Multiple Time Scale Dynamics

Multiple Time Scale Dynamics
Author: Christian Kuehn
Publisher: Springer
Total Pages: 816
Release: 2015-02-25
Genre: Mathematics
ISBN: 3319123165

This book provides an introduction to dynamical systems with multiple time scales. The approach it takes is to provide an overview of key areas, particularly topics that are less available in the introductory form. The broad range of topics included makes it accessible for students and researchers new to the field to gain a quick and thorough overview. The first of its kind, this book merges a wide variety of different mathematical techniques into a more unified framework. The book is highly illustrated with many examples and exercises and an extensive bibliography. The target audience of this book are senior undergraduates, graduate students as well as researchers interested in using the multiple time scale dynamics theory in nonlinear science, either from a theoretical or a mathematical modeling perspective.

Multiple-Time-Scale Dynamical Systems

Multiple-Time-Scale Dynamical Systems
Author: Christopher K.R.T. Jones
Publisher: Springer Science & Business Media
Total Pages: 278
Release: 2012-12-06
Genre: Mathematics
ISBN: 1461301173

Systems with sub-processes evolving on many different time scales are ubiquitous in applications: chemical reactions, electro-optical and neuro-biological systems, to name just a few. This volume contains papers that expose the state of the art in mathematical techniques for analyzing such systems. Recently developed geometric ideas are highlighted in this work that includes a theory of relaxation-oscillation phenomena in higher dimensional phase spaces. Subtle exponentially small effects result from singular perturbations implicit in certain multiple time scale systems. Their role in the slow motion of fronts, bifurcations, and jumping between invariant tori are all explored here. Neurobiology has played a particularly stimulating role in the development of these techniques and one paper is directed specifically at applying geometric singular perturbation theory to reveal the synchrony in networks of neural oscillators.

Multiple Time Scales

Multiple Time Scales
Author: Jeremiah U. Brackbill
Publisher: Academic Press
Total Pages: 457
Release: 2014-05-10
Genre: Mathematics
ISBN: 1483257568

Multiple Time Scales presents various numerical methods for solving multiple-time-scale problems. The selection first elaborates on considerations on solving problems with multiple scales; problems with different time scales; and nonlinear normal-mode initialization of numerical weather prediction models. Discussions focus on analysis of observations, nonlinear analysis, systems of ordinary differential equations, and numerical methods for problems with multiple scales. The text then examines the diffusion-synthetic acceleration of transport iterations, with application to a radiation hydrodynamics problem and implicit methods in combustion and chemical kinetics modeling. The publication ponders on molecular dynamics and Monte Carlo simulations of rare events; direct implicit plasma simulation; orbit averaging and subcycling in particle simulation of plasmas; and hybrid and collisional implicit plasma simulation models. Topics include basic moment method, electron subcycling, gyroaveraged particle simulation, and the electromagnetic direct implicit method. The selection is a valuable reference for researchers interested in pursuing further research on the use of numerical methods in solving multiple-time-scale problems.

Methods and Applications of Singular Perturbations

Methods and Applications of Singular Perturbations
Author: Ferdinand Verhulst
Publisher: Springer Science & Business Media
Total Pages: 332
Release: 2006-06-04
Genre: Mathematics
ISBN: 0387283137

Contains well-chosen examples and exercises A student-friendly introduction that follows a workbook type approach

Geometric Singular Perturbation Theory Beyond the Standard Form

Geometric Singular Perturbation Theory Beyond the Standard Form
Author: Martin Wechselberger
Publisher: Springer Nature
Total Pages: 143
Release: 2020-02-21
Genre: Mathematics
ISBN: 3030363996

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

Multi-Scale Dynamics Modeling of Brain Physiological Functions and Pathological Mechanisms

Multi-Scale Dynamics Modeling of Brain Physiological Functions and Pathological Mechanisms
Author: Ying Wu
Publisher: Frontiers Media SA
Total Pages: 120
Release: 2023-10-09
Genre: Science
ISBN: 2832535569

The mechanisms of brain physiological functions and pathological mechanisms are crucial for us to understand how the brain works in the normal function such as memory, information processing and attentional perception, or in pathological conditions such as epilepsy, Parkinson's disease, and Alzheimer's diseases. These brain physiological functions and pathological mechanisms generally involve multiple spatial scales of brains, ranging from micro molecules, cellular channels, and meso-scale neuronal networks to the brain regions. To comprehensively understand the neural mechanisms of brain physiological functions and pathological mechanisms, multiple-scale investigations are essential to carry, involving neuronal circuit modeling, neural field modeling, large-scale modeling, data-driven complex network modeling, etc.

