| Title:
|
On the limit cycle of the Liénard equation (English) |
| Author:
|
Odani, Kenzi |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
36 |
| Issue:
|
1 |
| Year:
|
2000 |
| Pages:
|
25-31 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the paper, we give an existence theorem of periodic solution for Liénard equation $\dot{x}=y-F(x)$, $\dot{y}=-g(x)$. As a result, we estimate the amplitude $\rho (\mu )$ (maximal $x$-value) of the limit cycle of the van der Pol equation $\dot{x}=y-\mu (x^3/3-x)$, $\dot{y}=-x$ from above by $\rho (\mu )<2.3439$ for every $\mu \ne 0$. The result is an improvement of the author’s previous estimation $\rho (\mu )<2.5425$. (English) |
| Keyword:
|
van der Pol equation |
| Keyword:
|
limit cycle |
| Keyword:
|
amplitude |
| MSC:
|
34C05 |
| idZBL:
|
Zbl 1048.34067 |
| idMR:
|
MR1751611 |
| . |
| Date available:
|
2008-06-06T22:25:05Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107715 |
| . |
| Reference:
|
[1] Alsholm P.: Existence of limit cycles for generalized Liénard equation.J. Math. Anal. Appl. 171 (1992), 242–255. MR 1192504 |
| Reference:
|
[2] Cartwright M. L.: Van der Pol’s equation for relaxation oscillation.In: Contributions to the Theory of Non-linear Oscillations II, S. Lefschetz, ed., Ann. of Math. Studies, vol. 29, Princeton Univ. Press, 1952, pp. 3–18. MR 0052617 |
| Reference:
|
[3] Giacomini H., Neukirch S.: On the number of limit cycles of Liénard equation.Physical Review E56 (1997), 3809-3813. MR 1476640 |
| Reference:
|
[4] van Horssen W. T.: A perturbation method based on integrating factors.SIAM J. Appl. Math. 59 (1999), 1427-1443. Zbl 0926.34043, MR 1692651 |
| Reference:
|
[5] Lefschetz S.: Differential Equations: Geometric Theory.2nd Ed., Interscience, 1963; reprint, Dover, New York, 1977. Zbl 0107.07101, MR 0153903 |
| Reference:
|
[6] Odani K.: The limit cycle of the van der Pol equation is not algebraic.J. Differential Equations 115 (1995), 146–152. Zbl 0816.34023, MR 1308609 |
| Reference:
|
[7] Odani K.: Existence of exactly $N$ periodic solutions for Liénard systems.Funkcialaj Ekvacioj 39 (1996), 217–234. Zbl 0864.34032, MR 1418722 |
| Reference:
|
[8] Odani K.: On the limit cycle of the van der Pol equation.In: Equadiff9 CD-ROM: Papers, Z. Došlá, J. Kuben, J. Vosmanský, eds., Masaryk Univ., Czech, 1998, pp. 229-235. |
| Reference:
|
[9] Ye Y.-Q., al.: Theory of Limit Cycles.Transl. of Math. Monographs, vol. 66, Amer. Math. Soc., 1986. (Eng. Transl.) Zbl 0588.34022, MR 0854278 |
| Reference:
|
[10] Zhang Z.-F., al.: Qualitative Theory of Differential Equations.Transl. of Math. Monographs, vol. 102, Amer. Math. Soc., 1992. (Eng. Transl.) Zbl 0779.34001, MR 1175631 |
| . |
