Memoirs of the American Mathematical Society 1996; 79 pp; softcover Volume: 119 ISBN10: 0821804685 ISBN13: 9780821804681 List Price: US$42 Individual Members: US$25.20 Institutional Members: US$33.60 Order Code: MEMO/119/570
 Onestep 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. Readership Research mathematicians. Table of Contents  Introduction and main results
 Discretization and rapid forcing
 Exponential smallness
 Genericity of positive splitting
 Estimating the chaotic wedge
 Numerical experiments
 Discussion
 Appendix
 References
