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Perturbations of a Hamiltonian family of cubic vector fields

Published online by Cambridge University Press:  17 April 2009

A.M. Urbina ,
M. Cañas ,
G. León de la Barra  and
M. León de la Barra
A.M. Urbina
Affiliation:
Department of Mathematics, Universidad Técnica, Federico Santa Maria, Casilla 100-V, Valparaiso, Chile
M. Cañas
Affiliation:
Department of Mathematics, Universidad Técnica, Federico Santa Maria, Casilla 100-V, Valparaiso, Chile
G. León de la Barra
Affiliation:
Department of Mathematics, Universidad Técnica, Federico Santa Maria, Casilla 100-V, Valparaiso, Chile
M. León de la Barra
Affiliation:
Department of Mathematics, Universidad Técnica, Federico Santa Maria, Casilla 100-V, Valparaiso, Chile
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Abstract

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This paper is related with the configurations of limit cycles for cubic polynomial vector fields in two variables (χ3).

It is an open question to decide whether every limit cycle configuration in χ3 can be obtained by perturbation of a corresponding Hamiltonian configuration of centres and graphs.

In this work, by considering perturbations of the Hamiltonian vector field XH = (Hy, − Hx), where H(x, y) = [a(x + h)2 + by2 − 1] [a(x − h)2 + by2 − 1], we make a global analysis of the possible cases.

The vector field XH has three centres (C−, C+ and the origin) and two saddles. By means of quadratic perturbations the centres become fine foci where C−and C+ have the same type of stability but opposed to that one of the origin and infinity. Further introducing cubic perturbations changes the stability of C−, C+ and the cycle at infinity and generates limit cycles. Lastly extra linear terms change the stability of the origin and generate another limit cycle.

Finally, we analyse the rupture of saddle connection of the Hamiltonian field under perturbation, via Melnikov's integral, in order to complete the study of the global phase portrait and to consider the possibility of new limit cycles emerging from the Hamiltonian graph.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 44 , Issue 1 , August 1991 , pp. 139 - 147
Copyright
Copyright © Australian Mathematical Society 1991

References

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