This book introduces the fractal interpolation functions (FIFs) in approximation theory to the readers and the concerned researchers in advanced level. FIFs can be used to precisely reconstruct the naturally occurring functions when compared with the classical interpolants. The book focuses on the construction of fractals in metric space through various iterated function systems. It begins by providing the Mathematical background behind the fractal interpolation functions with its graphical representations and then introduces the fractional integral and fractional derivative on fractal functions in various scenarios. Further, the existence of the fractal interpolation function with the countable iterated function system is demonstrated by taking suitable monotone and bounded sequences. It also covers the dimension of fractal functions and investigates the relationship between the fractal dimension and the fractional order of fractal interpolation functions. Moreover, this book explores the idea of fractal interpolation in the reconstruction scheme of illustrative waveforms and discusses the problems of identification of the characterizing parameters. In the application section, this research compendium addresses the signal processing and its Mathematical methodologies. A wavelet-based denoising method for the recovery of electroencephalogram (EEG) signals contaminated by nonstationary noises is presented, and the author investigates the recognition of healthy, epileptic EEG and cardiac ECG signals using multifractal measures. This book is intended for professionals in the field of Mathematics, Physics and Computer Science, helping them broaden their understanding of fractal functions and dimensions, while also providing the illustrative experimental applications for researchers in biomedicine and neuroscience.

This book discusses fundamentally new biomedical imaging methods, such as holography, holographic and resonant interferometry, and speckle optics. It focuses on the development of holographic interference microscopy and its use in the study of phase objects such as nerve and muscle fibers subjected to the influence of laser radiation, magnetic fields, and hyperbaric conditions. The book shows how the myelin sheath and even the axon itself exhibit waveguide properties, enabling a fresh new look at the mechanisms of information transmission in the human body. The book presents theoretically and experimentally tested holographic and speckle-optical methods and devices used for investigating complex, diffusely scattering surfaces such as skin and muscle tissue. Additionally, it gives broad discussion of the authors' own original fundamental and applied research dedicated to helping physicians introduce new contact-less methods of diagnosis and treatment of diseases of the cardiovascular and neuromuscular systems into medical practice. The book is aimed at a broad spectrum of scientific specialists in the fields of speckle optics, holography, laser physics, morphology and cytochemistry, as well as medical professionals such as physiologists, neuropathologists, neurosurgeons, cardiologists and dentists.

This monograph has arisen out of a number of attempts spanning almost five decades to understand how one might examine the evolution of densities in systems whose dynamics are described by differential delay equations. Though the authors have no definitive solution to the problem, they offer this contribution in an attempt to define the problem as they see it, and to sketch out several obvious attempts that have been suggested to solve the problem and which seem to have failed. They hope that by being available to the general mathematical community, they will inspire others to consider-and hopefully solve-the problem. Serious attempts have been made by all of the authors over the years and they have made reference to these where appropriate.

This is the proceedings of the IUTAM Symposium on Exploiting Nonlinear Dynamics for Engineering Systems that was held in Novi Sad, Serbia, from July 15th to 19th, 2018. The appearance of nonlinear phenomena used to be perceived as dangerous, with a general tendency to avoid them or control them. This perception has led to intensive research using various approaches and tailor-made tools developed over decades. However, the Nonlinear Dynamics of today is experiencing a profound shift of paradigm since recent investigations rely on a different strategy which brings good effects of nonlinear phenomena to the forefront. This strategy has a positive impact on different fields in science and engineering, such as vibration isolation, energy harvesting, micro/nano-electro-mechanical systems, etc. Therefore, the ENOLIDES Symposium was devoted to demonstrate the benefits and to unlock the potential of exploiting nonlinear dynamical behaviour in these but also in other emerging fields of science and engineering. This proceedings is useful for researchers in the fields of nonlinear dynamics of mechanical systems and structures, and in Mechanical and Civil Engineering.

This book analyzes a range of new developments in various fields concerning the concepts of chaos and complexity theory. The proceedings of the 7th International Symposium on Chaos, Complexity and Leadership feature newly developed concepts involving various research methodologies for identifying chaos and complexity in different fields of the sciences and leadership. In addition, it explores chaotic and complex systems from all fields of knowledge in order to stake a claim of prevalence of compatibility between knowledge fields. Particular emphasis is placed on exploring non-linearity in order to open a discussion on new approaches to and perspectives on chaos, complexity and leadership. Readers will find coverage of important events that have recently taken place in our world, regardless of whether they were social, political, economic or scientific in nature. The book explores diverse aspects of and issues related to the effects of chaos and complexity in the world; discusses the application of nonlinear dynamics in order to arrive at transformational policies; and offers projections of tomorrow's world using an interdisciplinary approach. Though primarily intended for readers with an interest in nonlinear science, thanks to its focus on the application of chaos and complexity to other disciplines, the book appeals to a broad readership.

