What every neuroscientist should know about the mathematical modeling of excitable cells. Combining empirical physiology and nonlinear dynamics, this text provides an introduction to the simulation and modeling of dynamic phenomena in cell biology and neuroscience. It introduces mathematical modeling techniques alongside cellular electrophysiology. Topics include membrane transport and diffusion, the biophysics of excitable membranes, the gating of voltage and ligand-gated ion channels, intracellular calcium signalling, and electrical bursting in neurons and other excitable cell types. It introduces mathematical modeling techniques such as ordinary differential equations, phase plane, and bifurcation analysis of single-compartment neuron models. With analytical and computational problem sets, this book is suitable for life sciences majors, in biology to neuroscience, with one year of calculus, as well as graduate students looking for a primer on membrane excitability and calcium signalling.

Nonlinear Structures & Systems, Volume 1: Proceedings of the 37th IMAC, A Conference and Exposition on Structural Dynamics, 2019, the first volume of eight from the Conference brings together contributions to this important area of research and engineering. The collection presents early findings and case studies on fundamental and applied aspects of Nonlinear Dynamics, including papers on: Nonlinear Reduced-order Modeling Jointed Structures: Identification, Mechanics, Dynamics Experimental Nonlinear Dynamics Nonlinear Model & Modal Interactions Nonlinear Damping Nonlinear Modeling & Simulation Nonlinearity & System Identification

Statistical Thermodynamics: An Engineering Approach covers in a practical, readily understandable manner the underlying meaning of entropy, temperature and other thermodynamic concepts, the foundations of quantum mechanics, and the physical basis of gas, liquid and solid phase properties. It presents simply the relationship between macroscopic and microscopic thermodynamics. In addition, the molecular basis of transport phenomena and chemical kinetics are explored, as are basic concepts in spectroscopy. Modern computational tools for solving thermodynamic problems are explored, and the student is assured that he or she will gain knowledge of practical usefulness. This essential text is suitable for mechanical or aerospace engineering graduate students who have a strong background in engineering thermodynamics, those entering advanced fields such as combustion, high temperature gas dynamics, environmental sciences, or materials processing, and those who wish to build a background for understanding advanced experimental diagnostic techniques in these or similar fields.

Nonlinear waves are pervasive in nature, but are often elusive when they are modelled and analysed. This book develops a natural approach to the problem based on phase modulation. It is both an elaboration of the use of phase modulation for the study of nonlinear waves and a compendium of background results in mathematics, such as Hamiltonian systems, symplectic geometry, conservation laws, Noether theory, Lagrangian field theory and analysis, all of which combine to generate the new theory of phase modulation. While the build-up of theory can be intensive, the resulting emergent partial differential equations are relatively simple. A key outcome of the theory is that the coefficients in the emergent modulation equations are universal and easy to calculate. This book gives several examples of the implications in the theory of fluid mechanics and points to a wide range of new applications.

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 reviews recent 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 4-volume compendium contains the verbatim hard copies of all color slides from the Chua Lecture Series presented at HP in Palo Alto, during the period from September 22 to November 24, 2015. Each lecture consists of 90 minutes, divided into a formal lecture, a discussion session, and an Encore of special trivia that the audience found mesmerizing.These lectures share some unique features of the classic Feynman Lectures on Physics, as much of the materials are presented in the unique style of the author, and the content is original as discovered or invented by the author himself. Unlike most technical books that suffer a notoriously short life span as their features could be superseded by superior models, this series of Chua lectures are intended to never be obsolete - many concepts and principles introduced are in fact new laws of nature, written in the language of sophomore-level mathematics, providing the foundation and the elan vital for initiating and nurturing future concepts and inventions.Volume I - covers everything that a researcher may want to know about memristors but is too afraid to ask.Volume II - shows that memristors can be either volatile or non-volatile, and effectively proving that synapses are non-volatile memristors, while action potentials are generated by locally-active memristors.Volume III - presents an overview of the fascinating phenomenon called chaos, while immersing the audience with the sights and sound of chaos from the Chua Circuit, invented in 1984 by Leon Chua, and has now become the standard textbook example of chaos exhibited by a real nonlinear electronic circuit, and not by computer simulations.Volume IV - surprises the audience with a new law of nature - dubbed the local activity principle, as discovered and proved mathematically in 1996 by Leon Chua. In particular, a Corollary of Chua's local activity theorem, dubbed the edge of chaos, is shown via insightful examples to be the originator of most complex phenomena, including intelligence, creativity, and deep learning. The edge of chaos is Mother Nature's tool for overcoming the tyranny of the second law of thermodynamics by providing an escape hatch for entropy to decrease over time. Indeed, the local activity principle which is profusely illustrated in the final volume, is widely recognized as a new law of thermodynamics, and is identified as the sine qua non of all complex phenomena, including life itself.Exclusive Access to the accompanying Video and Audio materials comes with the purchase of this book.

