Nowadays,theresearchersintoControlTheoryanditsApplicationsaswellas Matrix Analysisare well awareand recognizethe importanceof PositiveS- tems. This volume contains the proceedings of the "Third Multidisciplinary Symposium onPositiveSystems:Theory andApplications (POSTA09)" held in Valencia, Spain, September 2-4, 2009. At present, this is the only world congress whose main topic is focused on this ?eld. After this third event, we thinkthatwehaveestablishedthebasisofaregulartriennialeventsupported in the task of the previous organizing committees and hope to have met the requirements of their expectations. This POSTA09 meeting has been organized by members of the "Univ- sitat Polit' ecnica de Val' encia", who have taken their doctor's degree in the subjects of the conference. We are grateful to all of them for their ent- siasm, e?ort and especially their collaboration, without the help of which the outcome would have never been the same. Also, we highly appreciate Christian Commault for having his experience at our disposal. Besides that, we wish to thank the International Programme Committee components and additional referees for their outstanding work in the review process of the contributions. We are happy to say that their constructive suggestions and positive comments have improved the quality of the presentations. We are very much obliged to the following organizations for having- nancially backed this congress: "Ministerio de Educaci' on y Ciencia", "C- selleriad'Educaci' odelaGeneralitatValenciana","UniversitatPolit' ecnicade Val' encia(UPV)"and"SociedadEspano " ladeMatem' aticaAplicada(SEMA)".

Nature is inherently noisy and nonlinear. It is noisy in the sense that all macroscopic systems are subject to the fluctuations of their environments and also to internal fluctuations. It is nonlinear in the sense that the restoring force on a system displaced from equilibrium does not usually vary linearly with the size of the displacement. To calculate the properties of stochastic (noisy) nonlinear systems is in general extremely difficult, although considerable progress has been made in the past. The three volumes that make up Noise in Nonlinear Dynamical Systems comprise a collection of specially written authoritative reviews on all aspects of the subject, representative of all the major practitioners in the field. The first volume deals with the basic theory of stochastic nonlinear systems. It includes an historical overview of the origins of the field, chapters covering some developed theoretical techniques for the study of coloured noise, and the first English-language translation of the landmark 1933 paper by Pontriagin, Andronov and Vitt.

Quantum Chaos provides a comprehensive overview of our understanding of chaotic behaviour in a wide variety of quantum and semiclassical systems, and describes both experimental and theoretical investigations. A general introduction sets out the main features of chaos in quantum systems. Thereafter, in an authoritative collection of papers, prominent scientists put forward their particular interpretations of quantum chaos, with reference to a broad range of interesting physical systems. As yet, there is no universally accepted definition of quantum chaos. However, by dealing with such a wide range of topics from different branches of physics, this book provides an overview of this rapidly expanding field, and will be of great interest to graduate students and researchers in many areas of physics and chemistry.

This monograph, "Non-linear Cooperative Effects in Open Quantum Systems: Entanglement and Second Order Coherence" is dedicated to the large auditory of specialists interested in the modern approaches in quantum open systems, cooperative phenomena between excited atoms and the field of the non-linear interaction. Special attention is dedicated to the problems of non-linear interaction with vacuum fields and thermostat with finite temperature, but quantum aspects of laser generation of light in non-linear interaction with finite numbers of cavity modes remain the center of attention. In many situations, the limit to the traditional cooperative phenomena of open quantum systems and thermodynamics are taken into consideration. As the book contains the class of non-linear effects of generations of the particle in such cooperative phenomena, the author's aim was to describe squeezed problems and affect entanglement between the generation photons and phonons in cooperative processes. The new phenomenon of cooperative emission in the single- and two-quantum processes are carefully described for large audiences of specialists in the field of quantum optics and condensed matter physics, chemistry and biology.

Mixing processes occur in many technological and natural applications, with length and time scales ranging from the very small to the very large. The diversity of problems can give rise to a diversity of approaches. Are there concepts that are central to all of them? Are there tools that allow for prediction and quantification? The authors show how a variety of flows in very different settings possess the characteristic of streamline crossing. This notion can be placed on firm mathematical footing via Linked Twist Maps (LTMs), which is the central organizing principle of this book. The authors discuss the definition and construction of LTMs, provide examples of specific mixers that can be analyzed in the LTM framework and introduce a number of mathematical techniques which are then brought to bear on the problem of fluid mixing. In a final chapter, they present a number of open problems and new directions.