The mathematical models considered in this book can help to understand the swelling of mitochondria. For the first time, it presents new mathematical models of mitochondrial swelling that take into account, in particular, spatial effects. The results presented here could make it possible to predict properties of the underlying biological mechanisms. Taking into account that mitochondria could move within a cell, lead to a PDE-PDE model. The book discusses the well-posedness and long-term dynamics of solutions, depending on boundary conditions reflecting the in vitro and in vivo cases. These analytical and numerical results have inspired colleagues from the Institute of Pharmacology and Toxicology of the Helmholtz Center Munich to design new experiments justifying the theoretical and numerical results that are obtained. The book is intended for graduates students and researchers with a solid mathematical background and an interest in cell biology.

The 5th edition of this classic textbook covers the central concepts of practical optimization techniques, with an emphasis on methods that are both state-of-the-art and popular. One major insight is the connection between the purely analytical character of an optimization problem and the behavior of algorithms used to solve that problem. End-of-chapter exercises are provided for all chapters. The material is organized into three separate parts. Part I offers a self-contained introduction to linear programming. The presentation in this part is fairly conventional, covering the main elements of the underlying theory of linear programming, many of the most effective numerical algorithms, and many of its important special applications. Part II, which is independent of Part I, covers the theory of unconstrained optimization, including both derivations of the appropriate optimality conditions and an introduction to basic algorithms. This part of the book explores the general properties of algorithms and defines various notions of convergence. In turn, Part III extends the concepts developed in the second part to constrained optimization problems. Except for a few isolated sections, this part is also independent of Part I. As such, Parts II and III can easily be used without reading Part I and, in fact, the book has been used in this way at many universities. New to this edition are popular topics in data science and machine learning, such as the Markov Decision Process, Farkas' lemma, convergence speed analysis, duality theories and applications, various first-order methods, stochastic gradient method, mirror-descent method, Frank-Wolf method, ALM/ADMM method, interior trust-region method for non-convex optimization, distributionally robust optimization, online linear programming, semidefinite programming for sensor-network localization, and infeasibility detection for nonlinear optimization.

This book explores recent developments in theoretical research and data analysis of real-world complex systems, organized in three parts, namely Entropy, information, and complexity functions Multistability, oscillations, and rhythmic synchronization Diffusions, rotation, and convection in fluids The collection of works devoted to the memory of Professor Valentin Afraimovich provides a deep insight into the recent developments in complexity science by introducing new concepts, methods, and applications in nonlinear dynamical systems covering physical problems and mathematical modelling relevant to economics, genetics, engineering vibrations, as well as classic problems in physics, fluid and climate dynamics, and urban dynamics. The book facilitates a better understanding of the mechanisms and phenomena in nonlinear dynamics and develops the corresponding mathematical theory to apply nonlinear design to practical engineering. It can be read by mathematicians, physicists, complex systems scientists, IT specialists, civil engineers, data scientists, and urban planners.

This book focuses on theoretical aspects of dynamical systems in the broadest sense. It highlights novel and relevant results on mathematical and numerical problems that can be found in the fields of applied mathematics, physics, mechanics, engineering and the life sciences. The book consists of contributed research chapters addressing a diverse range of problems. The issues discussed include (among others): numerical-analytical algorithms for nonlinear optimal control problems on a large time interval; gravity waves in a reservoir with an uneven bottom; value distribution and growth of solutions for certain Painleve equations; optimal control of hybrid systems with sliding modes; a mathematical model of the two types of atrioventricular nodal reentrant tachycardia; non-conservative instability of cantilevered nanotubes using the Cell Discretization Method; dynamic analysis of a compliant tensegrity structure for use in a gripper application; and Jeffcott rotor bifurcation behavior using various models of hydrodynamic bearings.

This book discusses the realization and control problems of finite-dimensional dynamical systems which contain linear and nonlinear systems. The author focuses on algebraic methods for the discussion of control problems of linear and non-linear dynamical systems. The book contains detailed examples to showcase the effectiveness of the presented method. The target audience comprises primarily research experts in the field of control theory, but the book may also be beneficial for graduate students alike.

