**Proceedings of Symposia in Applied Mathematics**

Volume: 74;
2018;
224 pp;
Hardcover

MSC: Primary 37; 65; 35; 34;

**Print ISBN: 978-1-4704-2814-3
Product Code: PSAPM/74**

List Price: $110.00

AMS Member Price: $88.00

MAA Member Price: $99.00

**Electronic ISBN: 978-1-4704-4729-8
Product Code: PSAPM/74.E**

List Price: $110.00

AMS Member Price: $88.00

MAA Member Price: $99.00

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# Rigorous Numerics in Dynamics

Share this page *Edited by *
*Jan Bouwe van den Berg; Jean-Philippe Lessard*

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.

#### Readership

Graduate students and researchers interested in theoretical aspects and applications of numerical methods in dynamics.

# Table of Contents

## Rigorous Numerics in Dynamics

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Introduction to rigorous numerics in dynamics: General functional analytic setup and an example that forces chaos 110
- Validated numerics for equilibria of analytic vector fields: Invariant manifolds and connecting orbits 2736
- 1. Introduction 2736
- 2. Background: infinite multi-sequences 3342
- 3. The local problem: equilibria, stability, and local invariant manifolds 4554
- 4. The flow box problem: propagating and differentiating sets of initial conditions 6271
- 5. The global problem: connecting one local picture to another 7079
- 6. Acknowledgments 7483
- References 7483

- Continuation of solutions and studying delay differential equations via rigorous numerics 8190
- Computer-assisted bifurcation diagram validation and applications in materials science 123132
- Dynamics and chaos for maps and the Conley index 175184
- 1. Symbolic Dynamics and Topological Entropy 177186
- 2. Itineraries and Topological Semi-conjugacy 178187
- 3. Conley Index Theory and Surjectivity 180189
- 4. Combinatorial Outer Approximation 182191
- 5. Computational Conley Index Theory 184193
- 6. The Hénon Map: Sample Results 186195
- References 191200
- Index 193202

- Rigorous computational dynamics in the context of unknown nonlinearities 195204
- Back Cover Back Cover1226