Multiplicity of Time Scales in Complex Systems

Multiplicity of Time Scales in Complex Systems
Author: Bernhelm Booss
Publisher: Springer Nature
Total Pages: 477
Release: 2024
Genre: System theory
ISBN: 3031280490

Zusammenfassung: This highly interdisciplinary volume brings together a carefully curated set of case studies examining complex systems with multiple time scales (MTS) across a variety of fields: materials science, epidemiology, cell physiology, mathematics, climatology, energy transition planning, ecology, economics, sociology, history, and cultural studies. The book addresses the vast diversity of interacting processes underlying the behaviour of different complex systems, highlighting the multiplicity of characteristic time scales that are a common feature of many and showcases a rich variety of methodologies across disciplinary boundaries. Self-organizing, out-of-equilibrium, ever-evolving systems are ubiquitous in the natural and social world. Examples include the climate, ecosystems, living cells, epidemics, the human brain, and many socio-economic systems across history. Their dynamical behaviour poses great challenges in the pressing context of the climate crisis, since they may involve nonlinearities, feedback loops, and the emergence of spatial-temporal patterns, portrayed by resilience or instability, plasticity or rigidity; bifurcations, thresholds and tipping points; burst-in excitation or slow relaxation, and worlds of other asymptotic behaviour, hysteresis, and resistance to change. Chapters can be read individually by the reader with special interest in such behaviours of particular complex systems or in specific disciplinary perspectives. Read together, however, the case studies, opinion pieces, and meta-studies on MTS systems presented and analysed here combine to give the reader insights that are more than the sum of the book's individual chapters, as surprising similarities become apparent in seemingly disparate and unconnected systems. MTS systems call into question naïve perceptions of time and complexity, moving beyond conventional ways of description, analysis, understanding, modelling, numerical prediction, and prescription of the world around us. This edited collection presents new ways of forecasting, introduces new means of control, and - perhaps as the most demanding task - it singles out a sustainable description of an MTS system under observation, offering a more nuanced interpretation of the floods of quantitative data and images made available by high- and low-frequency measurement tools in our unprecedented era of information flows

Multi-Stage and Multi-Time Scale Feedback Control of Linear Systems with Applications to Fuel Cells

Multi-Stage and Multi-Time Scale Feedback Control of Linear Systems with Applications to Fuel Cells
Author: Verica Radisavljević-Gajić
Publisher: Springer
Total Pages: 220
Release: 2019-02-12
Genre: Technology & Engineering
ISBN: 3030103897

This book provides a comprehensive study of multi-stage and multi-time scale design of feedback controllers for linear dynamic systems. It examines different types of controllers as can be designed for different parts of the system (subsystems) using corresponding feedback gains obtained by performing calculations (design) only with subsystem (reduced-order) matrices.The advantages of the multi-stage/multi-time scale design are presented and conditions for implementation of these controllers are established. Complete derivations and corresponding design techniques are presented for two-stage/two-time-scale, three-stage/three-time scale, and four-stage/four-time-scale systems. The techniques developed have potential applications to a large number of real physical systems. The design techniques are demonstrated on examples of mathematical models of fuel cells, especially the proton exchange membrane fuel cell.

Nonlinear Dynamics and Chaos

Nonlinear Dynamics and Chaos
Author: J Hogan
Publisher: CRC Press
Total Pages: 370
Release: 2002-08-01
Genre: Mathematics
ISBN: 1420033832

Nonlinear dynamics has been successful in explaining complicated phenomena in well-defined low-dimensional systems. Now it is time to focus on real-life problems that are high-dimensional or ill-defined, for example, due to delay, spatial extent, stochasticity, or the limited nature of available data. How can one understand the dynamics of such sys

Numerical Continuation Methods for Dynamical Systems

Numerical Continuation Methods for Dynamical Systems
Author: Bernd Krauskopf
Publisher: Springer
Total Pages: 411
Release: 2007-11-06
Genre: Science
ISBN: 1402063563

Path following in combination with boundary value problem solvers has emerged as a continuing and strong influence in the development of dynamical systems theory and its application. It is widely acknowledged that the software package AUTO - developed by Eusebius J. Doedel about thirty years ago and further expanded and developed ever since - plays a central role in the brief history of numerical continuation. This book has been compiled on the occasion of Sebius Doedel's 60th birthday. Bringing together for the first time a large amount of material in a single, accessible source, it is hoped that the book will become the natural entry point for researchers in diverse disciplines who wish to learn what numerical continuation techniques can achieve. The book opens with a foreword by Herbert B. Keller and lecture notes by Sebius Doedel himself that introduce the basic concepts of numerical bifurcation analysis. The other chapters by leading experts discuss continuation for various types of systems and objects and showcase examples of how numerical bifurcation analysis can be used in concrete applications. Topics that are treated include: interactive continuation tools, higher-dimensional continuation, the computation of invariant manifolds, and continuation techniques for slow-fast systems, for symmetric Hamiltonian systems, for spatially extended systems and for systems with delay. Three chapters review physical applications: the dynamics of a SQUID, global bifurcations in laser systems, and dynamics and bifurcations in electronic circuits.