Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

The bifurcation set of the period function of the dehomogenized Loud's centers is bounded

Author(s): F. Mañosas; J. Villadelprat
Journal: Proc. Amer. Math. Soc. 136 (2008), 1631-1642.
MSC (2000): Primary 34C07, 34C23; Secondary 34C25
Posted: January 23, 2008
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: This paper is concerned with the behaviour of the period function of the quadratic reversible centers. In this context the interesting stratum is the family of the so-called Loud's dehomogenized systems, namely

\begin{displaymath} \left\{ \begin{array}{l} \dot x=-y+xy, [1pt] \dot y=x+Dx^2+Fy^2. \end{array}\right. \end{displaymath}

In this paper we show that the bifurcation set of the period function of these centers is contained in the rectangle $ K=(-7,2)\times(0,4).$ More concretely, we prove that if $ (D,F)\notin K$, then the period function of the center is monotonically increasing.

References:

1.
Changjian Liu, The monotonicity of the periods in some class of reversible quadratic centers, preprint (2006).

2.
C. Chicone, review in MathSciNet, ref. 94h:58072.

3.
R. Chouikha, Monotonicity of the period function for some planar differential systems. I. Conservative and quadratic systems, Appl. Math. 32 (2005), 305-325. MR 2213617 (2007d:34079)

4.
W.A. Coppel and L. Gavrilov, The period function of a Hamiltonian quadratic system, Differential Integral Equations 6 (1993), 1357-1365. MR 1235199 (94h:58072)

5.
A. Gasull, A. Guillamon and J. Villadelprat, The period function for second-order quadratic ODEs is monotone, Qual. Theory Dyn. Syst. 5 (2004), 201-224. MR 2129724 (2005k:34115)

6.
P. Mardešić, D. Marın and J. Villadelprat, On the time function of the Dulac map for fmailies of meromorphic vector fields, Nonlinearity 16 (2003), 855-881. MR 1975786 (2004k:37031)

7.
P. Mardešić, D. Marın and J. Villadelprat, The period function of reversible quadratic centers, J. Differential Equations 224 (2006), 120-171. MR 2220066 (2006m:34076)

8.
D. Marın and J. Villadelprat, On the return time function around monodromic polycycles, J. Differential Equations 228 (2006), 226-258. MR 2254430 (2007d:34061)

9.
F. Rothe, The periods of the Volterra-Lokta system, J. Reine Angew. Math. 355 (1985), 129-138. MR 772486 (86c:92026)

10.
F. Rothe, Remarks on periods of planar Hamiltonian systems, SIAM J. Math. Anal. 24 (1993), 129-154. MR 1199531 (93m:34058)

11.
R. Schaaf, Global behaviour of solution branches for some Neumann problems depending on one or several parameters, J. Reine Angew. Math. 346 (1984), 1-31. MR 727393 (85i:58035)

12.
R. Schaaf, A class of Hamiltonian systems with increasing periods, J. Reine Angew. Math. 363 (1985), 96-109. MR 814016 (87b:58029)

13.
J. Villadelprat, The period function of the generalized Lotka-Volterra centers, preprint (2006).

14.
J. Villadelprat, On the reversible quadratic centers with monotonic period function, Proc. Amer. Math. Soc. 135 (207), 2555-2565. MR 2302576

15.
J. Waldvogel, The period in the Lotka-Volterra system is monotonic, J. Math. Anal. Appl. 114 (1986), 178-184. MR 829122 (87j:92034)

16.
Y. Zhao, The monotonicity of period function for codimension four quadratic system $ Q_4,$ J. Differential Equations 185 (2002), 370-387. MR 1938124 (2003g:34057)

17.
Y. Zhao, The period function for quadratic integrable systems with cubic orbits, J. Math. Anal. Appl. 301 (2005), 295-312. MR 2105672 (2005h:34082)

18.
Y. Zhao, On the monotonicity of the period function of a quadratic system, Discrete Contin. Dyn. Syst. 13 (2005), 795-810. MR 2153144 (2006d:34072)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34C07, 34C23, 34C25

Retrieve articles in all Journals with MSC (2000): 34C07, 34C23, 34C25

Additional Information:

F. Mañosas
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

J. Villadelprat
Affiliation: Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain

DOI: 10.1090/S0002-9939-08-09131-4
PII: S 0002-9939(08)09131-4
Received by editor(s): October 18, 2006
Posted: January 23, 2008
Additional Notes: The authors were partially supported by the CONACIT through the grant 2005-SGR-00550 and by the DGES through the grant MTM-2005-06098-C02-1.
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2008, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google