Stretches across for chaos
In 2001, Kennedy et al. [Amer. Math. Mon. 108, 411-423 (2001) and Trans. Amer. Math. Soc. 353, 2513-2530 (2001)] showed a chaos lemma and stated the pseudoconjecture "stretches across implies existence of an invariant set." In this paper, we give a suitable definition of stretches across i...
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| Published in | Chaos (Woodbury, N.Y.) Vol. 29; no. 5; p. 053127 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
United States
01.05.2019
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| Online Access | Get more information |
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| Summary: | In 2001, Kennedy et al. [Amer. Math. Mon. 108, 411-423 (2001) and Trans. Amer. Math. Soc. 353, 2513-2530 (2001)] showed a chaos lemma and stated the pseudoconjecture "stretches across implies existence of an invariant set." In this paper, we give a suitable definition of stretches across in topological sense so that the conjecture has an affirmative answer. More precisely, we show that there must be an orbit through a sequence of stretches across. In particular, a closed loop of stretches across implies existence of a periodic orbit. We also give the geometric meaning of stretches across and its relation with the global implicit function theorem. |
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| ISSN: | 1089-7682 |
| DOI: | 10.1063/1.5091451 |