Towards a general theory of extremes for observables of chaotic dynamical systems

In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the cho- sen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan–Yorke dimension of the attractor. Preliminary numer- ical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.

Phase equilibria study of systems composed of refined babassu oil, lauric acid, ethanol, and water at 303.2 K

Deacidification of vegetable oils can be performed using liquid-liquid extraction as an alternative method to the classical chemical and physical refining processes. This paper reports experimental data for systems containing refined babassu oil, lauric acid, ethanol, and water at 303.2 K with different water mass fractions in the alcoholic solvent (0, 0.0557, 0.1045, 0.2029, and 0.2972). The dilution of solvent with water reduced the distribution coefficient values, which indicates a reduction in the loss of neutral oil. The experimental data were used to adjust the NRTL equation parameters. The global deviation between the observed and the estimated compositions was 0.0085, indicating that the model can accurately predict the behavior of the compounds at different levels of solvent hydration. (C) 2011 Elsevier Ltd. All rights reserved.

Semiclassical evolution of dissipative Markovian systems

A semiclassical approximation for an evolving density operator, driven by a `closed` Hamiltonian operator and `open` Markovian Lindblad operators, is obtained. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the Hamiltonian operator is a quadratic function and the Lindblad operators are linear functions of positions and momenta. Initially, the semiclassical formulae for the case of Hermitian Lindblad operators are reinterpreted in terms of a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra `open` term is added to the double Hamiltonian by the non-Hermitian part of the Lindblad operators in the general case of dissipative Markovian evolution. The particular case of generic Hamiltonian operators, but linear dissipative Lindblad operators, is studied in more detail. A Liouville-type equivariance still holds for the corresponding classical evolution in double phase space, but the centre subspace, which supports the Wigner function, is compressed, along with expansion of its conjugate subspace, which supports the chord function. Decoherence narrows the relevant region of double phase space to the neighbourhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by a propagator in a mixed representation, so that a further `small-chord` approximation leads to a simple generalization of the quadratic theory for evolving Wigner functions.

UNIFORM DISSIPATIVENESS, ROBUST SYNCHRONIZATION AND LOCATION OF THE ATTRACTOR OF PARAMETRIZED NONAUTONOMOUS DISCRETE SYSTEMS

In this series of papers, we study issues related to the synchronization of two coupled chaotic discrete systems arising from secured communication. The first part deals with uniform dissipativeness with respect to parameter variation via the Liapunov direct method. We obtain uniform estimates of the global attractor for a general discrete nonautonomous system, that yields a uniform invariance principle in the autonomous case. The Liapunov function is allowed to have positive derivative along solutions of the system inside a bounded set, and this reduces substantially the difficulty of constructing a Liapunov function for a given system. In particular, we develop an approach that incorporates the classical Lagrange multiplier into the Liapunov function method to naturally extend those Liapunov functions from continuous dynamical system to their discretizations, so that the corresponding uniform dispativeness results are valid when the step size of the discretization is small. Applications to the discretized Lorenz system and the discretization of a time-periodic chaotic system are given to illustrate the general results. We also show how to obtain uniform estimation of attractors for parametrized linear stable systems with nonlinear perturbation.

Birkhoffian Systems in Infinite Dimensional Manifolds

We generalize the theory of Kobayashi and Oliva (On the Birkhoff Approach to Classical Mechanics. Resenhas do Instituto de Matematica e Estatistica da Universidade de Sao Paulo, 2003) to infinite dimensional Banach manifolds with a view towards applications in partial differential equations.

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.

Spatial discretization of discrete dynamical systems

Investigates what can go wrong when dynamical systems are modelled with a computer. Number theoretic techniques were used to detail the effects "discretization" errors caused by computer round-off had on characteristics of a system. In particular, a relationship was established between the occurrence of long cycles in a system and the classical result known as Artin's conjecture. Algorithms were then developed which eliminated discretization errors.

