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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Period two implies chaos for a class of ODEs


Authors: Franco Obersnel and Pierpaolo Omari
Journal: Proc. Amer. Math. Soc. 135 (2007), 2055-2058
MSC (2000): Primary 34C25, 34A60
Published electronically: January 9, 2007
MathSciNet review: 2299480
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Abstract: We extend a result of J. Andres and K. Pastor concerning scalar time-periodic first order ordinary differential equations without uniqueness, by proving that the existence of just one subharmonic implies the existence of large sets of subharmonics of all given orders. Since these periodic solutions must coexist with complicated dynamics, we might paraphrase T. Y. Li and J. A. Yorke by loosely saying that in this setting even period two implies chaos. Similar results are obtained for a class of differential inclusions.


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Additional Information

Franco Obersnel
Affiliation: Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, Via A. Valerio 12/1, I-34127 Trieste, Italy
Email: obersnel@units.it

Pierpaolo Omari
Affiliation: Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, Via A. Valerio 12/1, I-34127 Trieste, Italy
Email: omari@units.it

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08700-X
PII: S 0002-9939(07)08700-X
Keywords: First order scalar ordinary differential equation, periodic solution, subharmonic solution, lower and upper solutions, differential inclusion
Received by editor(s): February 1, 2006
Received by editor(s) in revised form: February 28, 2006
Published electronically: January 9, 2007
Additional Notes: The first author acknowledges the support of G.N.A.M.P.A., in the setting of the project “Soluzioni periodiche di equazioni differenziali ordinarie”.
The second author acknowledges the support of M.I.U.R, in the setting of the P.R.I.N. project “Equazioni differenziali ordinarie e applicazioni”.
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.