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)".

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 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.

This book provides a complete understanding of chaotic dynamics in mathematics, physics, and the real world, with an explanation of why it is important and how it differs from the idea of randomness. The author draws on certain physical systems and phenomena, for example the weather forecast, a pendulumn, a coin toss, mass transit, politics, and the role of chaos in in gambling and the stock-market.

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. "

P. Habets: Stabilite asymptotique pour des problemes de perturbations singulieres.- J.K. Hale: Stability of linear systems with delays.- V. Lakshmikantham: Stability and asymptotic behaviour of solutions of differential equations in a Banach space.- P. Negrini: On a definition of total stability for continuous or discrete dynamical systems.-N. Rouche: Theorie de la stabilite dans les equations differentielles ordinaires.- E.O. Roxin: Stability and differential games.

This book presents a critical and modern analysis of the conceptual foundations of statistical mechanics as laid down in Boltzmann's works. The author emphasises the relation between microscopic reversibility and macroscopic irreversibility. Students will find a clear and detailed explanation of fundamental concepts such as equipartition, entropy and ergodicity. They will learn about Brownian motion, the modern treatment of the thermodynamic limit phase transitions, the microscopic and macroscopic theory of the coexistence of phases, statistical mechanics of stationary states, and fluctuations and dissipation in chaotic motions.

This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence.

Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits.

"Ergodic Theory with a view towards Number Theory" will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.

Establishing a new concept of local Lyapunov exponents the author brings together two separate theories, namely Lyapunov exponents and the theory of large deviations.

Specifically, a linear differential system is considered which is controlled by a stochastic process that during a suitable noise-intensity-dependent time is trapped near one of its so-called metastable states. The local Lyapunov exponent is then introduced as the exponential growth rate of the linear system on this time scale. Unlike classical Lyapunov exponents, which involve a limit as time increases to infinity in a fixed system, here the system itself changes as the noise intensity converges, too.

The present volume includes most of the material of the invited lectures delivered at the NATO Advanced Study Institute "Morphogenesis through the interplay of nonlinear chemical instabilities and elastic active media" held from 2th to 14th July 2007 at the Institut d'Etudes Scientifiques de Cargese (http: //www.iesc.univ-corse.fr/), in Corsica (France). This traditional place to organize Summer Schools and Workshops in a well equipped secluded location at the border of the Mediterranean sea has, over many years now, earned an increasing deserved reputation. Non-linear dynamics of non equilibrium systems has worked its way into a great number of fields and plays a key role in the understanding of se- organization and emergence phenomena in domains as diverse as chemical reactors, laser physics, fluid dynamics, electronic devices and biological morphogenesis. In the latter case, the viscoelastic properties of tissues are also known to play a key role. The control and formulation of soft responsive or "smart" materials has been a fast growing field of material science, specially in the area of po- mer networks, due to their growing applications in bio-science, chemical sensors, intelligent microfluidic devices, ... . Nature is an important p- vider of active materials whether at the level of tissues or at that of s- cellular structures. As a consequence, the fundamental understanding of the physical mechanisms at play in responsive materials also shines light in the understanding of biological artefacts."

This book is an often-requested reprint of two classic texts by H. Haken: "Synergetics. An Introduction" and "Advanced Synergetics." Synergetics, an interdisciplinary research program initiated by H. Haken in 1969, deals with the systematic and methodological approach to the rapidly growing field of complexity. Going well beyond qualitative analogies between complex systems in fields as diverse as physics, chemistry, biology, sociology and economics, Synergetics uses tools from theoretical physics and mathematics to construct an unifying framework within which quantitative descriptions of complex, self-organizing systems can be made. This may well explain the timelessness of H. Haken's original texts on this topic, which are now recognized as landmarks in the field of complex systems. They provide both the beginning graduate student and the seasoned researcher with solid knowledge of the basic concepts and mathematical tools. Moreover, they admirably convey the spirit of the pioneering work by the founder of Synergetics through the essential applications contained herein that have lost nothing of their paradigmatic character since they were conceived.

