This innovative volume introduces Trajectory Analysis, a new systems-based approach to measuring nonlinear dynamics in continuous change, to public health and epidemiology. It synthesizes influential strands of statistical and probability science (including chaos theory and catastrophe theory) to complement existing methods and models used in the health fields. The computational framework featured here pinpoints complex cause-and-effect processes in behavioral change as individuals and populations adjust to health interventions, with examples from neuroscience and cardiology. But this is no mere academic exercise, as the author illustrates how these methods can be harnessed toward finding real-world answers to longstanding public health problems, starting with treatment recidivism. Included in the coverage: * The universality of physical principles in the analysis of health and disease * The problem of recidivism in healthcare intervention studies * Stability and reversibility/irreversibility of health conditions * Chaos theory and sensitive dependence on initial conditions * Applications in health monitoring and geographic systems * Simulations, applications, and the challenge for public health A stimulating new take on statistics with powerful implications for future study, practice, and policy, Trajectory Analysis in Health Care should interest public health epidemiologists, researchers, clinicians, and policymakers.

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.

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 textbook presents the motion of pure nonlinear oscillatory systems and various solution procedures which give the approximate solutions of the strong nonlinear oscillator equations. It presents the author's original method for the analytical solution procedure of the pure nonlinear oscillator system. After an introduction, the physical explanation of the pure nonlinearity and of the pure nonlinear oscillator is given. The analytical solution for free and forced vibrations of the one-degree-of-freedom strong nonlinear system with constant and time variable parameters is considered. In this second edition of the book, the number of approximate solving procedures for strong nonlinear oscillators is enlarged and a variety of procedures for solving free strong nonlinear oscillators is suggested. A method for error estimation is also given which is suitable to compare the exact and approximate solutions. Besides the oscillators with one degree-of-freedom, the one and two mass oscillatory systems with two-degrees-of-freedom and continuous oscillators are considered. The chaos and chaos suppression in ideal and non-ideal mechanical systems is explained. In this second edition more attention is given to the application of the suggested methodologies and obtained results to some practical problems in physics, mechanics, electronics and biomechanics. Thus, for the oscillator with two degrees-of-freedom, a generalization of the solving procedure is performed. Based on the obtained results, vibrations of the vocal cord are analyzed. In the book the vibration of the axially purely nonlinear rod as a continuous system is investigated. The developed solving procedure and the solutions are applied to discuss the muscle vibration. Vibrations of an optomechanical system are analyzed using the oscillations of an oscillator with odd or even quadratic nonlinearities. The extension of the forced vibrations of the system is realized by introducing the Ateb periodic excitation force which is the series of a trigonometric function. The book is self-consistent and suitable for researchers and as a textbook for students and also professionals and engineers who apply these techniques to the field of nonlinear oscillations.

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.

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.

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 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.

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

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.

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."

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.

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.

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.

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.

Probably the first book to describe computational methods for numerically computing steady state and Hopf bifurcations. Requiring only a basic knowledge of calculus, and using detailed examples, problems, and figures, this is an ideal textbook for graduate students.

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 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 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 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.

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.

Main concepts of quasi-stationary distributions (QSDs) for killed processes are the focus of the present volume. For diffusions, the killing is at the boundary and for dynamical systems there is a trap. The authors present the QSDs as the ones that allow describing the long-term behavior conditioned to not being killed. Studies in this research area started with Kolmogorov and Yaglom and in the last few decades have received a great deal of attention. The authors provide the exponential distribution property of the killing time for QSDs, present the more general result on their existence and study the process of trajectories that survive forever. For birth-and-death chains and diffusions, the existence of a single or a continuum of QSDs is described. They study the convergence to the extremal QSD and give the classification of the survival process. In this monograph, the authors discuss Gibbs QSDs for symbolic systems and absolutely continuous QSDs for repellers. The findings described are relevant to researchers in the fields of Markov chains, diffusions, potential theory, dynamical systems, and in areas where extinction is a central concept. The theory is illustrated with numerous examples. The volume uniquely presents the distribution behavior of individuals who survive in a decaying population for a very long time. It also provides the background for applications in mathematical ecology, statistical physics, computer sciences, and economics.

This book deals with an effect in celestial mechanics that has become quite important in exoplanet research. The Lidov-Kozai effect reveals itself in coherent periodic variations (which can be very large) of the inclination and eccentricity of an orbiting body in the presence of an inclined perturber. The effect is known to be important in the motion of many asteroids and planetary satellites. What is more, now it attracts more and more interest in the astronomical and astrophysical community due to its relevance for many exoplanetary systems. Recent years witnessed major advancements in its theory. It would be no exaggeration to say that nowadays the Lidov-Kozai effect becomes one of the most studied astrophysical effects. This book covers the multitude of the Lidov-Kozai effect's modern applications and its theory developments. It will be useful for researchers and students working in astrophysics, celestial mechanics, stellar dynamics, theoretical mechanics, space missions design, depending on the interests of the reader. The book is self-contained. It provides the full detailed coverage of the effect's theory and applications.

Non-linear stochastic systems are at the center of many engineering disciplines and progress in theoretical research had led to a better understanding of non-linear phenomena. This book provides information on new fundamental results and their applications which are beginning to appear across the entire spectrum of mechanics. The outstanding points of these proceedings are Coherent compendium of the current state of modelling and analysis of non-linear stochastic systems from engineering, applied mathematics and physics point of view. Subject areas include: Multiscale phenomena, stability and bifurcations, control and estimation, computational methods and modelling. For the Engineering and Physics communities, this book will provide first-hand information on recent mathematical developments. The applied mathematics community will benefit from the modelling and information on various possible applications.