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.

The metabolic and cognitive energy costs of stabilising a high-energy interlimb coordination task

Kinematic (relative phase error), metabolic (oxygen consumption, heart rate) and attentional (baseline and cycling reaction times) variables were measured while participants practised a high energy-demanding, intrinsically unstable 90° relative phase coordination pattern on independent bicycle ergometers. The variables were found to be strongly inter-correlated, suggesting a link between emerging performance stability with practice and minimal metabolic and attentional cost. The effects of practice of 90° relative phase coordination on the performance of in-phase (0°-phase) and antiphase (180°-phase) coordination were investigated by measuring the relative phase attractor layouts and recording the metabolic and attentional cost of the three coordination patterns before and after practice. The attentional variables did not differ significantly between coordination patterns and did not change with practice. Before practice, the coordination performance was most accurate and stable for in-phase cycling, with antiphase next and 90°-phase the poorest. However, metabolic cost was lower for antiphase than either in-phase or 90°-phase cycling, and the pre-practice attractor layout deviated from that predicted on the basis of dynamic stability as an attractor state, revealing an attraction to antiphase cycling. After practice of 90°-phase cycling, in-phase cycling remained the most accurate and stable, with 90°-phase next and antiphase the poorest, but antiphase retained the lowest metabolic energy cost. The attractor layout had changed, with new attractors formed at the practised 90°-phase pattern and its symmetrical partner of 270°-phase. Considering both the pre- and post-practice results, attractors were formed at either a low metabolic energy cost but less stable (antiphase) pattern or at a more stable but higher metabolic energy cost (90°-phase) pattern, but in neither case at the most stable and accurate (in-phase) pattern. The results suggest that energetic factors affect coordination dynamics and that coordination modes lower in metabolic energy expenditure may compete with dynamically stable modes.

Experiential dimensions in internet marketing : an exploratory investigation

This paper examines the dimensions of an experience in product marketing conceptualised by Pine II and Gilmore (1998) – customer participation and customer connection – by extending the study of the dimensions to the online context. In online marketing, the business aim is to hold attention, have visitors more deeply penetrate a Website, purchase, and return to the site, among other objectives. The paper analyses and synthesises findings from a three-part study of Internet use, the WebQUAL Audit, and presents the proposition that the dimensions suggested by Pine II and Gilmore as attractors to many experiential offerings in the physical world, may not be applicable in the online environment populated by commercial Websites. The paper also suggests a future research agenda to reconcile the requirements of users and the perspective of Web designers and other contributors to commercial Websites as discussed in the literature review presented.

Moving beyond resistance to restraint minimization : a case study of change management in aged care

Aim: This case study describes a quality initiative to minimize restraint in an Australian residential aged care facility.

Approach: The process of improving practice is examined with reference to the literature on implementation of research into practice and change management. The differences between planned and emergent approaches to change management are discussed. The concepts of resistance and attractors are explored in relation to our experiences of managing the change process in this initiative. The importance of the interpersonal interactions that were involved in facilitating the change process is highlighted.

Implications: Recommendations are offered for dealing with change management processes in clinical environments, particularly the need to move beyond an individual mind-set to a systems-based approach for quality initiatives in residential aged care.

Energetics of interlimb coordination

While the traditional dependent variables of motor skill learning are accuracy and consistency of movement outcome, there has been increasing interest in aspects of motor performance that are described as reflecting the ‘energetics’ of motor behaviour. One defining characteristic of skilled motor performance is the ability to complete the task with minimum energy expenditure (Sparrow & Newell, 1998). A further consideration is that movements also have costs in terms of cognitive ‘effort’ or ‘energy’. The present project extends previous work on energy expenditure and motor skill learning within a coordination dynamics framework. From the dynamic pattern perspective, a coordination pattern lowest on the 11KB model potential curve (Haken, Kelso & Bunz, 1985) is more stable and least energy is required to maintain pattern stability (Temprado, Zanone, Monno & Laurent, 1999). Two experiments investigated the learning of stable and unstable coordination patterns with high metabolic energy demand. An experimental task was devised by positioning two cycle ergometers side-by-side, placing one foot on each, with the pedals free to move independently at any metronome-paced relative phase, Experiment 1 investigated practice-related changes to oxygen consumption, heart rate, relative phase, reaction time and muscle activation (EMG) as participants practiced anti-phase, in-phase and 90°-phase cycling. Across six practice trials metabolic energy cost reduced and AE and VE of relative phase declined. The trend in the metabolic and reaction time data and percent co-contraction of muscles was for the in-phase cycling to demonstrate the highest values, anti-phase the lowest and 90°-phase cycling in-between. It was found that anti- and in-phase cycling were both kinematically stable but anti-phase coordination revealed significantly lower metabolic energy cost. It was, therefore, postulated that of two equally stable coordination patterns, that associated with lower metabolic energy expenditure would constitute a stronger attractor. Experiment 2 was designed to determine whether a lower or higher energy-demanding coordination pattern was a stronger attractor by scanning the attractor layout at thirty-degree intervals from 0° to 330°. The initial attractor layout revealed that in-phase was most stable and accurate, but the remaining coordination patterns were attracted to the low energy cost anti-phase cycling. In Experiment 2 only 90°- phase cycling was practiced with a post-test attractor layout scan revealing that 90°-phase and its symmetrical partner 270°-phase had become attractors of other coordination patterns. Consistent with Experiment 1, practicing 90°-phase cycling revealed a decline in AE and VE and a reduction in metabolic and cognitive cost. Practicing 90°-phase cycling did not, however, destabilise the in-phase or anti-phase coordination patterns either kinematically or energetically. In summary, the findings suggest that metabolic and mental energy can be considered different representations of a ‘global’ energy expenditure or ‘energetic’ phenomenon underlying human coordination. The hypothesis that preferred coordination patterns emerge as stable, low-energy solutions to the problem of inter-and intra-limb coordination is supported here in showing that the low-energy minimum of coordination dynamics is also an energetic minimum.