This book, following the three published volumes of the book, provides the main purpose to collect research papers and review papers to provide an overview of the main issues, results, and open questions in the cutting-edge research on the fields of modeling, optimization, and dynamics and their applications to biology, economy, energy, industry, physics, psychology and finance. Assuming the scientific relevance of the presenting innovative applications as well as merging issues in these areas, the purpose of this book is to collect papers of the world experts in mathematics, economics, and other applied sciences that is seminal to the future research developments. The majority of the papers presented in this book is authored by the participants in The Joint Meeting 6th International Conference on Dynamics, Games, and Science - DGSVI - JOLATE and in the 21st ICABR Conference. The scientific scope of the conferences is focused on the fields of modeling, optimization, and dynamics and their applications to biology, economy, energy, industry, physics, psychology, and finance. Assuming the scientific relevance of the presenting innovative applications as well as merging issues in these areas, the purpose of the conference is to bring together some of the world experts in mathematics, economics, and other applied sciences that reinforce ongoing projects and establish future works and collaborations.

This book contains a collection of survey papers by leading researchers in ergodic theory, low-dimensional and topological dynamics and it comprises nine chapters on a range of important topics. These include: the role and usefulness of ultrafilters in ergodic theory, topological dynamics and Ramsey theory; topological aspects of kneading theory together with an analogous 2-dimensional theory called pruning; the dynamics of Markov odometers, Bratteli-Vershik diagrams and orbit equivalence of non-singular automorphisms; geometric proofs of Mather's connecting and accelerating theorems; recent results in one dimensional smooth dynamics; periodic points of nonexpansive maps; arithmetic dynamics; the defect of factor maps; entropy theory for actions of countable amenable groups.

The main focus of this book is on the development of the theory of Graph Directed Markov Systems. This far-reaching generalization of the theory of conformal iterated systems can be applied in many situations, including the theory of dynamical systems. Dan Mauldin and Mariusz Urbanski include much of the necessary background material to increase the appeal of this book to graduate students as well as researchers. They also include an extensive list of references for further reading.

This book is about algebro-geometric solutions of completely integrable nonlinear partial differential equations in (1+1)-dimensions; also known as soliton equations. Explicitly treated integrable models include the KdV, AKNS, sine-Gordon, and Camassa-Holm hierarchies as well as the classical massive Thirring system. An extensive treatment of the class of algebro-geometric solutions in the stationary and time-dependent contexts is provided. The formalism presented includes trace formulas, Dubrovin-type initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses techniques from the theory of differential equations, spectral analysis, and elements of algebraic geometry (most notably, the theory of compact Riemann surfaces).

Systems as diverse as clocks, singing crickets, cardiac pacemakers, firing neurons and applauding audiences exhibit a tendency to operate in synchrony. These phenomena are universal and can be understood within a common framework based on modern nonlinear dynamics. The first half of this book describes synchronization without formulae, and is based on qualitative intuitive ideas. The main effects are illustrated with experimental examples and figures, and the historical development is also outlined. The second half of the book presents the main effects of synchronization in a rigorous and systematic manner, describing both classical results on synchronization of periodic oscillators, and recent developments in chaotic systems, large ensembles, and oscillatory media.

In the new edition of this classic textbook Ed Ott has added much new material and has significantly increased the number of homework problems. The most important change is the addition of a completely new chapter on control and synchronization of chaos. Other changes include new material on riddled basins of attraction, phase locking of globally coupled oscillators, fractal aspects of fluid advection by Lagrangian chaotic flows, magnetic dynamos, and strange nonchaotic attractors.

This textbook on the theory of nonlinear dynamical systems for nonmathematical advanced undergraduate or graduate students is also a reference book for researchers in the physical and social sciences. It provides a comprehensive introduction including linear systems, stability theory of nonlinear systems, bifurcation theory, chaotic dynamics. Discussion of the measure--theoretic approach to dynamical systems and the relation between deterministic systems and stochastic processes is featured. There are a hundred exercises and an associated website provides a software program, computer exercises and answers to selected book exercises.

This textbook on the theory of nonlinear dynamical systems for nonmathematical advanced undergraduate or graduate students is also a reference book for researchers in the physical and social sciences. It provides a comprehensive introduction including linear systems, stability theory of nonlinear systems, bifurcation theory, chaotic dynamics. Discussion of the measure--theoretic approach to dynamical systems and the relation between deterministic systems and stochastic processes is featured. There are a hundred exercises and an associated website provides a software program, computer exercises and answers to selected book exercises.

