Publicado

1992-01-01

Melnikov deviations and limit cycles for Lienard equations

Palabras clave:

Vector, Function, Parameter (en)

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Autores/as

  • J. Billeke Universidad Nacional de Colombia
  • H. Burgos Universidad Nacional de Colombia
  • M. Wallace Universidad Nacional de Colombia

In this paper we analyze the method of small parameter and we calculate explicitly the derivatives of order two and three with respect to the parameter in the case of a homoclinic orbit and in the case of periodic orbits. Also, we apply this method to the Lienard equation of degree five on the plane:

Referencias

CHOW S. and HALE J., Methods of Bifurcations Theory, Springer-Verlag, 1982.

GUCKENHEIMER J. and HOLMES P., Nonlinear Oscillations, Dynamical, and Bifurcations of Vector Fields, SpringerVerlag, 1983.

UNS A., de MELO W.and PUGH C. C., On Liénard’s Equation, Lectures Notes in Math. 597 (1977).

LLOYD N. G. and LYNCH S., Small-amplitude limit cycles of certain Liénard Systems, Proc. R. Soc. London A. 418, 199- 208 (1988).

LLOYD N. G., Liénard Systems with several limit cycles, Math. Proc. Camb. Phil. Soc. 102, 565-572 (1987).

MELNIKOV V. K., On the stability of the center for time periodic perturbations, Trans. Moscow. Math. Soc., 12, 1- 57 (1963)

RYCHKOV G. S., The maximal number of limit cycles of the system

y ̅= -x ,-x =y- Σ i^2= 0 a i x^(2i+1)

is equal to two. Differentsial'nye Uravneniya 11, 390-391 (1975); Differential Equations 11, 301-302 (1975)

ZENG XIAN-WU, An existence and uniqueness theorem for limit cycles of the Lienard equation, Acta Math. Sinica 21, 263-269(1978).

ZHANG ZHI-FEN and HE QIMIN, A sufficient condition for the existence of no more than n limit cycles for the Lienard equation, Acta Math. Sinica 25, 585-594 (1982).

ZHANG ZHI-FEN, Proof of the uniqueness theorems of limit cycles of generalized Lienard equation, Applicable Analysis 23, 63-76 (1986).

J. Billeke U. de Santiago de Chile, Depto. de Matemática, Casilla 307, Santiago de Chile.

H. Burgos U. de la Frontera, Depto. de Matemática, Casilla 54-D, Temuco, Chile. M. Wallace U. de Concepción, Depto. de Matematica, Casilla 3-C, Concepción, Chile.

Cómo citar

APA

Billeke, J. ., Burgos, H. y Wallace, M. (1991). Melnikov deviations and limit cycles for Lienard equations. Revista Colombiana de Matemáticas, 26(1-4), 1–24. https://revistas.unal.edu.co/index.php/recolma/article/view/94130

ACM

[1]
Billeke, J. , Burgos, H. y Wallace, M. 1991. Melnikov deviations and limit cycles for Lienard equations. Revista Colombiana de Matemáticas. 26, 1-4 (ene. 1991), 1–24.

ACS

(1)
Billeke, J. .; Burgos, H.; Wallace, M. Melnikov deviations and limit cycles for Lienard equations. rev.colomb.mat 1991, 26, 1-24.

ABNT

BILLEKE, J. .; BURGOS, H.; WALLACE, M. Melnikov deviations and limit cycles for Lienard equations. Revista Colombiana de Matemáticas, [S. l.], v. 26, n. 1-4, p. 1–24, 1991. Disponível em: https://revistas.unal.edu.co/index.php/recolma/article/view/94130. Acesso em: 9 mar. 2025.

Chicago

Billeke, J., H. Burgos, y M. Wallace. 1991. «Melnikov deviations and limit cycles for Lienard equations». Revista Colombiana De Matemáticas 26 (1-4):1-24. https://revistas.unal.edu.co/index.php/recolma/article/view/94130.

Harvard

Billeke, J. ., Burgos, H. y Wallace, M. (1991) «Melnikov deviations and limit cycles for Lienard equations», Revista Colombiana de Matemáticas, 26(1-4), pp. 1–24. Disponible en: https://revistas.unal.edu.co/index.php/recolma/article/view/94130 (Accedido: 9 marzo 2025).

IEEE

[1]
J. . Billeke, H. Burgos, y M. Wallace, «Melnikov deviations and limit cycles for Lienard equations», rev.colomb.mat, vol. 26, n.º 1-4, pp. 1–24, ene. 1991.

MLA

Billeke, J. ., H. Burgos, y M. Wallace. «Melnikov deviations and limit cycles for Lienard equations». Revista Colombiana de Matemáticas, vol. 26, n.º 1-4, enero de 1991, pp. 1-24, https://revistas.unal.edu.co/index.php/recolma/article/view/94130.

Turabian

Billeke, J., H. Burgos, y M. Wallace. «Melnikov deviations and limit cycles for Lienard equations». Revista Colombiana de Matemáticas 26, no. 1-4 (enero 1, 1991): 1–24. Accedido marzo 9, 2025. https://revistas.unal.edu.co/index.php/recolma/article/view/94130.

Vancouver

1.
Billeke J, Burgos H, Wallace M. Melnikov deviations and limit cycles for Lienard equations. rev.colomb.mat [Internet]. 1 de enero de 1991 [citado 9 de marzo de 2025];26(1-4):1-24. Disponible en: https://revistas.unal.edu.co/index.php/recolma/article/view/94130

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