This text is an introduction to the spectral theory of the Laplacian on compact or finite area hyperbolic surfaces. For some of these surfaces, called "arithmetic hyperbolic surfaces", the eigenfunctions are of arithmetic nature, and one may use analytic tools as well as powerful methods in number theory to study them. After an introduction to the hyperbolic geometry of surfaces, with a special emphasis on those of arithmetic type, and then an introduction to spectral analytic methods on the Laplace operator on these surfaces, the author develops the analogy between geometry (closed geodesics) and arithmetic (prime numbers) in proving the Selberg trace formula. Along with important number theoretic applications, the author exhibits applications of these tools to the spectral statistics of the Laplacian and the quantum unique ergodicity property. The latter refers to the arithmetic quantum unique ergodicity theorem, recently proved by Elon Lindenstrauss. The fruit of several graduate level courses at Orsay and Jussieu, The Spectrum of Hyperbolic Surfaces allows the reader to review an array of classical results and then to be led towards very active areas in modern mathematics.

This book provides a challenging and stimulating introduction to the contemporary topics of complexity and criticality, and explores their common basis of scale invariance, a central unifying theme of the book.Criticality refers to the behaviour of extended systems at a phase transition where scale invariance prevails. The many constituent microscopic parts bring about macroscopic phenomena that cannot be understood by considering a single part alone. The phenomenology of phase transitions is introduced by considering percolation, a simple model with a purely geometrical phase transition, thus enabling the reader to become intuitively familiar with concepts such as scale invariance and renormalisation. The Ising model is then introduced, which captures a thermodynamic phase transition from a disordered to an ordered system as the temperature is lowered in zero external field. By emphasising analogies between percolation and the Ising model, the reader's intuition of phase transitions is developed so that the underlying theoretical formalism may be appreciated fully. These equilibrium systems undergo a phase transition only if an external agent finely tunes certain external parameters to particular values.Besides fractals and phase transitions, there are many examples in Nature of the emergence of such complex behaviour in slowly driven non-equilibrium systems: earthquakes in seismic systems, avalanches in granular media and rainfall in the atmosphere. A class of non-equilibrium systems, not constrained by having to tune external parameters to obtain critical behaviour, is addressed in the framework of simple models, revealing that the repeated application of simple rules may spontaneously give rise to emergent complex behaviour not encoded in the rules themselves. The common basis of complexity and criticality is identified and applied to a range of non-equilibrium systems. Finally, the reader is invited to speculate whether self-organisation in non-equilibrium systems might be a unifying concept for disparate fields such as statistical mechanics, geophysics and atmospheric physics.Visit for animations for the models in the book (available for Windows and Linux), solutions to exercises, as well as a list with corrections.

This book publishes select papers presented at the 4th International Conference on Frontiers in Industrial and Applied Mathematics (FIAM-2021), held at the Sant Longowal Institute of Engineering and Technology, Longowal, Punjab, India, from 21-22 December 2021. Most of the papers deal with mathematical theory embedded with its applications to engineering and sciences. This book illustrates numerical simulation of scientific problems and the state-of-the-art research in industrial and applied mathematics, including various computational and modeling techniques with case studies and concrete examples. Graduate students and researchers, who are interested in real applications of mathematics in the areas of computational and theoretical fluid dynamics, solid mechanics, optimization and operations research, numerical analysis, bio-mathematics, fuzzy, control and systems theory, dynamical systems and nonlinear analysis, algebra and approximation theory, will find the book useful.

Numerical minimization of an objective function is analyzed in this book to understand solution algorithms for optimization problems. Multiset-mappings are introduced to engineer numerical minimization as a repeated application of an iteration mapping. Ideas from numerical variational analysis are extended to define and explore notions of continuity and differentiability of multiset-mappings, and prove a fixed-point theorem for iteration mappings. Concepts from dynamical systems are utilized to develop notions of basin size and basin entropy. Simulations to estimate basins of attraction, to measure and classify basin size, and to compute basin are included to shed new light on convergence behavior in numerical minimization. Graduate students, researchers, and practitioners in optimization and mathematics who work theoretically to develop solution algorithms will find this book a useful resource.

This book is a remarkable contribution to the literature on dynamical systems and geometry. It consists of a selection of work in current research on Teichmuller dynamics, a field that has continued to develop rapidly in the past decades. After a comprehensive introduction, the author investigates the dynamics of the Teichmuller flow, presenting several self-contained chapters, each addressing a different aspect on the subject. The author includes innovative expositions, all the while solving open problems, constructing examples, and supplementing with illustrations. This book is a rare find in the field with its guidance and support for readers through the complex content of moduli spaces and Teichmuller Theory. The author is an internationally recognized expert in dynamical systems with a talent to explain topics that is rarely found in the field. He has created a text that would benefit specialists in, not only dynamical systems and geometry, but also Lie theory and number theory.

