Implementation of a new algorithm of computation of the Poincaré–Liapunov constants

Giné, Jaume; Santallusia, Xavier

doi:10.1016/j.cam.2003.08.043

Cited by 28 publications

(12 citation statements)

References 16 publications

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“…where V 2k are the focal values which are polynomials in the parameters of system (3), see [9,12]. The first nonzero focal value is V 4 = −a 2 + a 1 b 2 .…”

Section: Proof Of Theoremmentioning

confidence: 99%

Center Conditions for Polynomial Liénard Systems

Giné

2016

Qual. Theory Dyn. Syst.

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In this paper we study the center problem for polynomial Liénard systems of degree n with damping of degree n.Computing the focal values we find the center conditions for such systems for n = 5 and using modular arithmetics and Gröbner bases for n = 6. We also give some center conditions for polynomial Liénard systems of degree n with damping of degree n.2010 Mathematics Subject Classification. Primary 34C05. Secondary 37C10.

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“…where V 2k are the focal values which are polynomials in the parameters of system (3), see [9,12]. The first nonzero focal value is V 4 = −a 2 + a 1 b 2 .…”

Section: Proof Of Theoremmentioning

confidence: 99%

Center Conditions for Polynomial Liénard Systems

Giné

2016

Qual. Theory Dyn. Syst.

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“…The values α j give the so-called Lyapunov quantities and can be computed in many other ways, see for instance [2,6,14,18,19,21,22,26]. When all α j = 0, the weak focus is a center; otherwise, if α k = 0, is the first nonzero α j , then it is said that the origin is a weak focus of order k. It is well known that k is the maximum number of limit cycles (isolated periodic orbits) that bifurcate from this type of points and that this amount of limit cycles is attained for some analytic perturbations.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Highest weak focus order for trigonometric Liénard equations

Gasull

Giné

Valls

2019

Annali di Matematica

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Given a planar analytic differential equation with a critical point which is a weak focus of order k, it is well known that at most k limit cycles can bifurcate from it. Moreover, in case of analytic Liénard differential equations this order can be computed as one half of the multiplicity of an associated planar analytic map. By using this approach, we can give an upper bound of the maximum order of the weak focus of pure trigonometric Liénard equations only in terms of the degrees of the involved trigonometric polynomials. Our result extends to this trigonometric Liénard case a similar result known for polynomial Liénard equations.

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“…The centers of linear polynomial differential systems with homogeneous polynomial nonlinearities of degree k > 3 are not classified, but there are many partial results for k = 4, 5, 6, 7, 9 see [3,4,12,20,21,22,23]. In general the huge amount of computations which are necessary for obtaining the complete classification becomes the center problem intractable computationally, see for instance [16] and references quoted there.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%