Mathematical models are used to simulate, and sometimes control, the behavior of physical and artificial processes such as the weather and very large-scale integration (VLSI) circuits. The increasing need for accuracy has led to the development of highly complex models. However, in the presence of limited computational, accuracy, and storage capabilities, model reduction (system approximation) is often necessary. Approximation of Large-Scale Dynamical Systems provides a comprehensive picture of model reduction, combining system theory with numerical linear algebra and computational considerations. It addresses the issue of model reduction and the resulting trade-offs between accuracy and complexity. Special attention is given to numerical aspects, simulation questions, and practical applications.

This concise textbook gathers together key concepts and modern results on the theory of holomorphic foliations with singularities, offering a compelling vision on how the notion of foliation, usually linked to real functions and manifolds, can have an important role in the holomorphic world, as shown by modern results from mathematicians as H. Cartan, K. Oka, T. Nishino, and M. Suzuki. The text starts with a gentle presentation of the classical notion of foliations, advancing to holomorphic foliations and then holomorphic foliations with singularities. The theory behind reduction of singularities is described in detail, as well the cases for dynamics of a local diffeomorphism and foliations on complex projective spaces. A final chapter brings recent questions in the field, as holomorphic flows on Stein spaces and transversely homogeneous holomorphic foliations, along with a list of open questions for further study and research. Selected exercises at the end of each chapter help the reader to grasp the theory. Graduate students in Mathematics with a special interest in the theory of foliations will especially benefit from this book, which can be used as supplementary reading in Singularity Theory courses, and as a resource for independent study on this vibrant field of research.

H-Infinity control originated from an effort to codify classical control methods, where one shapes frequency response functions to meet certain objectives. H-Infinity control underwent tremendous development in the 1980s and made considerable strides toward systematizing classical control. This book addresses the next major issue of how this extends to nonlinear systems. At the core of nonlinear control theory lie two partial differential equations (PDEs). One is a first-order evolution equation called the information state equation, which constitutes the dynamics of the controller. One can view this equation as a nonlinear dynamical system. Much of this volume is concerned with basic properties of this system, such as the nature of trajectories, stability, and, most important, how it leads to a general solution of the nonlinear H-Infinity control problem. The second PDE actually builds on a classical type of partial differential inequality (PDI) called a Bellman-Isaacs inequality. While the information state PDE determines the dynamics of the controller, the PDI determines the output of the controller. The authors explore the system theoretic significance of the PDI and present its gross structure. These equations are only a few years old and their study is an expanding area of research. This book also emphasizes the theory effecting computer solvability of the information state equation, which at the outset looks numerically intractable, but which surprisingly is in many cases tractable. For example, the theory shows that careful initialization has a major influence on computer solvability. The authors keep the book self-contained by using the appendices to help explain certain prerequisite material. The reader should have a basic knowledge of control theory, real analysis and differential equations, nonlinear operator theory, and nonlinear PDEs.