This book concentrates on the modern theory of dynamical systems and its interactions with number theory and combinatorics. The greater part begins with a course in analytic number theory and focuses on its links with ergodic theory, presenting an exhaustive account of recent research on Sarnak's conjecture on Moebius disjointness. Selected topics involving more traditional connections between number theory and dynamics are also presented, including equidistribution, homogenous dynamics, and Lagrange and Markov spectra. In addition, some dynamical and number theoretical aspects of aperiodic order, some algebraic systems, and a recent development concerning tame systems are described.

Originating from research in the qualitative theory of ordinary differential equations, this book follows the authors' work on structurally stable planar quadratic polynomial differential systems. In the present work the authors aim at finding all possible phase portraits in the Poincare disc, modulo limit cycles, of planar quadratic polynomial differential systems manifesting the simplest level of structural instability. They prove that there are at most 211 and at least 204 of them.

This book offers a fundamental explanation of nonlinear oscillations in physical systems. Originally intended for electrical engineers, it remains an important reference for the increasing numbers of researchers studying nonlinear phenomena in physics, chemical engineering, biology, medicine, and other fields. Originally published in 1986. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Based on the third International Conference on Symmetries, Differential Equations and Applications (SDEA-III), this proceedings volume highlights recent important advances and trends in the applications of Lie groups, including a broad area of topics in interdisciplinary studies, ranging from mathematical physics to financial mathematics. The selected and peer-reviewed contributions gathered here cover Lie theory and symmetry methods in differential equations, Lie algebras and Lie pseudogroups, super-symmetry and super-integrability, representation theory of Lie algebras, classification problems, conservation laws, and geometrical methods. The SDEA III, held in honour of the Centenary of Noether's Theorem, proven by the prominent German mathematician Emmy Noether, at Istanbul Technical University in August 2017 provided a productive forum for academic researchers, both junior and senior, and students to discuss and share the latest developments in the theory and applications of Lie symmetry groups. This work has an interdisciplinary appeal and will be a valuable read for researchers in mathematics, mechanics, physics, engineering, medicine and finance.

This volume provides a comprehensive review of multiple-scale dynamical systems. Mathematical models of such multiple-scale systems are considered singular perturbation problems, and this volume focuses on the geometric approach known as Geometric Singular Perturbation Theory (GSPT). It is the first of its kind that introduces the GSPT in a coordinate-independent manner. This is motivated by specific examples of biochemical reaction networks, electronic circuit and mechanic oscillator models and advection-reaction-diffusion models, all with an inherent non-uniform scale splitting, which identifies these examples as singular perturbation problems beyond the standard form. The contents cover a general framework for this GSPT beyond the standard form including canard theory, concrete applications, and instructive qualitative models. It contains many illustrations and key pointers to the existing literature. The target audience are senior undergraduates, graduate students and researchers interested in using the GSPT toolbox in nonlinear science, either from a theoretical or an application point of view. Martin Wechselberger is Professor at the School of Mathematics & Statistics, University of Sydney, Australia. He received the J.D. Crawford Prize in 2017 by the Society for Industrial and Applied Mathematics (SIAM) for achievements in the field of dynamical systems with multiple time-scales.

Dieses Buch bietet eine erste Einfuhrung in die mathematische Theorie der dynamischen Systeme, die fur Studierende des letzten Studienjahres des Bachelor Studiums und fur das Master Studium geeignet ist. Aufbauend auf den Grundbegriffen der Topologischen Dynamik und der Ergodentheorie in den ersten beiden Kapiteln behandelt das dritte Kapitel den fur die Ergodentheorie zentralen Begriff der Entropie, der seinen Ursprung in der statistischen Physik und der Informationstheorie hat, und der die Komplexitat eines masstheoretischen dynamischen Systems quantifiziert. Das vierte Kapitel ist ebenfalls der Entropie gewidmet, diesmal aber im Rahmen der topologischen Dynamik, bei derEntropie einen quantitativen Ausdruck fur die Verformung eines kompakten metrischen Raumes durch eine stetige Transformation darstellt. Das funfte und letzte Kapitel gibt einen kleinen Einblick in aktuelle Entwicklungen der Theorie der dynamischen Systeme mit ihren mehrparametrischen Verallgemeinerungen des klassischen Konzepts der Zeitentwicklung und den daraus entspringenden und zum Teil uberraschenden Querverbindungen zu anderen mathematischen Disziplinen.
Das in Vorlesungen erprobte Material dieses Buches kann durch eine den Interessen der Studierenden angepasste Themenauswahl wahlweise fur eine ein- oder zweisemestrige Vorlesung eingesetzt werden. "