| Title:
|
Uniqueness of limit cycles bounded by two invariant parabolas (English) |
| Author:
|
Sáez, Eduardo |
| Author:
|
Szántó, Iván |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
57 |
| Issue:
|
5 |
| Year:
|
2012 |
| Pages:
|
521-529 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we consider a class of cubic polynomial systems with two invariant parabolas and prove in the parameter space the existence of neighborhoods such that in one the system has a unique limit cycle and in the other the system has at most three limit cycles, bounded by the invariant parabolas. (English) |
| Keyword:
|
stability |
| Keyword:
|
limit cycles |
| Keyword:
|
center |
| Keyword:
|
bifurcation |
| Keyword:
|
Matlab |
| MSC:
|
34C05 |
| MSC:
|
37C75 |
| MSC:
|
37N25 |
| MSC:
|
58F14 |
| MSC:
|
58F21 |
| MSC:
|
92B05 |
| MSC:
|
92D25 |
| idZBL:
|
Zbl 1262.92003 |
| idMR:
|
MR2984617 |
| DOI:
|
10.1007/s10492-012-0030-y |
| . |
| Date available:
|
2012-08-19T22:06:53Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142914 |
| . |
| Reference:
|
[1] Burnside, W. S., Panton, A. W.: The Theory of Equations, Vol. 1.Dover Publications New York (1960). MR 0115987 |
| Reference:
|
[2] Chavarriga, J., Sáez, E., Szántó, I., Grau, M.: Coexistence of limit cycles and invariant algebraic curves on a Kukles system.Nonlinear Anal., Theory Methods Appl. 59 (2004), 673-693. MR 2096323 |
| Reference:
|
[3] Cherkas, L. A., Zhilevich, L. I.: The limit cycles of certain differential equations.Differ. Uravn. 8 (1972), 1207-1213 Russian. |
| Reference:
|
[4] Chicone, C.: Bifurcations of nonlinear oscillations and frequency entrainment near resonance.SIAM J. Math. Anal. 23 (1992), 1577-1608. Zbl 0765.58018, MR 1185642, 10.1137/0523087 |
| Reference:
|
[5] Christopher, C.: Quadratic systems having a parabola as an integral curve.Proc. R. Soc. Edinb., Sect. A 112 (1989), 113-134. Zbl 0677.34034, MR 1007539, 10.1017/S0308210500028195 |
| Reference:
|
[6] Guoren, D., Songlin, W.: Closed orbits and straight line invariants in $E_3$ systems.Acta Math. Sci. 9 (1989), 251-261 Chinese. |
| Reference:
|
[7] Lloyd, N. G., Pearson, J. M., Sáez, E., Szántó, I.: A cubic Kolmogorov system with six limit cycles.Comput. Math. Appl. 44 (2002), 445-455. Zbl 1210.34048, MR 1912841, 10.1016/S0898-1221(02)00161-X |
| Reference:
|
[8] : MATLAB: The Language of technical computing Using MATLAB (version 7.0).MatWorks Natwick (2004). |
| Reference:
|
[9] Sáez, E., Szántó, I.: A cubic system with a limit cycle bounded by two invariant parabolas.Differ. Equations Dyn. Syst. 17 (2009), 163-168. Zbl 1207.34038, MR 2550235, 10.1007/s12591-009-0012-z |
| Reference:
|
[10] Yang, X.: A survey of cubic systems.Ann. Differ. Equations 7 (1991), 323-363. Zbl 0747.34019, MR 1139341 |
| . |
