|
Periodic segment implies infinitely many periodic solutions
Author(s):
Waclaw
Marzantowicz;
Klaudiusz
Wójcik
Journal:
Proc. Amer. Math. Soc.
135
(2007),
2637-2647.
MSC (2000):
Primary 54H25, 37B35, 37B55
Posted:
March 21, 2007
MathSciNet review:
2302587
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
In this note we show that the existence of a periodic segment for a non-autonomous ODE with periodic coefficients implies the existence of infinitely many periodic solutions inside this segment provided that a sequence of Lefschetz numbers of iterations of an associated map is not constant. In the case when this sequence is bounded we have to impose a geometric condition on the segment to get solutions by use of symbolic dynamics.
References:
-
- [BB]
- I. K. Babienko, S. A. Bogatyi, Behaviour of the index of periodic points under iterations of mapping. Izv.-Akad.-Nauk-SSSR-Ser.-Mat., 55 No. 1, (1991), 3-31.
- [D]
- A. Dold, Fixed point indices of iterated maps. Invent. Math., 74, (1983), 419-435. MR 0724012 (85c:54077)
- [G]
- G. Graff, Minimal periods of maps of rational exterior spaces, Fundamenta Math., 163, (2000), 99-115. MR 1752098 (2001d:37018)
- [JM]
- J. Jezierski, W. Marzantowicz, Homotopy Methods in Topological Fixed and Periodic Point Theory. Series: Topological Fixed Point Theory and Its Applications, Vol. 3, Springer, (2005), XI, 319 p., ISBN: 1-4020-3930-1. MR 2189944 (2006i:55003)
- [MP]
- W. Marzantowicz, P. M. Przygodzki, Finding periodic points of a map by use of a
 -adic expansion. Discrete and Cont. Dyn. Sys., vol 5, (1999), 495-514. MR 1696325 (2000c:37024) - [PW]
- L. Pieniazek, K. Wójcik, Complicated dynamics in non-autonomous ODE's. Universitatis Iagellonicae Acta Mathematica XLI, (2003), 163-179. MR 2084760 (2005g:34101)
- [ShSu]
- M. Shub and P. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps, Topology, 13, (1974), 189-191. MR 0350782 (50:3274)
- [S1]
- R. Srzednicki, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations, Nonlinear Anal. 22, (1994), 707-737. MR 1270166 (95c:34076)
- [S2]
- R. Srzednicki, Wazewski method and the Conley index. in Handbook of Differential Equations vol 1, Edited by A. Canada, P. Drabek, A. Fonda, 591-684. MR 2166495 (2006j:37014)
- [SW]
- R. Srzednicki, K. Wójcik, A geometric method for detecting chaotic dynamics., J. Diff. Eq., 135, (1997), 66-82. MR 1434915 (98f:58140)
- [SWZ]
- R. Srzednicki, K. Wójcik, P. Zgliczynski, Fixed point results based on Wazewski method. in "Handbook of topological fixed point theory" Ed: R. Brown, M. Furi, L. Górniewicz, B. Jiang, 903-941, Springer, Dordrecht, (2005). MR 2171125 (2006g:37015)
- [Su]
- Zhi-Wei Sun, Combinatorial identities in dual sequences. European J. Combin. 24, (2003), no.6, 709-718. MR 1995582 (2004g:05017)
- [Wi]
- H. S. Wilf, Generatingfunctionology (2nd edition), Academic Press, 1994. MR 1277813 (95a:05002)
- [W]
- K. Wójcik, On detecting periodic solutions and chaos in the time periodically forced ODE's. Nonlinear Anal. TMA 45, (2001), 19-27. MR 1828064 (2003b:34092)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (2000):
54H25, 37B35, 37B55
Retrieve articles in all Journals with
MSC (2000):
54H25, 37B35, 37B55
Additional Information:
Waclaw
Marzantowicz
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz Universiy of Poznan, Umultowska 67, 61-614 Poznan, Poland
Klaudiusz
Wójcik
Affiliation:
PWSZ Nowy Sacz, Institute of Pedagogy, Ul. Chruslicka 6, 33-300 Nowy Sacz, Poland and Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
DOI:
10.1090/S0002-9939-07-08750-3
PII:
S 0002-9939(07)08750-3
Keywords:
Periodic solution,
periodic segment,
Lefschetz number,
fixed point index.
Received by editor(s):
January 2, 2006
Received by editor(s) in revised form:
April 7, 2006
Posted:
March 21, 2007
Additional Notes:
The first author's research was supported by KBN grant 2P03A 03929
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.
|
