Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
van der Pol equation; limit cycle; amplitude
van der Pol equation; limit cycle; amplitude
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 $\rho (\mu )<2.5425$.
References:
[1] Alsholm P.: Existence of limit cycles for generalized Liénard equation. J. Math. Anal. Appl. 171 (1992), 242–255. MR 1192504
[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
[3] Giacomini H., Neukirch S.: On the number of limit cycles of Liénard equation. Physical Review E56 (1997), 3809-3813. MR 1476640
[4] van Horssen W. T.: A perturbation method based on integrating factors. SIAM J. Appl. Math. 59 (1999), 1427-1443. MR 1692651 | Zbl 0926.34043
[5] Lefschetz S.: Differential Equations: Geometric Theory. 2nd Ed., Interscience, 1963; reprint, Dover, New York, 1977. MR 0153903 | Zbl 0107.07101
[6] Odani K.: The limit cycle of the van der Pol equation is not algebraic. J. Differential Equations 115 (1995), 146–152. MR 1308609 | Zbl 0816.34023
[7] Odani K.: Existence of exactly $N$ periodic solutions for Liénard systems. Funkcialaj Ekvacioj 39 (1996), 217–234. MR 1418722 | Zbl 0864.34032
[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.
[9] Ye Y.-Q., al.: Theory of Limit Cycles. Transl. of Math. Monographs, vol. 66, Amer. Math. Soc., 1986. (Eng. Transl.) MR 0854278 | Zbl 0588.34022
[10] Zhang Z.-F., al.: Qualitative Theory of Differential Equations. Transl. of Math. Monographs, vol. 102, Amer. Math. Soc., 1992. (Eng. Transl.) MR 1175631 | Zbl 0779.34001
Search
Partner of
