Discretization of Homoclinic Orbits, Rapid Forcing and Invisible Chaos (Memoirs of the AMS) (Memoirs of the American Mathematical Society) - Softcover

Fiedler, Bernold; Scheurle, Jurgen

Softcover

ISBN 10: 0821804685 ISBN 13: 9780821804681

Publisher: American Mathematical Society, 1996

Synopsis

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About this title

Synopsis

One-step discretization 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 non autonomous 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.

Edition details

• Publisher: American Mathematical Society
• Publication date: 1996
• Language: English
• ISBN 10: 0821804685
• ISBN 13: 9780821804681
• Binding: Paperback
• Number of pages: 79