Higher order limit cycle bifurcations from non-degenerate centers

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Abstract

The computational problems which appear in the computation of the Poincaré–Liapunov constants and the determination of their functionally independent number has led in recent works to consider only the lowest terms of these constants. In this work we improve the results obtained in this direction for polynomials systems of the form x˙=-y+Pn(x,y),y˙=x+Qn(x,y), where Pn and Qn are a homogeneous polynomial of degree n. We use center bifurcation to estimate the cyclicity of a unique singular point of focus-center type for different values of n and compare with the results given by the conjecture presented in [15].

Section snippets

Introduction and statement of the main result

We consider planar polynomial system of the formx˙=-y+P(x,y),y˙=x+Q(x,y),where P,Q 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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      Taking a look at all analyzed systems, it is clear that we need new good examples to get higher lower bounds for the local cyclicity. The main difficulty is to know how to get them to ensure that only with developments of first-order it is enough to get the value originally conjectured by Giné ([19,20]) and recently updated in [21]. In the language of Żoła̧dek, see [32], this is equivalent to find systems with maximal codimension.

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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.

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