This book focuses on the approximation of nonlinear equations using iterative methods. Nine contributions are presented on the construction and analysis of these methods, the coverage encompassing convergence, efficiency, robustness, dynamics, and applications. Many problems are stated in the form of nonlinear equations, using mathematical modeling. In particular, a wide range of problems in Applied Mathematics and in Engineering can be solved by finding the solutions to these equations. The book reveals the importance of studying convergence aspects in iterative methods and shows that selection of the most efficient and robust iterative method for a given problem is crucial to guaranteeing a good approximation. A number of sample criteria for selecting the optimal method are presented, including those regarding the order of convergence, the computational cost, and the stability, including the dynamics. This book will appeal to researchers whose field of interest is related to nonlinear problems and equations, and their approximation.

This volume collects contributions related to selected presentations from the 12th IFAC Workshop on Time Delay Systems, Ann Arbor, June 28-30, 2015. The included papers present novel techniques and new results of delayed dynamical systems. The topical spectrum covers control theory, numerical analysis, engineering and biological applications as well as experiments and case studies. The target audience primarily comprises research experts in the field of time delay systems, but the book may also be beneficial for graduate students alike.

This is the first book to systematically state the fundamental theory of integrability and its development of ordinary differential equations with emphasis on the Darboux theory of integrability and local integrability together with their applications. It summarizes the classical results of Darboux integrability and its modern development together with their related Darboux polynomials and their applications in the reduction of Liouville and elementary integrabilty and in the center-focus problem, the weakened Hilbert 16th problem on algebraic limit cycles and the global dynamical analysis of some realistic models in fields such as physics, mechanics and biology. Although it can be used as a textbook for graduate students in dynamical systems, it is intended as supplementary reading for graduate students from mathematics, physics, mechanics and engineering in courses related to the qualitative theory, bifurcation theory and the theory of integrability of dynamical systems.

The book covers nonlinear physical problems and mathematical modeling, including molecular biology, genetics, neurosciences, artificial intelligence with classical problems in mechanics and astronomy and physics. The chapters present nonlinear mathematical modeling in life science and physics through nonlinear differential equations, nonlinear discrete equations and hybrid equations. Such modeling can be effectively applied to the wide spectrum of nonlinear physical problems, including the KAM (Kolmogorov-Arnold-Moser (KAM)) theory, singular differential equations, impulsive dichotomous linear systems, analytical bifurcation trees of periodic motions, and almost or pseudo- almost periodic solutions in nonlinear dynamical systems.

Recently, a great deal of progress has been made in the modeling and understanding of processes with nonlinear dynamics, even when only time series data are available. Modern reconstruction theory deals with creating nonlinear dynamical models from data and is at the heart of this improved understanding. Most of the work has been done by dynamicists, but for the subject to reach maturity, statisticians and signal processing engineers need to provide input both to the theory and to the practice. The book brings together different approaches to nonlinear time series analysis in order to begin a synthesis that will lead to better theory and practice in all the related areas. This book describes the state of the art in nonlinear dynamical reconstruction theory. The chapters are based upon a workshop held at the Isaac Newton Institute, Cambridge University, UK, in late 1998. The book's chapters present theory and methods topics by leading researchers in applied and theoretical nonlinear dynamics, statistics, probability, and systems theory. Features and topics: * disentangling uncertainty and error: the predictability of nonlinear systems * achieving good nonlinear models * delay reconstructions: dynamics vs. statistics * introduction to Monte Carlo Methods for Bayesian Data Analysis * latest results in extracting dynamical behavior via Markov Models * data compression, dynamics and stationarity Professionals, researchers, and advanced graduates in nonlinear dynamics, probability, optimization, and systems theory will find the book a useful resource and guide to current developments in the subject.

Is it possible to "guide" the process of self-organisation towards specific patterns and outcomes?Wouldn t this be self-contradictory?After all, a self-organising process assumes a transition into a more organised form, or towards a more structured functionality, in the "absence" of centralised control.Then how can we place the guiding elements so that they do not override rich choices potentially discoverable by an uncontrolled process?