What is so ‘primitive’ about ‘primitive democracy’? Comparing the ancient Middle East and classical Athens

This chapter seeks to delve deeper into the ancient history of democracy than is normally permitted, back to a time preceding the developments of classical Athens, when the earliest signs of organized society and complex governmental systems emerged across the ancient Middle East. It then seeks to compare and contrast these ancient Middle Eastern examples with those of classical Athens and to offer new insights into, and questions about, the nature and history of democracy. Building on some recent work (Fleming, 2004; Isakhan, 2007a; Keane, 2009: 78–155), this chapter also hopes to move the discussion beyond the phrase usually associated with ancient Middle Eastern democracies, that of ‘primitive democracy’. This chapter also argues that, while the Middle Eastern experiments were less rigid and formalized, they were in no measurable sense more ‘primitive’ than the later example offered by classical Athens. However, this essay also cautiously notes that, while not all of the elements which made ancient Athens significant occurred in the same way and at the same time in the ancient Middle East, all of them did exist at varying times and in varying guises across these earlier civilizations. To demonstrate this thesis, the remainder of the chapter utilizes several of the key criteria by which we commonly measure Athenian democracy – the functioning of its assembly, the mechanisms of justice and of the law, the varying voting and elective procedures, the rights and freedoms of the citizens, and the systematic exclusion of ‘non-citizens’ – and discusses precedents and parallels drawn from the extant evidence concerning the ancient Middle East.

A Framework supporting collaboration on the distributed design of integrated systems

The work described in this thesis aims to support the distributed design of integrated systems and considers specifically the need for collaborative interaction among designers. Particular emphasis was given to issues which were only marginally considered in previous approaches, such as the abstraction of the distribution of design automation resources over the network, the possibility of both synchronous and asynchronous interaction among designers and the support for extensible design data models. Such issues demand a rather complex software infrastructure, as possible solutions must encompass a wide range of software modules: from user interfaces to middleware to databases. To build such structure, several engineering techniques were employed and some original solutions were devised. The core of the proposed solution is based in the joint application of two homonymic technologies: CAD Frameworks and object-oriented frameworks. The former concept was coined in the late 80's within the electronic design automation community and comprehends a layered software environment which aims to support CAD tool developers, CAD administrators/integrators and designers. The latter, developed during the last decade by the software engineering community, is a software architecture model to build extensible and reusable object-oriented software subsystems. In this work, we proposed to create an object-oriented framework which includes extensible sets of design data primitives and design tool building blocks. Such object-oriented framework is included within a CAD Framework, where it plays important roles on typical CAD Framework services such as design data representation and management, versioning, user interfaces, design management and tool integration. The implemented CAD Framework - named Cave2 - followed the classical layered architecture presented by Barnes, Harrison, Newton and Spickelmier, but the possibilities granted by the use of the object-oriented framework foundations allowed a series of improvements which were not available in previous approaches: - object-oriented frameworks are extensible by design, thus this should be also true regarding the implemented sets of design data primitives and design tool building blocks. This means that both the design representation model and the software modules dealing with it can be upgraded or adapted to a particular design methodology, and that such extensions and adaptations will still inherit the architectural and functional aspects implemented in the object-oriented framework foundation; - the design semantics and the design visualization are both part of the object-oriented framework, but in clearly separated models. This allows for different visualization strategies for a given design data set, which gives collaborating parties the flexibility to choose individual visualization settings; - the control of the consistency between semantics and visualization - a particularly important issue in a design environment with multiple views of a single design - is also included in the foundations of the object-oriented framework. Such mechanism is generic enough to be also used by further extensions of the design data model, as it is based on the inversion of control between view and semantics. The view receives the user input and propagates such event to the semantic model, which evaluates if a state change is possible. If positive, it triggers the change of state of both semantics and view. Our approach took advantage of such inversion of control and included an layer between semantics and view to take into account the possibility of multi-view consistency; - to optimize the consistency control mechanism between views and semantics, we propose an event-based approach that captures each discrete interaction of a designer with his/her respective design views. The information about each interaction is encapsulated inside an event object, which may be propagated to the design semantics - and thus to other possible views - according to the consistency policy which is being used. Furthermore, the use of event pools allows for a late synchronization between view and semantics in case of unavailability of a network connection between them; - the use of proxy objects raised significantly the abstraction of the integration of design automation resources, as either remote or local tools and services are accessed through method calls in a local object. The connection to remote tools and services using a look-up protocol also abstracted completely the network location of such resources, allowing for resource addition and removal during runtime; - the implemented CAD Framework is completely based on Java technology, so it relies on the Java Virtual Machine as the layer which grants the independence between the CAD Framework and the operating system. All such improvements contributed to a higher abstraction on the distribution of design automation resources and also introduced a new paradigm for the remote interaction between designers. The resulting CAD Framework is able to support fine-grained collaboration based on events, so every single design update performed by a designer can be propagated to the rest of the design team regardless of their location in the distributed environment. This can increase the group awareness and allow a richer transfer of experiences among them, improving significantly the collaboration potential when compared to previously proposed file-based or record-based approaches. Three different case studies were conducted to validate the proposed approach, each one focusing one a subset of the contributions of this thesis. The first one uses the proxy-based resource distribution architecture to implement a prototyping platform using reconfigurable hardware modules. The second one extends the foundations of the implemented object-oriented framework to support interface-based design. Such extensions - design representation primitives and tool blocks - are used to implement a design entry tool named IBlaDe, which allows the collaborative creation of functional and structural models of integrated systems. The third case study regards the possibility of integration of multimedia metadata to the design data model. Such possibility is explored in the frame of an online educational and training platform.

