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 | References | Similar Articles | Additional Information

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

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.