This book presents different approaches to resolving this "paradox."In doing so, the presented studies address a broad range of phenomena, ranging from autopoietic systems to morphological computation, and from small-world networks to information cascades in swarms.A large variety of methods is employed, from spontaneous symmetry breaking to information dynamics to evolutionary algorithms, creating a rich spectrum reflecting this emerging field.

Demonstrating several foundational theories and frameworks, as well as innovative practical implementations, "Guided Self-Organisation: Inception," will be an invaluable tool for advanced students and researchers in a multiplicity of fields across computer science, physics and biology, including information theory, robotics, dynamical systems, graph theory, artificial life, multi-agent systems, theory of computation and machine learning.

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This is the first comprehensive presentation of the quantum non-linear sigma-models. The original papers consider in detail geometrical properties and renormalization of a generic non-linear sigma-model, illustrated by explicit multi-loop calculations in perturbation theory.

This comprehensive reference text gives an overview of the current state of nonlinear wave mechanics in both elastic and fluid media. Consisting of self-contained chapters, the book covers new aspects on strong discontinuities (shock waves) and localized self- preserving (permanent) shapes (solitary waves and solitons). Special attention is devoted to the kinematics and dynamics of permanent waves when dissipative effects are added to the original balance between nonlinearity and dispersion. Key features include: * survey chapters written in an accessible style by leading specialists * coverage of emerging topics in the field * interdisciplinary approach integrating mathematical theory and physical applications of nonlinear waves in elastic and fluid media * treatment of the intrinsic mechanisms of propagation of different types of nonlinear waves * presentation of analytical methods for solving wave propagation problems in elastic and fluid media * user-friendly index 'Selected Topics in Nonlinear Wave Mechanics' provides readers with recent developments in the nonlinear propagation and scattering of waves in both elastic solids and liquids. The book is useful for applied mathematicians, physicists, mechanical, civil and aerospace engineers, as well as graduate students in those fields. Contributors: R.M. Axel, C.I. Christov, A. Guran, J.B. Haddow, G.A. Maugin, A. Morro, A. Nagl, P.K. Newton, A.V. Porubov, R.J. Tait, H. sberall, M.G. Velarde, W.B. Zimmerman

This volume presents selected contributions by top researchers in the field of operations research, originating from the XVI Congress of APDIO. It provides interesting findings and applications of operations research methods and techniques in a wide variety of problems. The contributions address complex real-world problems, including inventory management with lateral transshipments, sectors and routes in solid-waste collection and production planning for perishable food products. It also discusses the latest techniques, making the volume a valuable tool for researchers, students and practitioners who wish to learn about current trends. Of particular interest are the applications of nonlinear and mixed-integer programming, data envelopment analysis, clustering techniques, hybrid heuristics, supply chain management and lot sizing, as well as job scheduling problems. This biennial conference, organized by APDIO, the Portuguese Association of Operational Research, held in Braganca, Portugal, in June 2013, presented a perfect opportunity to discuss the latest development in this field and to narrow the gap between academic researchers and practitioners.

This book provides a complete exposition of equidistribution and counting problems weighted by a potential function of common perpendicular geodesics in negatively curved manifolds and simplicial trees. Avoiding any compactness assumptions, the authors extend the theory of Patterson-Sullivan, Bowen-Margulis and Oh-Shah (skinning) measures to CAT(-1) spaces with potentials. The work presents a proof for the equidistribution of equidistant hypersurfaces to Gibbs measures, and the equidistribution of common perpendicular arcs between, for instance, closed geodesics. Using tools from ergodic theory (including coding by topological Markov shifts, and an appendix by Buzzi that relates weak Gibbs measures and equilibrium states for them), the authors further prove the variational principle and rate of mixing for the geodesic flow on metric and simplicial trees-again without the need for any compactness or torsionfree assumptions. In a series of applications, using the Bruhat-Tits trees over non-Archimedean local fields, the authors subsequently prove further important results: the Mertens formula and the equidistribution of Farey fractions in function fields, the equidistribution of quadratic irrationals over function fields in their completions, and asymptotic counting results of the representations by quadratic norm forms. One of the book's main benefits is that the authors provide explicit error terms throughout. Given its scope, it will be of interest to graduate students and researchers in a wide range of fields, for instance ergodic theory, dynamical systems, geometric group theory, discrete subgroups of locally compact groups, and the arithmetic of function fields.