This book reviews topics on the areas of fixed point theory, convex and set-valued analysis, variational inequality and complementarity problem theory, non-linear ergodic theory, difference, differential and integral equations, control and optimisation theory, dynamic system theory, inequality theory, stochastic analysis and probability theory, and their applications.

The physics and mathematics of nonlinear dynamics and chaotic and complex systems constitute some of the most fascinating developments of late twentieth-century science. It turns out that chaotic behaviour can be understood, and even utilized, to a far greater degree than had been suspected. Surprisingly, universal constants have been discovered. The implications have changed our understanding of important phenomena in physics, biology, chemistry, economics, medicine and numerous other fields of human endeavour. In this book, two dozen scientists and mathematicians who were deeply involved in the 'nonlinear revolution' cover most of the basic aspects of the field. The book is divided into five parts: dynamical systems, bifurcation theory and chaos; spatially extended systems; dynamical chaos, quantum physics and the foundations of statistical mechanics; evolutionary and cognitive systems; and complex systems as an interface between the sciences.

The papers in this volume address current topics of research in nonlinear mathematics, including nonlinear dynamics with application to fluid mechanics, boundary layer transition, driven oscillators and waves. There are also papers on problems in nonlinear elasticity and mathematical biology. The book forms a coherent and accessible account of recent advances in nonlinear mathematics for students in applied mathematics, physics, and engineering.

This book represents a comprehensive overview of our present understanding of chaotic behavior in a wide variety of quantum and semiclassical systems, and describes both experimental and theoretical investigations. A general introduction sets out the main features of chaos in quantum systems. Thereafter, in an authoritative collection of new or previously published papers, prominent scientists put forward their particular interpretations of quantum chaos with reference to a broad range of interesting physical systems.

One of the most unexpected results in science in recent years is that quite ordinary systems obeying simple laws can give rise to complex, nonlinear or chaotic, behavior. In this book, the author presents a unified treatment of the concepts and tools needed to analyze nonlinear phenomena and to outline some representative applications drawn from the physical, engineering, and biological sciences. Some of the interesting topics covered include: dynamical systems with a finite number of degrees of freedom, linear stability analysis of fixed points, nonlinear behavior of fixed points, bifurcation analysis, spatially distributed systems, broken symmetries, pattern formation, and chaotic dynamics. The author makes a special effort to provide a logical connection between ordinary dynamical systems and spatially extended systems, and to balance the emphasis on chaotic behavior and more classical nonlinear behavior. He also develops a statistical approach to complex systems and compares it to traditional deterministic phase space descriptions. This book is suitable for senior undergraduate and graduate students taking nonlinear courses from many different perspectives including physics, chemistry, biology, and engineering.

A coherent treatment of nonlinear systems covering chaos, fractals, and bifurcation, as well as equilibrium, stability, and nonlinear oscillations. The systems treated are mostly of difference and differential equations. The author introduces the mathematical properties of nonlinear systems as an integrated theory, rather than simply presenting isolated fashionable topics. The topics are discussed in as concrete a way as possible, worked examples and problems are used to motivate and illustrate the general principles. More advanced parts of the text are denoted by asterisks, thus making it ideally suited to both undergraduate and graduate courses.

The ""infinite-dimensional groups"" in the title refer to unitary groups of Hilbert spaces, the infinite symmetric group, groups of homeomorphisms of manifolds, groups of transformations of measure spaces, etc. The book presents an approach to the study of such groups based on ideas from geometric functional analysis and from exploring the interplay between dynamical properties of those groups, combinatorial Ramsey-type theorems, and the phenomenon of concentration of measure. The dynamics of infinite-dimensional groups is very much unlike that of locally compact groups. For instance, every locally compact group acts freely on a suitable compact space (Veech). By contrast, a 1983 result by Gromov and Milman states that whenever the unitary group of a separable Hilbert space continuously acts on a compact space, it has a common fixed point. In the book, this new fast-growing theory is built strictly from well-understood examples up. The book has no close counterpart and is based on recent research articles. At the same time, it is organized so as to be reasonably self-contained. The topic is essentially interdisciplinary and will be of interest to mathematicians working in geometric functional analysis, topological and ergodic dynamics, Ramsey theory, logic and descriptive set theory, representation theory, topological groups, and operator algebras.

