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Period two implies chaos for a class of ODEs
Author(s):
Franco
Obersnel;
Pierpaolo
Omari
Journal:
Proc. Amer. Math. Soc.
135
(2007),
2055-2058.
MSC (2000):
Primary 34C25, 34A60
Posted:
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.
References:
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- 1.
- J. Andres, J. Fišer and L. Jüttner, On a multivalued version of the Sharkovskii Theorem and its application to differential inclusions, Set-Valued Anal. 10 (2002), 1-14. MR 1888453 (2002m:37057)
- 2.
- J. Andres, L. Jüttner and K. Pastor, On a multivalued version of the Sharkovskii Theorem and its application to differential inclusions II, Set-Valued Anal. 13 (2005), 47-68.MR 2128697 (2006c:37018)
- 3.
- J. Andres and K. Pastor, A version of Sharkovskii Theorem for differential equations, Proc. Amer. Math. Soc. 133 (2005), 449-453. MR 2093067 (2005e:34124)
- 4.
- A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Dordrecht, 1988. MR 1028776 (90i:34002)
- 5.
- T.Y. Li and J.A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985-992. MR 0385028 (52:5898)
- 6.
- F. Obersnel and P. Omari, Old and new results for first order periodic ODEs without uniqueness: a comprehensive study by lower and upper solutions, Advanced Nonlinear Studies 4 (2004), 323-376.MR 2079818 (2005g:34093)
- 7.
- F. Obersnel and P. Omari, Period two implies any period for a class of differential inclusions, Quaderni Matematici Ser. II Univ. Trieste 575, 2006. (http://www.dmi.units.it/ pubblicazioni/Quaderni_Matematici/2006.html)
- 8.
- A. N. Sharkovskii, Coexistence of cycles of a continuous map of a line into itself (Russian), Ukrainian Math. J. 16 (1964), 61-71.MR 0159905 (28:3121)
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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:
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
Posted:
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
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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