The volume contains carefully selected papers presented at the International Conference on Differential & Difference Equations and Applications held in Ponta Delgada - Azores, from July 4-8, 2011 in honor of Professor Ravi P. Agarwal. The objective of the gathering was to bring together researchers in the fields of differential & difference equations and to promote the exchange of ideas and research. The papers cover all areas of differential and difference equations with a special emphasis on applications.

This book deals with the economic aspects of changing attitudes in arts and sciences. The effects of the public good character of culture, along with the very long production period and lifetime for its products, are emphasized, since both contribute to the failure of normal market solutions. Embodiment of ideas and the consequences of modern reproduction technology for protection of property rights are closely examined. The evolution within arts and sciences, which often seems to return to previously scrapped ideals, is illustrated by detailed case studies, in which the importance of changing tastes, rather than progress proper, is emphasized. The author attempts an understanding for this using Darwinian evolution in combination with modern mathematical complexity theory, expressed in terms accessible to the general reader. The second edition is extended and updated especially as regards the illustration material.

Synchronization of chaotic systems, a patently nonlinear phenomenon, has emerged as a highly active interdisciplinary research topic at the interface of physics, biology, applied mathematics and engineering sciences. In this connection, time-delay systems described by delay differential equations have developed as particularly
suitable tools for modeling specific dynamical systems. Indeed, time-delay is ubiquitous in many physical systems, for example due to finite
switching speeds of amplifiers in electronic circuits, finite lengths of vehicles in traffic flows, finite signal propagation times in biological networks and circuits, and quite generally whenever memory effects are relevant.
This monograph presents the basics of chaotic time-delay systems and their synchronization with an emphasis on the effects of time-delay feedback which give rise to new collective dynamics.
Special attention is devoted to scalar chaotic/hyperchaotic time-delay
systems, and some higher order models, occurring in different branches of science and technology as well as to the synchronization of their coupled versions.

Last but not least, the presentation as a whole strives for a balance between the necessary mathematical description of the basics
and the detailed presentation of real-world applications.

The idea of optimization runs through most parts of control theory. The simplest optimal controls are preplanned (programmed) ones. The problem of constructing optimal preplanned controls has been extensively worked out in literature (see, e. g., the Pontrjagin maximum principle giving necessary conditions of preplanned control optimality). However, the concept of op timality itself has a restrictive character: it is limited by what one means under optimality in each separate case. The internal contradictoriness of the preplanned control optimality ("the better is the enemy of the good") yields that the practical significance of optimal preplanned controls proves to be not great: such controls are usually sensitive to unregistered disturbances (includ ing the round-off errors which are inevitable when computer devices are used for forming controls), as there is the effect of disturbance accumulation in the control process which makes controls to be of little use on large time inter vals. This gap is mainly provoked by oversimplified settings of optimization problems. The outstanding result of control theory established in the end of the first half of our century is that controls in feedback form ensure the weak sensitivity of closed loop systems with respect to "small" unregistered internal and external disturbances acting in them (here we do not need to discuss performance indexes, since the considered phenomenon is of general nature). But by far not all optimal preplanned controls can be represented in a feedback form."

In mathematical modeling of processes one often encounters optimization problems involving more than one objective function, so that Multiobjective Optimization (or Vector Optimization) has received new impetus. The growing interest in multiobjective problems, both from the theoretical point of view and as it concerns applications to real problems, asks for a general scheme which embraces several existing developments and stimulates new ones. In this book the authors provide the newest results and applications of this quickly growing field. This book will be of interest to graduate students in mathematics, economics, and engineering, as well as researchers in pure and applied mathematics, economics, engineering, geography, and town planning. A sound knowledge of linear algebra and introductory real analysis should provide readers with sufficient background for this book.

