Far East Journal of Dynamical Systems
Volume 4, Issue 1, Pages 17 - 25
(June 2002)
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THE LIMIT SETS OF THE EULER METHOD WITH VARIABLE STEPSIZES
May-Ru Chen (Taiwan) and Ming-Chai Li (Taiwan)
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Abstract:
In
[Bull. Austral. Math. Soc. 55 (1997), 63-71], Garay
and Hofbauer considered approximation schemes with a
variable stepsize sequence approaching zero for
numerical solving ODEs and showed that the chain
recurrent set of the exact solution flow contains
numerical w-limit sets.
In this paper, we give an alternative proof of the
result for the Euler method by using the techniques
recently developed by Benaim and Hirsch in the study
of stochastic dynamics. As its applications, we show
that for stable systems, such as gradient-like,
Morse-Smale and Axiom A systems, and the systems
with hyperbolic chain recurrent sets, every Euler w-limit set is a fixed
point, a closed orbit, or is a basic set. |
| Keywords and phrases: w-limit set, chain recurrent set, Euler method,
ordinary differential equation, gradient-like system,
Morse-Smale system, Axiom A system. |
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