18. Euler's Finite Difference Scheme and Chaos - Project Euclid
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18. Euler's Finite Difference Scheme and Chaos - Project Euclid
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Proc. Japan Acad., 58, Ser. A (1979)
78
18.
[Vol. 55 (A),
Euler’ s Finite Difference Scheme and Chaos By Masaya YAMAGUTI and Hiroshi VATAN0 Kyoto University
(Communicated by K6saku YOSIDA,
M. J. A.,
March 12, 1979)
1. Introduction. Despite the simple dynamical structure of scalar ordinary differential equations., the corresponding difference equations. (Euler’s scheme) sometimes exhibit a very complicated dynamical behavior. Such phenomena have recently been understood somewhat systematically from the viewpoint of "chaos" theory, which is in course of powerful development since the work of Li-Yorke [1]. In this short note we shall make clear the importance of the role which asymptotically stable equilibrium points play in the process of chaotic phenomena. In other words., it will be shown that a best-settled equilibrium point in the original differential equation is actually apt to turn into a source of chaos in the corresponding difference equation. Our research here was inspired by R. M. May’s example of du dt =u(-hu). 2, Notation and theorem, Let us consider scalar differential equations of the orm du (1) f(u), dt where f(u) is continuous in R We assume that (1) has at least two equilibrium points one o which is asymptotically stable. As is easily seen, this assumption reduces (after a linear transformation of the unknown i necessary) to the conditions for some > O, (*) f(O) f() 0 (*) (O