Nonlinear and Hybrid Systems in Automotive Control will enable researchers, control engineers and automotive engineers to understand the engine and whole-vehicle models necessary for control. A new generation of control strategies has become necessary because of the increasingly rigorous requirements of vehicle and engine control systems for accuracy, ride comfort, safety, complexity, functionality and emission levels. In contrast with earlier systems, these new control systems are based on dynamic physical models and the principles of advanced nonlinear control. The contributors to this work come from both academic and industrial backgrounds and the subjects they cover include: suspension control; modelling of driver position and behaviour; anti-lock braking systems and optimal braking control; stability analysis of hybrid systems; Hamiltonian formulation of bond graphs; approximation of maximal controlled safe sets for hybrid systems. This book should be of use to academic researchers and graduate students as well as to engineers in the automotive industry.

This book offers a comprehensive treatment of the theory of measures of noncompactness. It discusses various applications of the theory of measures of noncompactness, in particular, by addressing the results and methods of fixed-point theory. The concept of a measure of noncompactness is very useful for the mathematical community working in nonlinear analysis. Both these theories are especially useful in investigations connected with differential equations, integral equations, functional integral equations and optimization theory. Thus, one of the book's central goals is to collect and present sufficient conditions for the solvability of such equations. The results are established in miscellaneous function spaces, and particular attention is paid to fractional calculus.

This book focuses on bifurcation and stability in nonlinear discrete systems, including monotonic and oscillatory stability. It presents the local monotonic and oscillatory stability and bifurcation of period-1 fixed-points on a specific eigenvector direction, and discusses the corresponding higher-order singularity of fixed-points. Further, it explores the global analysis of monotonic and oscillatory stability of fixed-points in 1-dimensional discrete systems through 1-dimensional polynomial discrete systems. Based on the Yin-Yang theory of nonlinear discrete systems, the book also addresses the dynamics of forward and backward nonlinear discrete systems, and the existence conditions of fixed-points in said systems. Lastly, in the context of local analysis, it describes the normal forms of nonlinear discrete systems and infinite-fixed-point discrete systems. Examining nonlinear discrete systems from various perspectives, the book helps readers gain a better understanding of the nonlinear dynamics of such systems.

This book contains a selected collection of papers providing an overview of the state of the art in the study of dynamical systems. A broad range of aspects of dynamical systems is covered, focusing on discrete and continuous dynamical systems, bifurcation theory, celestial mechanics, delay difference and differential equations, Hamiltonian systems and also the classic challenges in planar vector fields. Particular attention has been posed on real-world applications of dynamical systems, showing the constant interaction of the field with other sciences. The authors have made a special effort in placing the reader at the frontiers of current knowledge in the discipline. In this way, recent advances and new trends become available. The papers are based on talks given at the International Conference Dynamical Systems: 100 years after Poincare held at the University of Oviedo, Gijon (Spain), on September 3-7, 2012. Recent advances and new trends have been discussed during the meeting, including applications to a wide range of disciplines such as Biology, Chemistry, Physics and Economics, among others. The memory of Poincare, who laid the foundations of dynamical systems, provided the backdrop for the discussion of the new challenges 100 years after his death.

This book presents a new degree theory for maps which commute with a group of symmetries. This degree is no longer a single integer but an element of the group of equivariant homotopy classes of maps between two spheres and depends on the orbit types of the spaces. The authors develop completely the theory and applications of this degree in a self-contained presentation starting with only elementary facts. The first chapter explains the basic tools of representation theory, homotopy theory and differential equations needed in the text. Then the degree is defined and its main abstract properties are derived. The next part is devoted to the study of equivariant homotopy groups of spheres and to the classification of equivariant maps in the case of abelian actions. These groups are explicitely computed and the effects of symmetry breaking, products and composition are thorougly studied. The last part deals with computations of the equivariant index of an isolated orbit and of an isolated loop of stationary points. Here differential equations in a variety of situations are considered: symmetry breaking, forcing, period doubling, twisted orbits, first integrals, gradients etc. Periodic solutions of Hamiltonian systems, in particular spring-pendulum systems, are studied as well as Hopf bifurcation for all these situations.

