approximation of the pullback attraction in the original continuous dynamical system. The nature of the results regarding discretisation provide a non-autonomous extension to the work initiated by A. Stuart and J. Humphries [34, 35] for the numerical approximation of semi-group attractors within autonomous systems. . Of particular importance is the effect on the system's asymptotic behaviour over non-finite intervals of discretisation.

Reliable and valid survey design for structural equation modelling

My quantitative study asks how Chinese Australians’ “Chineseness” and their various resources influence their Chinese language proficiency, using online survey and snowball sampling. ‘Operationalization’ is a challenging process which ensures that the survey design talks back to the informing theory and forwards to the analysis model. It requires the attention to two core methodological concerns, namely ‘validity’ and ‘reliability’. Construction of a high-quality questionnaire is critical to the achievement of valid and reliable operationalization. A series of strategies were chosen to ensure the quality of the questions, and thus the eventual data. These strategies enable the use of structural equation modelling to examine how well the data fits the theoretical framework, which was constructed in light of Bourdieu’s theory of habitus, capital and field.

Heritage language for Chinese Australians : negotiating 'Chineseness' and, capitalising on resources in the lived world

The ethnic identity and commitment of Heritage Language Learners play salient roles in Heritage Language learning process. The mutually constitutive effect amongst Heritage Language Learner's ethnic identity, commitment, and Heritage Language proficiency has been well documented in social psychological and poststructuralist literatures. Both social psychological and poststructural schools offer meaningful insights into particular contexts but receive critiques from other contexts. In addition, the two schools largely oppose each other. This study uses Bourdieu's sociological triad of habitus, capital, and field to reconcile the two schools through the examination of Chinese Heritage Language Learners in Australia, an idiosyncratic social, cultural, and historical context for these learners. Specifically, this study investigates how young Chinese Australian adults (18-35 in age) negotiate their 'Chineseness' and capitalise on resources through Chinese Heritage Language learning in the lived world. The study adopts an explanatory mixed methods design to combine the quantitative approach with the qualitative approach. The initial quantitative phase addresses the first research question: Is Chinese Heritage Language proficiency of young Chinese Australian adults influenced by their investment of capital, the strength of their habitus of 'Chineseness', or both? The subsequent qualitative phase addresses the second research question: How do young Chinese Australian adults understand their Chinese Heritage Language learning in relation to (potential) profits produced by this linguistic capital in given fields? The initial quantitative phase applies Structural Equation Modelling to analyse the data from an online survey with 230 respondents. Findings indicate the statistically significant positive contribution made by the habitus of 'Chineseness' and by investment of capital to Chinese Heritage Language proficiency (r = .71 and r = .86 respectively). Subsequent multiple regression analysis demonstrates that 62% of the variance of Chinese Heritage Language proficiency can be accounted for by the joint contribution of 'Chineseness' and 'capital'. The qualitative phase of the study uses multiple interviews with five participants. It reveals that Chinese Heritage Language offers meaningful benefits for participants in the forms of capital production and habitus capture or recapture. Findings from the two phases talk to each other in terms of the inherent entanglement amongst habitus of 'Chineseness', investment of capital, and Chinese Heritage Language proficiency. The study offers important contributions. Theoretically, by virtue of Bourdieu's signature concepts of habitus, capital, and field, the study provides answers to questions that both social psychological and poststructuralist theories have long been struggling to answer. Methodologically, the position of 'pluralism' talks back to Bourdieu's theory and forwards to the mixed methods design. Particularly, the study makes a methodological breakthrough: A set of instruments was developed and validated to quantify Bourdieu's key concepts of capital and habitus within certain social fields. Practically, understanding Chinese Australians' heterogeneity and the potential drivers behind Chinese Heritage Language learning contributes to the growing interest in Chinese Australians' contemporary life experiences and helps to better accommodate linguistically diverse Chinese Heritage Language Learners in Chinese language courses. In addition, this study is very timely. It resonates with the recently released Australia in the Asian Century White Paper: Chinese Australians, with sound knowledge of Chinese culture and language obtained through negotiating their 'Chineseness' and capitalising on diverse resources for learning, will help to serve Australia's economic, social, and political needs in unique ways.

Streamline topology and dilute particle dynamics in a Karman vortex street flow

Three types of streamline topology in a Karman vortex street flow are shown under the variation of spatial parameters. For the motion of dilute particles in the Karman vortex street flow, there exist a route of bifurcation to a chaotic orbit and more attractors in a bifurcation diagram for the proportion of particle density to fluid density. Along with the increase of spatial parameters in the flow field, the bifurcation process is suspended, as well as more and more attractors emerge. In the motion of dilute particles, a drag term and gravity term dominate and result in the bifurcation phenomenon.

Forward pricing and the housing market: the pre-sales housing system in Taiwan

In countries that have experienced rapid economic development, the need to establish more efficient markets in which private property can be constructed has induced some innovative solutions. One such solution is the phenomenon of a pre-sales market of the kind that can be observed in Taiwan, Korea, and more recently in China. Developers sell their property before building is started in order to acquire financing for the development companies. This paper discusses the process and, by recognising the analogy between the pre-sales market and forwards markets, analyses the implications for developers

Models of complexity in Robert Coover's John's wife and the adventures of Lucky Pierre

This doctoral dissertation analyzes two novels by the American novelist Robert Coover as examples of hypertextual writing on the book bound page, as tokens of hyperfiction. The complexity displayed in the novels, John's Wife and The Adventures of Lucky Pierre, integrates the cultural elements that characterize the contemporary condition of capitalism and technologized practices that have fostered a different subjectivity evidenced in hypertextual writing and reading, the posthuman subjectivity. The models that account for the complexity of each novel are drawn from the concept of strange attractors in Chaos Theory and from the concept of rhizome in Nomadology. The transformations the characters undergo in the degree of their corporeality sets the plane on which to discuss turbulence and posthumanity. The notions of dynamic patterns and strange attractors, along with the concept of the Body without Organs and Rhizome are interpreted, leading to the revision of narratology and to analytical categories appropriate to the study of the novels. The reading exercised throughout this dissertation enacts Daniel Punday's corporeal reading. The changes in the characters' degree of materiality are associated with the stages of order, turbulence and chaos in the story, bearing on the constitution of subjectivity within and along the reading process. Coover's inscription of planes of consistency to counter linearity and accommodate hypertextual features to the paper supported narratives describes the characters' trajectory as rhizomatic. The study led to the conclusion that narrative today stands more as a regime in a rhizomatic relation with other regimes in cultural practice than as an exclusively literary form and genre. Besides this, posthuman subjectivity emerges as class identity, holding hypertextual novels as their literary form of choice.