H-infinity control design for time-delay linear systems: a rational transfer function based approach

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Identification of Mechanical Systems through Kautz Filter

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Colombeau's theory and shock wave solutions for systems of PDEs

In this article we study the existence of shock wave solutions for systems of partial differential equations of hydrodynamics with viscosity in one space dimension in the context of Colombeau's theory of generalized functions. This study uses the equality in the strict sense and the association of generalized functions (that is the weak equality). The shock wave solutions are given in terms of generalized functions that have the classical Heaviside step function as macroscopic aspect. This means that solutions are sought in the form of sequences of regularizations to the Heaviside function that have to satisfy part of the equations in the strict sense and part of the equations in the sense of association.

Electronic ballast for multiple fluorescent lamp systems

An electronic ballast for multiple tubular fluorescent lamp systems is presented. The proposed structure has a high value for the power factor, a dimming capability, and soft switching of the semiconductor devices operated at high frequencies. A zero-current switching pulse width modulated SEPIC converter is used as the rectifying stage and it is controlled using the instantaneous average input current technique. The inverting stage consists of classical resonant half-bridge converter with series-resonant parallel-loaded filters. The dimming control technique is based on varying the switching frequency and monitoring the phase shift of the current drained by the filters and lamps in order to establish a closed loop control. Experimental results are presented that validate the theoretical analysis.

Uniform estimates of attracting sets of extended Lurie systems using LMIs

This research presents a systematic procedure to obtain estimates, via extended Lyapunov functions, of attracting sets of a class of nonlinear systems, as well as an estimate of their stability regions. The considered class of nonlinear systems, called in this note the extended Lurie system, consists of nonlinear systems like those of the Lurie problem where one of the nonlinear functions can violate the sector conditions of the Lurie problem around the origin. In case of nonautonomous systems the concept of absolute stability is extended and uniform estimates of the attracting set are obtained. Two classical nonlinear systems, the forced duffing equation and the Van der Pol system, are analyzed with the proposed procedure.

Necessary and sufficient condition for generalized passivity, passification and application to multivariable adaptive systems

Recently, a generalized passivity concept for linear multivariable systems was obtained which allows circumventing the restrictiveness of the usual passivity concept. The latter is associated with the classical SPR (Strictly Positive Real) condition whereas the new concept of passivity is associated with the so called WSPR condition and its advantage in multivariable systems is that it does not require a restrictive symmetry condition of SPR systems. As a result, it allows the design of multivariable adaptive control that, unlike some existing factorization approaches, does not imply in additional overparameterization of the adaptive controller. In this paper, we complete a previously established WSPR sufficient condition and prove that it is also necessary. We also propose some methods of passification by either premultiplying the system output tracking error vector or the system input vector by an adequate passifying matrix multiplier, so that the resulting input/output transfer function becomes WSPR. The efficiency of our proposals are illustrated by simulation utilizing a well known robotics adaptive visual servoing problem. © 2011 IFAC.