This book presents the recently introduced and already widely referred semi-discretization method for the stability analysis of delayed dynamical systems. Delay differential equations often come up in different fields of engineering, like feedback control systems, machine tool vibrations, balancing/stabilization with reflex delay. The behavior of such systems is often counter-intuitive and closed form analytical formulas can rarely be given even for the linear stability conditions. If parametric excitation is coupled with the delay effect, then the governing equation is a delay differential equation with time periodic coefficients, and the stability properties are even more intriguing. The semi-discretization method is a simple but efficient method that is based on the discretization with respect to the delayed term and the periodic coefficients only. The method can effectively be used to construct stability diagrams in the space of system parameters.

This book deals with the application of mathematics in modeling and understanding physiological systems, especially those involving rhythms. It is divided roughly into two sections. In the first part of the book, the authors introduce ideas and techniques from nonlinear dynamics that are relevant to the analysis of biological rhythms. The second part consists of five in-depth case studies in which the authors use the theoretical tools developed earlier to investigate a number of physiological processes: the dynamics of excitable nerve and cardiac tissue, resetting and entrainment of biological oscillators, the effects of noise and time delay on the pupil light reflex, pathologies associated with blood cell replication, and Parkinsonian tremor. One novel feature of the book is the inclusion of classroom-tested computer exercises throughout, designed to form a bridge between the mathematical theory and physiological experiments. This book will be of interest to students and researchers in the natural and physical sciences wanting to learn about the complexities and subtleties of physiological systems from a mathematical perspective. The authors are members of the Centre for Nonlinear Dynamics in Physiology and Medicine. The material in this book was developed for use in courses and was presented in three Summer Schools run by the authors in Montreal.

This monograph contains an in-depth analysis of the dynamics given by a linear Hamiltonian system of general dimension with nonautonomous bounded and uniformly continuous coefficients, without other initial assumptions on time-recurrence. Particular attention is given to the oscillation properties of the solutions as well as to a spectral theory appropriate for such systems. The book contains extensions of results which are well known when the coefficients are autonomous or periodic, as well as in the nonautonomous two-dimensional case. However, a substantial part of the theory presented here is new even in those much simpler situations. The authors make systematic use of basic facts concerning Lagrange planes and symplectic matrices, and apply some fundamental methods of topological dynamics and ergodic theory. Among the tools used in the analysis, which include Lyapunov exponents, Weyl matrices, exponential dichotomy, and weak disconjugacy, a fundamental role is played by the rotation number for linear Hamiltonian systems of general dimension. The properties of all these objects form the basis for the study of several themes concerning linear-quadratic control problems, including the linear regulator property, the Kalman-Bucy filter, the infinite-horizon optimization problem, the nonautonomous version of the Yakubovich Frequency Theorem, and dissipativity in the Willems sense. The book will be useful for graduate students and researchers interested in nonautonomous differential equations; dynamical systems and ergodic theory; spectral theory of differential operators; and control theory.

This book is about morphogenesis as the genesis of forms. It is not restricted to plants growing from seed or animals developing from an embryo (although these do supply the most abundant examples) but also addresses kindred processes, from inorganic to social to biomorphic technology. It is about our morphogenetic universe: unplanned, unfair and frustratingly complicated but benevolent in allowing us to emerge, survive, and inquire into its laws.

This book treats essentials from neurophysiology (Hodgkin-Huxley equations, synaptic transmission, prototype networks of neurons) and related mathematical concepts (dimensionality reductions, equilibria, bifurcations, limit cycles and phase plane analysis). This is subsequently applied in a clinical context, focusing on EEG generation, ischaemia, epilepsy and neurostimulation. The book is based on a graduate course taught by clinicians and mathematicians at the Institute of Technical Medicine at the University of Twente. Throughout the text, the author presents examples of neurological disorders in relation to applied mathematics to assist in disclosing various fundamental properties of the clinical reality at hand. Exercises are provided at the end of each chapter; answers are included. Basic knowledge of calculus, linear algebra, differential equations and familiarity with MATLAB or Python is assumed. Also, students should have some understanding of essentials of (clinical) neurophysiology, although most concepts are summarized in the first chapters. The audience includes advanced undergraduate or graduate students in Biomedical Engineering, Technical Medicine and Biology. Applied mathematicians may find pleasure in learning about the neurophysiology and clinic essentials applications. In addition, clinicians with an interest in dynamics of neural networks may find this book useful, too.