Rigorous numerics in dynamics : AMS Short Course, Rigorous Numerics in Dynamics, January 4-5, 2016, Seattle, Washington / Jan Bouwe van den Berg, Jean-Philippe Lessard, editors.

Format:

Book

Contributor:

Berg, Jan Bouwe van den, 1973- editor.

Lessard, Jean-Philippe, 1978- editor.

Series:

Proceedings of symposia in applied mathematics ; Volume 74.

Proceedings of symposia in applied mathematics ; Volume 74

Language:

English

Subjects (All):

Nonlinear mechanics.

Topological dynamics.

Physical Description:

1 online resource (226 pages).

Edition:

1st ed.

Other Title:

AMS short course, rigorous numerics in dynamics

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2018]

Summary:

This volume is based on lectures delivered at the 2016 AMS Short Course "Rigorous Numerics in Dynamics", held January 4-5, 2016, in Seattle, Washington. Nonlinear dynamics shapes the world around us, from the harmonious movements of celestial bodies, via the swirling motions in fluid flows, to the complicated biochemistry in the living cell. Mathematically these phenomena are modeled by nonlinear dynamical systems, in the form of ODEs, PDEs and delay equations. The presence of nonlinearities complicates the analysis, and the difficulties are even greater for PDEs and delay equations, which are naturally defined on infinite dimensional function spaces. With the availability of powerful computers and sophisticated software, numerical simulations have quickly become the primary tool to study the models. However, while the pace of progress increases, one may ask: just how reliable are our computations? Even for finite dimensional ODEs, this question naturally arises if the system under study is chaotic, as small differences in initial conditions (such as those due to rounding errors in numerical computations) yield wildly diverging outcomes. These issues have motivated the development of the field of rigorous numerics in dynamics, which draws inspiration from ideas in scientific computing, numerical analysis and approximation theory. The articles included in this volume present novel techniques for the rigorous study of the dynamics of maps via the Conley-index theory; periodic orbits of delay differential equations via continuation methods; invariant manifolds and connecting orbits; the dynamics of models with unknown nonlinearities; and bifurcations diagrams.

Contents:

Cover

Title page

Contents

Preface

Introduction to rigorous numerics in dynamics: General functional analytic setup and an example that forces chaos

1. Introduction

2. The general setup

3. Example of a rigorous computation that forces chaos

4. The computational proof

References

Validated numerics for equilibria of analytic vector fields: Invariant manifolds and connecting orbits

2. Background: infinite multi-sequences

3. The local problem: equilibria, stability, and local invariant manifolds

4. The flow box problem: propagating and differentiating sets of initial conditions

5. The global problem: connecting one local picture to another

6. Acknowledgments

Continuation of solutions and studying delay differential equations via rigorous numerics

1. Introduction, Motivation and Examples

2. Rigorous Continuation of Solutions

Computer-assisted bifurcation diagram validation and applications in materials science

2. Two Examples from Materials Science

3. Branches and the Constructive Implicit Function Theorem

4. Direct Localization of Bifurcation Points

Index

Dynamics and chaos for maps and the Conley index

1. Symbolic Dynamics and Topological Entropy

2. Itineraries and Topological Semi-conjugacy

3. Conley Index Theory and Surjectivity

4. Combinatorial Outer Approximation

5. Computational Conley Index Theory

6. The Hénon Map: Sample Results

Rigorous computational dynamics in the context of unknown nonlinearities

2. Concepts from Dynamical Systems

3. Combinatorial Dynamics

4. Approximating Dynamics

5. Extracting Dynamics from Outer Approximations

6. Conley Index

Back Cover.

Notes:

Includes bibliographical references and index.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

1-4704-4729-0

OCLC:

1048780419