This book collects recent advances in the field of nonlinear dynamics in biological systems. Focusing on medical applications as well as more fundamental questions in biochemistry, it presents recent findings in areas such as control in chemically driven reaction-diffusion systems, electrical wave propagation through heart tissue, neural network growth, chiral symmetry breaking in polymers and mechanochemical pattern formation in the cytoplasm, particularly in the context of cardiac cells. It is a compilation of works, including contributions from international scientists who attended the "2nd BCAM Workshop on Nonlinear Dynamics in Biological Systems," held at the Basque Center for Applied Mathematics, Bilbao in September 2016. Embracing diverse disciplines and using multidisciplinary approaches - including theoretical concepts, simulations and experiments - these contributions highlight the nonlinear nature of biological systems in order to be able to reproduce their complex behavior. Edited by the conference organizers and featuring results that represent recent findings and not necessarily those presented at the conference, the book appeals to applied mathematicians, biophysicists and computational biologists.

This thesis presents the application of non-perturbative, or functional, renormalization group to study the physics of critical stationary states in systems out-of-equilibrium. Two different systems are thereby studied. The first system is the diffusive epidemic process, a stochastic process which models the propagation of an epidemic within a population. This model exhibits a phase transition peculiar to out-of-equilibrium, between a stationary state where the epidemic is extinct and one where it survives. The present study helps to clarify subtle issues about the underlying symmetries of this process and the possible universality classes of its phase transition. The second system is fully developed homogeneous isotropic and incompressible turbulence. The stationary state of this driven-dissipative system shows an energy cascade whose phenomenology is complex, with partial scale-invariance, intertwined with what is called intermittency. In this work, analytical expressions for the space-time dependence of multi-point correlation functions of the turbulent state in 2- and 3-D are derived. This result is noteworthy in that it does not rely on phenomenological input except from the Navier-Stokes equation and that it becomes exact in the physically relevant limit of large wave-numbers. The obtained correlation functions show how scale invariance is broken in a subtle way, related to intermittency corrections.

The book offers a novel approach to the study of the complex dynamics of cities. It is based on (1) Synergetics as a science of cooperation and selforganization, (2) information theory including semantic and pragmatic aspects, and optimization principles, (3) a theory of steady state maintenance, and of (4) phase transition, i.e. qualitative changes of structure or behavior. From this novel theoretical vantage point, the book addresses particularly three issues that stand at the core of current discourse on cities: Urban Scaling, Smart Cities and City Planning. An important consequence of "the 21st century as the age of cities", is that the study of cities currently attracts scientists from a variety of disciplines, ranging from physics, mathematics and computer science, through urban studies, architecture, planning and human geography, to economics, psychology, sociology, public administration and more. The book is thus likely to attract scholars, researchers and students of these research domains, of complexity theories of cities, as well as of general complexity theory. In addition, it is directed also to practitioners of urbanism, city planning and urban design.

This book presents a unified study of dynamically coupled systems involving a rigid body and an ideal fluid flow from the perspective of Lagrangian and Hamiltonian mechanics. It compiles theoretical investigations on the topic of dynamically coupled systems using a framework grounded in Kirchhoff's equations. The text achieves a balance between geometric mechanics, or the modern theories of reduction of Lagrangian and Hamiltonian systems, and classical fluid mechanics, with a special focus on the applications of these principles. Following an introduction to Kirchhoff's equations of motion, the book discusses several extensions of Kirchhoff's work, particularly related to vortices. It addresses the equations of motions of these systems and their Lagrangian and Hamiltonian formulations. The book is suitable to mathematicians, physicists and engineers with a background in Lagrangian and Hamiltonian mechanics and theoretical fluid mechanics. It includes a brief introductory overview of geometric mechanics in the appendix.

This book constitutes the proceedings of the 6th International Symposium on Chaos, Complexity and Leadership (ICCLS). Written by interdisciplinary researchers and students from the fields of mathematics, physics, education, economics, political science, statistics, the management sciences and social sciences, the peer-reviewed contributions explore chaotic and complex systems, as well as chaos and complexity theory in the context of their applicability to management and leadership. The book discusses current topics, such as complexity leadership in the healthcare fields and tourism industry, conflict management and organization intelligence, and presents practical applications of theoretical concepts, making it a valuable resource for managers and leaders.