This volume is the proceedings of the 14th MSJ International Research Institute "Asymptotic Analysis and Singularity", which was held at Sendai, Japan in July 2005. The proceedings contain survey papers and original research papers on nonlinear partial differential equations, dynamical systems, calculus of variations and mathematical physics.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

This volume collects, in written form, eight plenary lectures and twenty-five selected contributions from invited and contributed lectures delivered at the International Workshop on Potential Theory 2004. The workshop was held at Shimane University, Matsue, Japan, from 23 to 28 August, 2004. The topic of the workshop was Potential Theory and its related fields. There were stimulus talks from classical potential theory to pluri-potential theory and probabilistic potential theory.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

This book provides a comprehensive presentation of classical and advanced topics in estimation and control of dynamical systems with an emphasis on stochastic control. Many aspects which are not easily found in a single text are provided, such as connections between control theory and mathematical finance, as well as differential games. The book is self-contained and prioritizes concepts rather than full rigor, targeting scientists who want to use control theory in their research in applied mathematics, engineering, economics, and management science. Examples and exercises are included throughout, which will be useful for PhD courses and graduate courses in general.Dr. Alain Bensoussan is Lars Magnus Ericsson Chair at UT Dallas and Director of the International Center for Decision and Risk Analysis which develops risk management research as it pertains to large-investment industrial projects that involve new technologies, applications and markets. He is also Chair Professor at City University Hong Kong.

The spectra of transfer operators associated to dynamical systems, when acting on suitable Banach spaces, contain key information about the ergodic properties of the systems. Focusing on expanding and hyperbolic maps, this book gives a self-contained account on the relation between zeroes of dynamical determinants, poles of dynamical zeta functions, and the discrete spectra of the transfer operators. In the hyperbolic case, the first key step consists in constructing a suitable Banach space of anisotropic distributions. The first part of the book is devoted to the easier case of expanding endomorphisms, showing how the (isotropic) function spaces relevant there can be studied via Paley-Littlewood decompositions, and allowing easier access to the construction of the anisotropic spaces which is performed in the second part. This is the first book describing the use of anisotropic spaces in dynamics. Aimed at researchers and graduate students, it presents results and techniques developed since the beginning of the twenty-first century.

This volume is the proceedings of the 14th MSJ International Research Institute "Asymptotic Analysis and Singularity", which was held at Sendai, Japan in July 2005. The proceedings contain survey papers and original research papers on nonlinear partial differential equations, dynamical systems, calculus of variations and mathematical physics.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

Many processes in materials science and engineering, such as the load deformation behaviour of certain structures, exhibit nonlinear characteristics. The computer simulation of such processes therefore requires a deep understanding of both the theoretical aspects of nonlinearity and the associated computational techniques. This book provides a complete set of exercises and solutions in the field of theoretical and computational nonlinear continuum mechanics and is the perfect companion to Nonlinear Continuum Mechanics for Finite Element Analysis, where the authors set out the theoretical foundations of the subject. It employs notation consistent with the theory book and serves as a great resource to students, researchers and those in industry interested in gaining confidence by practising through examples. Instructors of the subject will also find the book indispensable in aiding student learning.

With the rapid development of science and technology, the computer has become an important tool in many science fields. Particularly, symbolic computation, which is one of the most exciting and challenging areas. It has been applied in many sciences such as mathematics, physics, chemistry, biology, mechanics, engineering, etc., in particular, non-linear sciences ad complex sciences. Nowadays, many symbolic computation softwares have been used to deal with these problems. Up to now, there have existed many non-linear differential/difference systems related to solitons and chaos in the non-linear science field. In order to understand these complex physical phenomena, it is important to research some of their basic properties. Because of the complexity of these non-linear systems, with the symbolic computation, this new book presents important research on non-linear differential/difference systems, related to solitons and chaos as well as other non-linear sciences in views of constructive algorithms.

Non-linear analysis is a broad, interdisciplinary field characterised by a mixture of analysis, topology, and applications. Its concepts and techniques provide the tools for developing more realistic and accurate models for a variety of phenomena encountered in fields ranging from engineering and chemistry to economics and biology. This book presents recent and important research in the field.

This new book focuses on recent and significant research on non-linear, or chaotic behaviour which in real world systems has been reported in electronic circuits and communications systems, chemical reactions, biological behaviour. The applications include solitons, integrable systems, cellular automata, pattern formation, qualitative structure and bifurcation theory, onset of chaos and turbulence, analytic dynamics, and transport phenomena.

