Multiple Time Scale Dynamics with Two Fast Variables and One Slow Variable

Multiple Time Scale Dynamics with Two Fast Variables and One Slow Variable
Author: Christian Kuehn
Publisher:
Total Pages: 0
Release: 2010
Genre:
ISBN:

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This thesis considers dynamical systems that have multiple time scales. The focus lies on systems with two fast variables and one slow variable. The twoparameter bifurcation structure of the FitzHugh-Nagumo (FHN) equation is analyzed in detail. A singular bifurcation diagram is constructed and invariant manifolds of the problem are computed. A boundary-value approach to compute slow manifolds of saddle-type is developed. Interactions of classical invariant manifolds and slow manifolds explain the exponentially small turning of a homoclinic bifurcation curve in parameter space. Mixed-mode oscillations and maximal canards are detected in the FHN equation. An asymptotic formula to find maximal canards is proved which is based on the first Lyapunov coefficient at a singular Hopf bifurcation.

Multiple Time Scale Dynamics

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

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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

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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.

Neuronal Dynamics

Neuronal Dynamics
Author: Wulfram Gerstner
Publisher: Cambridge University Press
Total Pages: 591
Release: 2014-07-24
Genre: Computers
ISBN: 1107060834

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This solid introduction uses the principles of physics and the tools of mathematics to approach fundamental questions of neuroscience.

Multiplicity of Time Scales in Complex Systems

Multiplicity of Time Scales in Complex Systems
Author: Bernhelm Booss
Publisher: Springer Nature
Total Pages: 514
Release: 2024
Genre: System theory
ISBN: 3031451058

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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

Phase Space Approach To Nuclear Dynamics - Proceedings Of The Topical Meeting

Phase Space Approach To Nuclear Dynamics - Proceedings Of The Topical Meeting
Author: Massimo Di Toro
Publisher: World Scientific
Total Pages: 814
Release: 1986-03-01
Genre:
ISBN: 9813201843

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This proceedings volume is devoted to the interplay of symmetry and perturbation theory, as well as to cognate fields such as integrable systems, normal forms, n-body dynamics and choreographies, geometry and symmetry of differential equations, and finite and infinite dimensional dynamical systems. The papers collected here provide an up-to-date overview of the research in the field, and have many leading scientists in the field among their authors, including: D Alekseevsky, S Benenti, H Broer, A Degasperis, M E Fels, T Gramchev, H Hanssmann, J Krashil'shchik, B Kruglikov, D Krupka, O Krupkova, S Lombardo, P Morando, O Morozov, N N Nekhoroshev, F Oliveri, P J Olver, J A Sanders, M A Teixeira, S Terracini, F Verhulst, P Winternitz, B Zhilinskii.

Structured Population Models in Biology and Epidemiology

Structured Population Models in Biology and Epidemiology
Author: Pierre Magal
Publisher: Springer
Total Pages: 314
Release: 2008-04-12
Genre: Mathematics
ISBN: 3540782737

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In this new century mankind faces ever more challenging environmental and publichealthproblems,suchaspollution,invasionbyexoticspecies,theem- gence of new diseases or the emergence of diseases into new regions (West Nile virus,SARS,Anthrax,etc.),andtheresurgenceofexistingdiseases(in?uenza, malaria, TB, HIV/AIDS, etc.). Mathematical models have been successfully used to study many biological, epidemiological and medical problems, and nonlinear and complex dynamics have been observed in all of those contexts. Mathematical studies have helped us not only to better understand these problems but also to ?nd solutions in some cases, such as the prediction and control of SARS outbreaks, understanding HIV infection, and the investi- tion of antibiotic-resistant infections in hospitals. Structuredpopulationmodelsdistinguishindividualsfromoneanother- cording to characteristics such as age, size, location, status, and movement, to determine the birth, growth and death rates, interaction with each other and with environment, infectivity, etc. The goal of structured population models is to understand how these characteristics a?ect the dynamics of these models and thus the outcomes and consequences of the biological and epidemiolo- cal processes. There is a very large and growing body of literature on these topics. This book deals with the recent and important advances in the study of structured population models in biology and epidemiology. There are six chapters in this book, written by leading researchers in these areas.

The Ecological Basis of Conservation

The Ecological Basis of Conservation
Author: Steward Pickett
Publisher: Springer Science & Business Media
Total Pages: 477
Release: 2012-12-06
Genre: Science
ISBN: 1461560039

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From its inception, the U.S. Department of the Interior has been charged with a conflicting mission. One set of statutes demands that the department must develop America's lands, that it get our trees, water, oil, and minerals out into the marketplace. Yet an opposing set of laws orders us to conserve these same resources, to preserve them for the long term and to consider the noncommodity values of our public landscape. That dichotomy, between rapid exploitation and long-term protection, demands what I see as the most significant policy departure of my tenure in office: the use of science-interdisciplinary science-as the primary basis for land management decisions. For more than a century, that has not been the case. Instead, we have managed this dichotomy by compartmentalizing the American landscape. Congress and my predecessors handled resource conflicts by drawing enclosures: "We'll create a national park here," they said, "and we'll put a wildlife refuge over there." Simple enough, as far as protection goes. And outside those protected areas, the message was equally simplistic: "Y'all come and get it. Have at it." The nature and the pace of the resource extraction was not at issue; if you could find it, it was yours.

Pursuit-Evasion Differential Games

Pursuit-Evasion Differential Games
Author: Y. Yavin
Publisher: Elsevier
Total Pages: 352
Release: 2014-06-28
Genre: Mathematics
ISBN: 1483295931

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Twenty papers are devoted to the treatment of a wide spectrum of problems in the theory and applications of dynamic games with the emphasis on pursuit-evasion differential games. The problem of capturability is thoroughly investigated, also the problem of noise-corrupted (state) measurements. Attention is given to aerial combat problems and their attendant modelling issues, such as variable speed of the combatants, the three-dimensionality of physical space, and the combat problem, i.e. problems related to 'role determination'.

Cooperation and Its Evolution

Cooperation and Its Evolution
Author: Kim Sterelny
Publisher: MIT Press
Total Pages: 587
Release: 2024-08-06
Genre: Science
ISBN: 0262552787

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Essays from a range of disciplinary perspectives show the central role that cooperation plays in structuring our world. This collection reports on the latest research on an increasingly pivotal issue for evolutionary biology: cooperation. The chapters are written from a variety of disciplinary perspectives and utilize research tools that range from empirical survey to conceptual modeling, reflecting the rich diversity of work in the field. They explore a wide taxonomic range, concentrating on bacteria, social insects, and, especially, humans. Part I ("Agents and Environments") investigates the connections of social cooperation in social organizations to the conditions that make cooperation profitable and stable, focusing on the interactions of agent, population, and environment. Part II ("Agents and Mechanisms") focuses on how proximate mechanisms emerge and operate in the evolutionary process and how they shape evolutionary trajectories. Throughout the book, certain themes emerge that demonstrate the ubiquity of questions regarding cooperation in evolutionary biology: the generation and division of the profits of cooperation; transitions in individuality; levels of selection, from gene to organism; and the "human cooperation explosion" that makes our own social behavior particularly puzzling from an evolutionary perspective. Bradford Books imprint