This book presents the Proceedings of the 54th Winter School of Theoretical Physics on Simplicity of Complexity in Economic and Social Systems, held in Ladek Zdroj, Poland, from 18 to 24 February 2018. The purpose of the book is to introduce the new interdisciplinary research that links statistical physics, and particular attention is given to link physics of complex systems, with financial analysis and sociology. The main tools used in these areas are numerical simulation of agents behavior and the interpretation of results with the help of complexity methods, therefore a background in statistical physics and in physics of phase transition is necessary to take the first steps towards these research fields called econophysics and sociophysics. In this perspective, the book is intended to graduated students and young researchers who want to begin the study of this established new area, which connects physicists, economists, sociologists and IT professionals, to better understand complexity phenomena existing not only in physics but also in complex systems being seemingly far from traditional view at physics.

The idea of structure-preserving algorithms appeared in the 1980's. The new paradigm brought many innovative changes. The new paradigm wanted to identify the long-time behaviour of the solutions or the existence of conservation laws or some other qualitative feature of the dynamics. Another area that has kept growing in importance within Geometric Numerical Integration is the study of highly-oscillatory problems: problems where the solutions are periodic or quasiperiodic and have to be studied in time intervals that include an extremely large number of periods. As is known, these equations cannot be solved efficiently using conventional methods. A further study of novel geometric integrators has become increasingly important in recent years. The objective of this monograph is to explore further geometric integrators for highly oscillatory problems that can be formulated as systems of ordinary and partial differential equations. Facing challenging scientific computational problems, this book presents some new perspectives of the subject matter based on theoretical derivations and mathematical analysis, and provides high-performance numerical simulations. In order to show the long-time numerical behaviour of the simulation, all the integrators presented in this monograph have been tested and verified on highly oscillatory systems from a wide range of applications in the field of science and engineering. They are more efficient than existing schemes in the literature for differential equations that have highly oscillatory solutions. This book is useful to researchers, teachers, students and engineers who are interested in Geometric Integrators and their long-time behaviour analysis for differential equations with highly oscillatory solutions.

This book addresses the fascinating phenomena associated with nonlinear waves and spatio-temporal patterns. These appear almost everywhere in nature from sand bed forms to brain patterns, and yet their understanding still presents fundamental scientific challenges. The reader will learn here, in particular, about the current state-of-the art and new results in: Nonlinear water waves: resonance, solitons, focusing, Bose-Einstein condensation, as well as and their relevance for the sea environment (sea-wind interaction, sand bed forms, fiber clustering) Pattern formation in non-equilibrium media: soap films, chimera patterns in oscillating media, viscoelastic Couette-Taylor flow, flow in the wake behind a heated cylinder, other pattern formation. The editors and authors dedicate this book to the memory of Alexander Ezersky, Professor of Fluid Mechanics at the University of Caen Normandie (France) from September 2007 to July 2016. Before 2007, he had served as a Senior Scientist at the Institute of Applied Physics of the Russian Academy of Sciences in Nizhny Novgorod (Russia). The chapters have been written by leading scientists in Nonlinear Physics, and the topics chosen so as to cover all the fields to which Prof. Ezersky himself contributed, by means of experimental, theoretical and numerical approaches. The volume will appeal to advanced students and researchers studying nonlinear waves and pattern dynamics, as well as other scientists interested in their applications in various natural media.

This book contains the proceedings of the Seventh National Conference of the Italian Systems Society. The title, Systemics of Incompleteness and Quasi-Systems, aims to underline the need for Systemics and Systems Science to deal with the concepts of incompleteness and quasiness. Classical models of Systemics are intended to represent comprehensive aspects of phenomena and processes. They consider the phenomena in their temporal and spatial completeness. In these cases, possible incompleteness in the modelling is assumed to have a provisional or practical nature, which is still under study, and because there is no theoretical reason why the modelling cannot be complete. In principle, this is a matter of non-complex phenomena, to be considered using the concepts of the First Systemics. When dealing with emergence, there are phenomena which must be modelled by systems having multiple models, depending on the aspects being taken into consideration. Here, incompleteness in the modelling is intrinsic, theoretically relating changes in properties, structures, and status of system. Rather than consider the same system parametrically changing over time, we consider sequences of systems coherently. We consider contexts and processes for which modelling is incomplete, being related to only some properties, as well as those for which such modelling is theoretically incomplete-as in the case of processes of emergence and for approaches considered by the Second Systemics. In this regard, we consider here the generic concept of quasi explicating such incompleteness. The concept of quasi is used in various disciplines including quasi-crystals, quasi-particles, quasi-electric fields, and quasi-periodicity. In general, the concept of quasiness for systems concerns their continuous structural changes which are always meta-stable, waiting for events to collapse over other configurations and possible forms of stability; whose equivalence depends on the type of phenomenon under study. Interest in the concept of quasiness is not related to its meaning of rough approximation, but because it indicates an incompleteness which is structurally sufficient to accommodate processes of emergence and sustain coherence or generate new, equivalent or non-equivalent, levels. The conference was devoted to identifying, discussing and understanding possible interrelationships of theoretical disciplinary improvements, recognised as having prospective fundamental roles for a new Quasi-Systemics. The latter should be able to deal with problems related to complexity in more general and realistic ways, when a system is not always a system and not always the same system. In this context, the inter-disciplinarity should consist, for instance, of a constructionist, incomplete, non-ideological, multiple, contradiction-tolerant, Systemics, always in progress, and in its turn, emergent.

