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.

Rock slope stability analyses using extreme learning neural network and terminal steepest descent algorithm

The analysis of rock slope stability is a classical problem for geotechnical engineers. However, for practicing engineers, proper software is not usually user friendly, and additional resources capable of providing information useful for decision-making are required. This study developed a convenient tool that can provide a prompt assessment of rock slope stability. A nonlinear input-output mapping of the rock slope system was constructed using a neural network trained by an extreme learning algorithm. The training data was obtained by using finite element upper and lower bound limit analysis methods. The newly developed techniques in this study can either estimate the factor of safety for a rock slope or obtain the implicit parameters through back analyses. Back analysis parameter identification was performed using a terminal steepest descent algorithm based on the finite-time stability theory. This algorithm not only guarantees finite-time error convergence but also achieves exact zero convergence, unlike the conventional steepest descent algorithm in which the training error never reaches zero.

Dynamical analysis of neural networks with time-varying delays using the LMI approach

This study is concerned with the delay-range-dependent stability analysis for neural networks with time-varying delay and Markovian jumping parameters. The time-varying delay is assumed to lie in an interval of lower and upper bounds. The Markovian jumping parameters are introduced in delayed neural networks, which are modeled in a continuous-time along with finite-state Markov chain. Moreover, the sufficient condition is derived in terms of linear matrix inequalities based on appropriate Lyapunov-Krasovskii functionals and stochastic stability theory, which guarantees the globally asymptotic stable condition in the mean square. Finally, a numerical example is provided to validate the effectiveness of the proposed conditions.

Young women with diabetes: using internet communication to create stability during life transitions

Aims and objectives. The aim of the current study was to explore and describe the strategies young women with Type 1 diabetes used to manage transitions in their lives. This paper will describe one aspect of the findings of how women with Type 1 diabetes used the Internet to interact with other people with diabetes and create stability in their lives.

Background. Individuals living with diabetes develop a range of different strategies to create stability in their lives and enhance their well-being. Changing social and emotional conditions during life transitions have a major impact on diabetes management. Although the literature indicates that strategies enabling the individuals to cope with transitions are important, they remain under-researched.

Design. Using grounded theory, interviews were conducted with 20 women with Type 1 diabetes. Constant comparative data analysis was used to analyse the data and develop an understanding of how young women with Type 1 diabetes used the Internet to create stability in their lives.

Findings. The findings revealed that the women valued their autonomy and being in control of when and to whom they reveal their diabetic status, especially during life transitions and at times of uncertainty. However, during these times they also required health and social information and interacting with other people. One of the women's main strategies in managing transitions was to use Internet chat lines as a way of obtaining information and communicating with others. This strategy gave women a sense of autonomy, enabled them to maintain their anonymity and interact with other people on their own terms.

Conclusions. Having meaningful personal interactions, social support and being able to connect with others were fundamental to the women's well being. Most importantly, preserving autonomy and anonymity during such interactions were integral to the way the women with Type 1 diabetes managed life transitions.

Relevance to clinical practice. Health professionals need to explore and incorporate Internet communication process or anonymous help lines into their practice as a way to assist people manage their diabetes.

Genre theory and language change in a community of practice

Business letters are complex sites of interaction. Discussions of genre theory from different disciplinary perspectives (psychology, applied linguistics, rhetoric) highlight tension between stability and change in writing as a social activity. The research extends the use of ethnographic methodology and Communities of Practice in examining writing and a writing community.

Structure and phase stability of NiMnGa ferromagnetic shape memory alloys by experimental and ab-initio techniques

This paper summarizes some of our recent results on crystal structure, microstructure, orientation relationship between martensitic variants and crystallographic features of martensitic transformation in Ni-Mn-Ga FSMAs. It was shown that Ni53Mn25Ga22 has a tetragonal I4/mmm martensitic structure at room temperature. The neighboring martensitic variants in Ni53Mn25Ga22 have a compound twinning relationship with the twinning elements K1={112}, K2={11-2}, η1=<11-1>, η2=<111>, P={1-10} and s=0.379. The ratio of the relative amounts of twins within the same initial austenite grain is ~1.70. The main orientation relationship between austenite and martensite is Kurdjumov-Sachs (K-S) relationship. Based on the crystallographic phenomenological theory, the calculated habit plane is {0.690 -0.102 0.716}A (5.95° from {101}A), and the magnitude, direction and shear angle of the macroscopic transformation shear are 0.121, <-0.709 0.105 0.698>A (6.04° from <-101>A) and 6.88°, respectively.

Transient stability enhancement of multimachine power systems using nonlinear observer-based excitation controller

This paper presents an approach to design a nonlinear observer-based excitation controller for multimachine power systems to enhance the transient stability. The controller is designed based on the partial feedback linearization of a nonlinear power system model which transforms the model into a reducedorder linear one with an autonomous dynamical part. Then a linear state feedback stabilizing controller is designed for the reduced-order linear power system model using optimal control theory which enhances the stability of the entire system. The states of the feedback stabilizing controller are obtained from the nonlinear observer and the performance of this observer-based controller is independent of the operating points of power systems. The performance of the proposed observer-based controller is compared to that of an exact feedback linearizing observer-based controller and a partial feedback linearizing controller without observer under different operating conditions.