| Title:
|
Bautin bifurgation of a modified generalized Van der Pol-Mathieu equation (English) |
| Author:
|
Kadeřábek, Zdeněk |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
52 |
| Issue:
|
1 |
| Year:
|
2016 |
| Pages:
|
49-70 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The modified generalized Van der Pol-Mathieu equation is generalization of the equation that is investigated by authors Momeni et al. (2007), Veerman and Verhulst (2009) and Kadeřábek (2012). In this article the Bautin bifurcation of the autonomous system associated with the modified generalized Van der Pol-Mathieu equation has been proved. The existence of limit cycles is studied and the Lyapunov quantities of the autonomous system associated with the modified Van der Pol-Mathieu equation are computed. (English) |
| Keyword:
|
Van der Pol-Mathieu equation |
| Keyword:
|
periodic solutions |
| Keyword:
|
autonomous system |
| Keyword:
|
generalized Hopf bifurcation |
| Keyword:
|
Bautin bifurcation |
| Keyword:
|
averaging method |
| Keyword:
|
limit cycles |
| MSC:
|
34C05 |
| MSC:
|
34C23 |
| MSC:
|
34C25 |
| MSC:
|
34C29 |
| MSC:
|
34D05 |
| idZBL:
|
Zbl 06562208 |
| idMR:
|
MR3475112 |
| DOI:
|
10.5817/AM2016-1-49 |
| . |
| Date available:
|
2016-02-29T18:33:11Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144839 |
| . |
| Reference:
|
[1] Kadeřábek, Z.: The autonomous system derived from Van der Pol-Mathieu equation.Aplimat - J. Appl. Math., Slovak Univ. Tech., Vol. 5 (2), vol. 5, 2012, pp. 85–96. |
| Reference:
|
[2] Kalas, J., Kadeřábek, Z.: Periodic solutions of a generalized Van der Pol-Mathieu differential equation.Appl. Math. Comput. 234 (2014), 192–202. Zbl 1309.34068, MR 3190531, 10.1016/j.amc.2014.01.161 |
| Reference:
|
[3] Kuznetsov, N.V., Leonov, G.A.: Computation of Lyapunov quantities.Proceedings of the 6th EUROMECH Nonlinear Dynamics Conference, 2008, IPACS Electronic Library, pp. 1–10. |
| Reference:
|
[4] Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory.2nd ed., Springer-Verlag New York, 1998. Zbl 0914.58025, MR 1711790 |
| Reference:
|
[5] Momeni, I., Moslehi-Frad, M., Shukla, P.K.: A Van der Pol-Mathieu equation for the dynamics of dust grain charge in dusty plasmas.J. Phys. A: Math. Theor. 40 (2007), F473–F481. MR 2345462, 10.1088/1751-8113/40/24/F06 |
| Reference:
|
[6] Perko, L.: Differential Equations and Dynamical Systems.2nd ed., Springer, 1996. Zbl 0854.34001, MR 1418638, 10.1007/978-1-4684-0249-0 |
| Reference:
|
[7] Veerman, F., Verhulst, F.: Quasiperiodic phenomena in the Van der Pol-Mathieu equation.J. Sound Vibration 326 (1–2) (2009), 314–320. 10.1016/j.jsv.2009.04.040 |
| Reference:
|
[8] Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems.2nd ed., Springer, 2006. MR 1036522 |
| . |