Most physical systems lose or gain stability through bifurcation behavior. This book explains a series of experimentally found bifurcation phenomena by means of the methods of static bifurcation theory.

This book highlights recent advances of optical spatial solitons in photorefractive materials ranging broadly from the coupling, modulation instability, effect of pyroelectricity, and the stability of photorefractive solitons, among other topics. Photorefractive solitons have been at the forefront of research because of their formation at low laser powers and unique saturable nonlinearity present in photorefractive materials which supports solitons in (2+1) D. There has been a spurt in research on photorefractive solitons recently, which has contributed to a greater understanding of the theoretical foundation of photorefractive solitons as also of their various interesting and practical applications. The book elucidates the diversity of photorefractive solitons and provides a good resource for students, researchers, and professionals in the area of nonlinear optics.

This classic text provides an excellent introduction to a new and rapidly developing field of research. Now well established as a textbook in this rapidly developing field of research, the new edition is much enlarged and covers a host of new results.

This book provides a unique survey displaying the power of Riccati equations to describe reversible and irreversible processes in physics and, in particular, quantum physics. Quantum mechanics is supposedly linear, invariant under time-reversal, conserving energy and, in contrast to classical theories, essentially based on the use of complex quantities. However, on a macroscopic level, processes apparently obey nonlinear irreversible evolution equations and dissipate energy. The Riccati equation, a nonlinear equation that can be linearized, has the potential to link these two worlds when applied to complex quantities. The nonlinearity can provide information about the phase-amplitude correlations of the complex quantities that cannot be obtained from the linearized form. As revealed in this wide ranging treatment, Riccati equations can also be found in many diverse fields of physics from Bose-Einstein-condensates to cosmology. The book will appeal to graduate students and theoretical physicists interested in a consistent mathematical description of physical laws.

This book discusses many of the common scaling properties observed in some nonlinear dynamical systems mostly described by mappings. The unpredictability of the time evolution of two nearby initial conditions in the phase space together with the exponential divergence from each other as time goes by lead to the concept of chaos. Some of the observables in nonlinear systems exhibit characteristics of scaling invariance being then described via scaling laws. From the variation of control parameters, physical observables in the phase space may be characterized by using power laws that many times yield into universal behavior. The application of such a formalism has been well accepted in the scientific community of nonlinear dynamics. Therefore I had in mind when writing this book was to bring together few of the research results in nonlinear systems using scaling formalism that could treated either in under-graduation as well as in the post graduation in the several exact programs but no earlier requirements were needed from the students unless the basic physics and mathematics. At the same time, the book must be original enough to contribute to the existing literature but with no excessive superposition of the topics already dealt with in other text books. The majority of the Chapters present a list of exercises. Some of them are analytic and others are numeric with few presenting some degree of computational complexity.

Here is an accurate and readable translation of a seminal article by Henri Poincare that is a classic in the study of dynamical systems popularly called chaos theory. In an effort to understand the stability of orbits in the solar system, Poincare applied a Hamiltonian formulation to the equations of planetary motion and studied these differential equations in the limited case of three bodies to arrive at properties of the equations' solutions, such as orbital resonances and horseshoe orbits. Poincare wrote for professional mathematicians and astronomers interested in celestial mechanics and differential equations. Contemporary historians of math or science and researchers in dynamical systems and planetary motion with an interest in the origin or history of their field will find his work fascinating.