This work is aimed at mathematics and engineering graduate students and researchers in the areas of optimization, dynamical systems, control sys tems, signal processing, and linear algebra. The motivation for the results developed here arises from advanced engineering applications and the emer gence of highly parallel computing machines for tackling such applications. The problems solved are those of linear algebra and linear systems the ory, and include such topics as diagonalizing a symmetric matrix, singular value decomposition, balanced realizations, linear programming, sensitivity minimization, and eigenvalue assignment by feedback control. The tools are those, not only of linear algebra and systems theory, but also of differential geometry. The problems are solved via dynamical sys tems implementation, either in continuous time or discrete time, which is ideally suited to distributed parallel processing. The problems tackled are indirectly or directly concerned with dynamical systems themselves, so there is feedback in that dynamical systems are used to understand and optimize dynamical systems. One key to the new research results has been the recent discovery of rather deep existence and uniqueness results for the solution of certain matrix least squares optimization problems in geomet ric invariant theory. These problems, as well as many other optimization problems arising in linear algebra and systems theory, do not always admit solutions which can be found by algebraic methods."

This book is dedicated to fundamentals of a new theory, which is an analog of affine algebraic geometry for (nonlinear) partial differential equations. This theory grew up from the classical geometry of PDE's originated by S. Lie and his followers by incorporating some nonclassical ideas from the theory of integrable systems, the formal theory of PDE's in its modern cohomological form given by D. Spencer and H. Goldschmidt and differential calculus over commutative algebras (Primary Calculus). The main result of this synthesis is Secondary Calculus on diffieties, new geometrical objects which are analogs of algebraic varieties in the context of (nonlinear) PDE's. Secondary Calculus surprisingly reveals a deep cohomological nature of the general theory of PDE's and indicates new directions of its further progress.Recent developments in quantum field theory showed Secondary Calculus to be its natural language, promising a nonperturbative formulation of the theory. In addition to PDE's themselves, the author describes existing and potential applications of Secondary Calculus ranging from algebraic geometry to field theory, classical and quantum, including areas such as characteristic classes, differential invariants, theory of geometric structures, variational calculus, control theory, etc. This book, focused mainly on theoretical aspects, forms a natural dipole with ""Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Volume 182"" in this same series, ""Translations of Mathematical Monographs"", and shows the theory 'in action'.

Linear-Quadratic Controls in Risk-Averse Decision Making cuts across control engineering (control feedback and decision optimization) and statistics (post-design performance analysis) with a common theme: reliability increase seen from the responsive angle of incorporating and engineering multi-level performance robustness beyond the long-run average performance into control feedback design and decision making and complex dynamic systems from the start. This monograph provides a complete description of statistical optimal control (also known as cost-cumulant control) theory. In control problems and topics, emphasis is primarily placed on major developments attained and explicit connections between mathematical statistics of performance appraisals and decision and control optimization. Chapter summaries shed light on the relevance of developed results, which makes this monograph suitable for graduate-level lectures in applied mathematics and electrical engineering with systems-theoretic concentration, elective study or a reference for interested readers, researchers, and graduate students who are interested in theoretical constructs and design principles for stochastic controlled systems. "

Covering one of the fastest growing areas of applied mathematics, Nonlinear Dynamics and Chaos: Second Edition, is a fully updated edition of this highly respected text. Covering a breadth of topics, ranging from the basic concepts to applications in the physical sciences, the book is highly illustrated and written in a clear and comprehensible style.

• Provides a self-contained introduction to the theory and applications of nonlinear dynamics and chaos.

• Introduces the concepts of instabilities, bifurcations, and catastrophes.

• Each idea is carefully explained and supported by examples.

• Features many applications to a wide variety of scientific fields.

• Includes an illustrated glossary of geometrical dynamics.

• Features a supplementary bibliography of further reading.

• Assumes minimal background knowledge.

The book presents a systematic and unified study of geometric nonlinear functional analysis. This area has its classical roots in the beginning of the twentieth century and is now a very active research area, having close connections to geometric measure theory, probability, classical analysis, combinatorics, and Banach space theory. The main theme of the book is the study of uniformly continuous and Lipschitz functions between Banach spaces (e.g., differentiability, stability, approximation, existence of extensions, fixed points, etc.). This study leads naturally also to the classification of Banach spaces and of their important subsets (mainly spheres) in the uniform and Lipschitz categories. Many recent rather deep theorems and delicate examples are included with complete and detailed proofs. Challenging open problems are described and explained, and promising new research directions are indicated.