Non-linear, or chaotic behaviour in real world systems has been reported in electronic circuits and communications systems, chemical reactions, biological behaviour. Applications include solitons, integrable systems, cellular automata, pattern formation, qualitative structure and bifurcation theory, onset of chaos and turbulence, analytic dynamics, and transport phenomena. This book presents important new research in this dynamic field.

Nonlinear dynamics of complex processes is an active research field with large numbers of publications in basic research, and broad applications from diverse fields of science. Nonlinear dynamics as manifested by deterministic and stochastic evolution models of complex behavior has entered statistical physics, physical chemistry, biophysics, geophysics, astrophysics, theoretical ecology, semiconductor physics and -optics, etc. This field of research has induced a new terminology in science connected with new questions, problems, solutions and methods. New scenarios have emerged for spatio-temporal structures in dynamical systems far from equilibrium. Their analysis and possible control are intriguing and challenging aspects of the current research.The duality of fundamental and applied research is a focal point of its main attractivity and fascination. Basic topics and foundations are always linked to concrete and precise examples. Models and measurements of complex nonlinear processes evoke and provoke new fundamental questions that diversify and broaden the mathematical concepts and tools. In return, new mathematical approaches to modeling and analysis enlarge the scope and efficiency of applied research.

Although the mathematical theory of nonlinear waves and solitons has made great progress, its applications to concrete physical problems are rather poor, especially when compared with the classical theory of linear dispersive waves and nonlinear fluid motion. The Whitham method, which describes the combining action of the dispersive and nonlinear effects as modulations of periodic waves, is not widely used by applied mathematicians and physicists, though it provides a direct and natural way to treat various problems in nonlinear wave theory. Therefore it is topical to describe recent developments of the Whitham theory in a clear and simple form suitable for applications in various branches of physics.This book develops the techniques of the theory of nonlinear periodic waves at elementary level and in great pedagogical detail. It provides an introduction to a Whitham's theory of modulation in a form suitable for applications. The exposition is based on a thorough analysis of representative examples taken from fluid mechanics, nonlinear optics and plasma physics rather than on the formulation and study of a mathematical theory. Much attention is paid to physical motivations of the mathematical methods developed in the book. The main applications considered include the theory of collisionless shock waves in dispersive systems and the nonlinear theory of soliton formation in modulationally unstable systems. Exercises are provided to amplify the discussion of important topics such as singular perturbation theory, Riemann invariants, the finite gap integration method, and Whitham equations and their solutions.

This textbook presents the most efficient analytical techniques in the local bifurcation theory of vector fields. It is centered on the theory of normal forms and its applications, including interaction with symmetries.

The first part of the book reviews the center manifold reduction and introduces normal forms (with complete proofs). Basic bifurcations are studied together with bifurcations in the presence of symmetries. Special attention is given to examples with reversible vector fields, including the physical example given by the water waves. In this second edition, many problems with detailed solutions are added at the end of the first part (some systems being in infinite dimensions). The second part deals with the Couette-Taylor hydrodynamical stability problem, between concentric rotating cylinders. The spatial structure of various steady or unsteady solutions results directly from the analysis of the reduced system on a center manifold. In this part we also study bifurcations (simple here) from group orbits of solutions in an elementary way (avoiding heavy algebra). The third part analyzes bifurcations from time periodic solutions of autonomous vector fields. A normal form theory is developed, covering all cases, and emphasizing a partial Floquet reduction theory, which is applicable in infinite dimensions. Studies of period doubling as well as Arnold's resonance tongues are included in this part.

Here is an in-depth, up-to-date analysis of wave interactions for general systems of hyperbolic and viscous conservation laws. This self-contained study of shock waves explains the new wave phenomena from both a physical and a mathematical standpoint. The analysis is useful for the study of various physical situations, including nonlinear elasticity, magnetohydrodynamics, multiphase flows, combustion, and classical gas dynamics shocks. The central issue throughout the book is the understanding of nonlinear wave interactions. The book describes the qualitative theory of shock waves. It begins with the basics of the theory for scalar conservation law and Lax's solution of the Reimann problem. For hyperbolic conservation laws, the Glimm scheme and wave tracing techniques are presented and used to study the regularity and large-time behavior of solutions. Viscous nonlinear waves are studied via the recent approach to pointwise estimates.