| eBook ISBN: | 978-1-4704-0149-8 |
| Product Code: | MEMO/119/570.E |
| List Price: | $42.00 |
| MAA Member Price: | $37.80 |
| AMS Member Price: | $25.20 |
| eBook ISBN: | 978-1-4704-0149-8 |
| Product Code: | MEMO/119/570.E |
| List Price: | $42.00 |
| MAA Member Price: | $37.80 |
| AMS Member Price: | $25.20 |
-
Book DetailsMemoirs of the American Mathematical SocietyVolume: 119; 1996; 79 ppMSC: Primary 34; 58; 65
One-step discretizations of order \(p\) and step size \(e\) of autonomous ordinary differential equations can be viewed as time-\(e\) maps of a certain first order ordinary differential equation that is a rapidly forced nonautonomous system.
Fiedler and Scheurle study the behavior of a homoclinic orbit for \(e = 0\), under discretization. Under generic assumptions they show that this orbit becomes transverse for positive \(e\). Likewise, the region where complicated, “chaotic” dynamics prevail is under certain conditions estimated to be exponentially small.
These results are illustrated by high precision numerical experiments. The experiments show that, due to exponential smallness, homoclinic transversality is already practically invisible under normal circumstances, for only moderately small discretization steps.
ReadershipResearch mathematicians.
-
Table of Contents
-
Chapters
-
1. Introduction and main results
-
2. Discretization and rapid forcing
-
3. Exponential smallness
-
4. Genericity of positive splitting
-
5. Estimating the chaotic wedge
-
6. Numerical experiments
-
7. Discussion
-
8. Appendix
-
-
RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Requests
One-step discretizations of order \(p\) and step size \(e\) of autonomous ordinary differential equations can be viewed as time-\(e\) maps of a certain first order ordinary differential equation that is a rapidly forced nonautonomous system.
Fiedler and Scheurle study the behavior of a homoclinic orbit for \(e = 0\), under discretization. Under generic assumptions they show that this orbit becomes transverse for positive \(e\). Likewise, the region where complicated, “chaotic” dynamics prevail is under certain conditions estimated to be exponentially small.
These results are illustrated by high precision numerical experiments. The experiments show that, due to exponential smallness, homoclinic transversality is already practically invisible under normal circumstances, for only moderately small discretization steps.
Research mathematicians.
-
Chapters
-
1. Introduction and main results
-
2. Discretization and rapid forcing
-
3. Exponential smallness
-
4. Genericity of positive splitting
-
5. Estimating the chaotic wedge
-
6. Numerical experiments
-
7. Discussion
-
8. Appendix
