Higher order limit cycle bifurcations from non-degenerate centers☆
Section snippets
Introduction and statement of the main result
We consider planar polynomial system of the formwhere are polynomials with real coefficients without constants and linear terms. It is well-known that the above system always has either a center or a fine focus at the origin. The center problem consists in distinguishing between a center and a focus at the origin of system (1).
From Poincaré [29] who defined the notion of center for a real system of differential equations in the plane, the center problem has been
The method
To find the bifurcations of limit cycles from a center we will use the method developed in [8]. Naïvely, we would expect the number of limit cycles to be estimated by one less the maximum codimension of a component of the center variety. When we take into account only the linear terms of the focal values the used method is the following. We choose a point on the center variety, and we linearize the focal values about this point. We would hope that the point chosen on a component of the center
The implementation
In this section we describe the implementation of the method to compute the leading terms of the focal values in order to obtain the lower bounds given in Theorem 2.
Consider a system of the form (1) where P and Q a homogeneous polynomials and with a center at the origin. We perturb these systems inside the class of homogeneous system of the same degree in order to compute the leading terms of each focal value.
Liapunov [25] established that the computation of focal values of system (1) can be
Acknowledgments
The author is grateful to Prof. Colin Christopher who kindly give him its implementation to do some computations of the present work and for their valuable remarks and comments. The author is also grateful to the referees for their valuable remarks which helped to improve the manuscript.
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The author is partially supported by a MICINN/FEDER Grant number MTM2011-22877 and by a Generalitat de Catalunya Grant number 2009SGR 381.