Chaos rules : an exploration of the work of instructional designers in distance education

This thesis provides an examination of the work of instructional designers in distance education, through the conceptual lens of chaos theory. Chaos theory was chosen as an analytical tool because of its ability to reveal the patterns and processes of complex systems as they move between order and turbulence. Recent work in the social sciences, specifically literary theory, has provided impetus for applications of chaos theory to educational settings. Specifically, chaos theory is used to analyse eight case studies of projects volunteered by instructional designers working in five institutions in Hong Kong and Australia. Data were gathered over a period of months with each participant, chiefly through interviews, but also involving diary accounts, electronic mail and letters. The methodology was thus qualitative, specifically informed by Eisner's vision of the ‘critical connoisseur’. Eisner equates an ‘enlightened eye’ with attainment of the skills of a critical connoisseur. First, an effective qualitative researcher must develop connoisseurship, the art of appreciation. On its own, though, connoisseurship is not enough; it is a private act, and thus needs a public face or presence. Criticism is this link, criticism being the art of disclosure. The critical connoisseur aims to help others to increase perception and deepen understanding of an educational situation or event. In addition to the empirical work, a parallel strand of this thesis investigates the theory and reported practice of instructional design. A brief history of instructional design is presented, along with discussion of acknowledged deficiencies of current theory and approaches. Recent reported investigations of both theory and practice are analysed from the viewpoint of chaos theory. Examination of key contributions in the literature of instructional design and distance education reveals considerable resonance between these contributions and the fundamental properties of chaotic systems. Links are made, in both the theoretical and empirical strands, between instructional design and the behaviour of dissipative structures, attractors and the process of bifurcation. Use is also made of the time-dependent nature of chaos theory as a theory of becoming, rather than one of being. The thesis comprises eight chapters, two appendices and a references section. The introductory chapter explains the research problem, and outlines the structure of the thesis. Methodological considerations are left until after an assessment of instructional design literature and (reported) practice. This deliberately theoretical investigation (Chapters 2 and 3) comprises the first of the parallel strands that are presented. The basic conclusions are that instructional design theory has not been particularly helpful to or used by instructional designers, and that chaos theory might provide an alternative way of viewing instructional design practice. The other parallel strand is the empirical work, which for four chapters outlines the methodology and my findings concerning the role of instructional designers in distance education. The methodology is detailed in Chapter 4. Chapter 5 establishes the contexts of the participants, by examining their backgrounds and introductions to their roles. It also investigates their views on their role and status within their institutions and with working colleagues. Chapter 6 is an exploration of the major issues that influenced the work of the instructional designers. These are the issues that arose naturally in the interviews as the participants outlined the development and interactions that took place on a day to day basis. Time emerges as a key influence in their work, and its effects on the projects are outlined and analysed. The ways that instructional designers give advice to those with whom they work is also investigated. The next chapter continues consideration of their work, but this time as they reflect on their role and its demands. This includes their reactions to the various metaphors that have appeared in the literature, along with those that they introduced into our discussions. The links that are established between the two parallel strands are drawn more explicitly in the final chapter, Chapter 8, which is a notion of what a model of instructional design based on my conclusions might resemble. It summarises the evidence that it is not necessarily by striving for order—in fact quite the opposite — during key periods of course development, that leads to creative outcomes. The introduction of uncertainty and turbulence does, in some cases and under some conditions, move the system to a higher level. The image that is offered from chaos theory is that of time-bound dissipative structures, interacting with their open environment at far-from-equilibrium conditions, and transforming themselves from disorder to order through bifurcation. The role of strange or chaotic attractors is highlighted in the process. The first appendix gives background information in terms of the methodology. The second is the heart of the data upon which the thesis draws. That is, the second appendix outlines the case studies of the participants. Most are short summaries, but the final one is a detailed study, tracing the progress of the design and development of a subject in distance education.

Remix the social: remix the learning

This paper reports on how the findings from an eighteen month secondary school action research study, in which social media was integrated into face-to-face classroom practice, was used to inform a fourth year undergraduate teacher-education unit at Deakin University in Australia. The school action research study was conducted in an Australian public secondary school. Students were aged between 13 and 16 years of age and a total of thirteen classes were involved. In each of the three semesters of data collection, one online social network was shared with up to seven classes and each class had approximately 25 students. Blogs, Groups, Chats, Discussion Forums, Web 2.0 tools and a wide range of student-generated content were shared online, within a class and between classes. Students were encouraged to interact and to share their thoughts and ideas about planning as well as using their out-of-school skills and knowledge. Each topic, within each class, was one action research cycle, using Armstrong and Moore’s (2004) framework. By following Graham Nuthall’s lens on learning, the researcher was able to focus on teaching as being about sensitivity and adaptation: adjusting to the here-and-now circumstances of particular students (Nuthall 2007). Elements of self organisation with spontaneous and strange attractors were identified throughout the study and these made links to Doll’s (1993) post-modern perspective of chaotic behaviour and the complexity of Hayles’ (1990) ‘disorderly order’.