Period two implies all periods for a class of ODEs: A multivalued map approach
Authors:
Jan Andres, Tomás Fürst and Karel Pastor
Journal:
Proc. Amer. Math. Soc. 135 (2007), 31873191
MSC (2000):
Primary 34A60, 34C25, 37E05, 47H04
Published electronically:
February 28, 2007
MathSciNet review:
2322749
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We present an elementary proof that, for a multivalued map with nonempty connected values and monotone margins, the existence of a periodic orbit of any order implies the existence of periodic orbits of all orders. This generalizes a very recent result of this type in terms of scalar ordinary differential equations without uniqueness, due to F. Obersnel and P. Omari, obtained by means of lower and upper solutions techniques.
 [AFJ]
J.
Andres, J.
Fišer, and L.
Jüttner, On a multivalued version of the Sharkovskii theorem
and its application to differential inclusions, SetValued Anal.
10 (2002), no. 1, 1–14. MR 1888453
(2002m:37057), http://dx.doi.org/10.1023/A:1014488216807
 [AG]
Jan
Andres and Lech
Górniewicz, Topological fixed point principles for boundary
value problems, Topological Fixed Point Theory and Its Applications,
vol. 1, Kluwer Academic Publishers, Dordrecht, 2003. MR 1998968
(2005a:47102)
 [AJP]
Jan
Andres, Libor
Jüttner, and Karel
Pastor, On a multivalued version of the Sharkovskii theorem and its
application to differential inclusions. II, SetValued Anal.
13 (2005), no. 1, 47–68. MR 2128697
(2006c:37018), http://dx.doi.org/10.1007/s112280048200z
 [AP]
Jan
Andres and Karel
Pastor, A version of Sharkovskii’s
theorem for differential equations, Proc. Amer.
Math. Soc. 133 (2005), no. 2, 449–453. MR 2093067
(2005e:34124), http://dx.doi.org/10.1090/S0002993904076270
 [AS]
J. Andres and P. Šnyrychová: Hyperchaos induced by multivalued maps, in preparation.
 [F]
A.
F. Filippov, Differential equations with discontinuous righthand
sides, Mathematics and its Applications (Soviet Series), vol. 18,
Kluwer Academic Publishers Group, Dordrecht, 1988. Translated from the
Russian. MR
1028776 (90i:34002)
 [LY]
Tien
Yien Li and James
A. Yorke, Period three implies chaos, Amer. Math. Monthly
82 (1975), no. 10, 985–992. MR 0385028
(52 #5898)
 [OO1]
Franco
Obersnel and Pierpaolo
Omari, Old and new results for first order periodic ODEs without
uniqueness: a comprehensive study by lower and upper solutions, Adv.
Nonlinear Stud. 4 (2004), no. 3, 323–376. MR 2079818
(2005g:34093)
 [OO2]
F. Obersnel and P. Omari: Period two implies chaos for a class of ODEs, Proc. Amer. Math. Soc., posted on January 9, 2007, PII: 50029937(07)08700X (to appear in print).
 [S]
A.
N. Sharkovskiĭ, Coexistence of cycles of a continuous map of
the line into itself, Proceedings of the Conference “Thirty
Years after Sharkovskiĭ’s Theorem: New Perspectives”
(Murcia, 1994), 1995, pp. 1263–1273. Translated from the Russian
[Ukrain.\ Mat.\ Zh.\ {16} (1964), no.\ 1, 61–71; MR0159905 (28
#3121)] by J. Tolosa. MR 1361914
(96j:58058), http://dx.doi.org/10.1142/S0218127495000934
 [AFJ]
 J. Andres, J. Fišer and L. Jüttner: On a multivalued version of the Sharkovskii theorem and its application to differential inclusions, SetValued Anal. 10 (2002), 114. MR 1888453 (2002m:37057)
 [AG]
 J. Andres and L. Górniewicz: Topological Fixed Point Principles for Boundary Value Problems, Kluwer, Dordrecht, 2003. MR 1998968 (2005a:47102)
 [AJP]
 J. Andres, L. Jüttner and K. Pastor: On a multivalued version of the Sharkovskii theorem and its application to differential inclusions, II, SetValued Anal. 13 (2005), 4768. MR 2128697 (2006c:37018)
 [AP]
 J. Andres and K. Pastor: A version of Sharkovskii's theorem for differential equations, Proc. Amer. Math. Soc, 133 (2005), 449453.MR 2093067 (2005e:34124)
 [AS]
 J. Andres and P. Šnyrychová: Hyperchaos induced by multivalued maps, in preparation.
 [F]
 A.F. Filippov: Differential Equations with Discontinuous RightHand Sides, Kluwer, Dordrecht, 1988.MR 1028776 (90i:34002)
 [LY]
 T.Y. Li and J.A. Yorke: Period three implies chaos, Amer. Math. Monthly 82 (1975), 985992.MR 0385028 (52:5898)
 [OO1]
 F. Obersnel and P. Omari: Old and new results for first order periodic ODEs without uniqueness: a comprehensive study by lower and upper solutions, Adv. Nonlin. Stud. 4 (2004), 323376. MR 2079818 (2005g:34093)
 [OO2]
 F. Obersnel and P. Omari: Period two implies chaos for a class of ODEs, Proc. Amer. Math. Soc., posted on January 9, 2007, PII: 50029937(07)08700X (to appear in print).
 [S]
 A.N. Sharkovskii: Coexistence of cycles of a continuous map of a line into itself, Ukrain. Math. J. 16(1964), 6171 (Russian); English translation: Int. J. Bifurc. Chaos. 5 (1995), 12631273. MR 1361914 (96j:58058)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
34A60,
34C25,
37E05,
47H04
Retrieve articles in all journals
with MSC (2000):
34A60,
34C25,
37E05,
47H04
Additional Information
Jan Andres
Affiliation:
Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, Tomkova 40, 779 00 OlomoucHejčín, Czech Republic
Email:
andres@inf.upol.cz
Tomás Fürst
Affiliation:
Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, Tomkova 40, 779 00 OlomoucHejčín, Czech Republic
Email:
tomas.furst@seznam.cz
Karel Pastor
Affiliation:
Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, Tomkova 40, 779 00 OlomoucHejčín, Czech Republic
Email:
pastor@inf.upol.cz
DOI:
http://dx.doi.org/10.1090/S0002993907088855
PII:
S 00029939(07)088855
Keywords:
Periodic orbits,
multivalued maps,
monotone margins,
Sharkovskii's theorem,
ordinary differential equations without uniqueness,
subharmonic solutions.
Received by editor(s):
June 14, 2006
Published electronically:
February 28, 2007
Additional Notes:
This work was supported by the Council of Czech Government (MSM 6198959214).
Communicated by:
Carmen C. Chicone
Article copyright:
© Copyright 2007
American Mathematical Society
