Quadratic Hamiltonian Vector Fields

“…(2,3,1) p + (1, 0, 1) q = 0, M 1 L = 0, 0,1,2 = = 0, 3 K 1 < 0(2, 1, 1) p + (1, 2, 1) q = 0, M = 0, 0,1,2 = = L = 0, 1 = 0, 3 K 1 < 0 Fig. 10 (2, 2, 2) p + (1, 0, 1) q = 0, M = 0, 0,1 = = 1 = 0, 2 < 0, L > 0, K < 0Fig.…”

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confidence: 98%

Geometry of quadratic differential systems in the neighborhood of infinity

Schlomiuk

Vulpe

2005

Journal of Differential Equations

117

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In this article we give a complete global classification of the class QS ess of planar, essentially quadratic differential systems (i.e. defined by relatively prime polynomials and whose points at infinity are not all singular), according to their topological behavior in the vicinity of infinity. This class depends on 12 parameters but due to the action of the affine group and re-scaling of time, the family actually depends on five parameters. Our classification theorem (Theorem 7.1) gives us a complete dictionary connecting very simple integer-valued invariants which encode the geometry of the systems in the vicinity of infinity, with algebraic invariants and comitants which are a powerful tool for computer algebra computations helpful in the route to obtain the full topological classification of the class QS of all quadratic differential systems.

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“…In short, applying our approach we know the bifurcation diagram of the studied system, providing an alternative proof to that of [4].…”

Section: Applicationsmentioning

confidence: 93%

“…The phase portraits of the quadratic Hamiltonian differential systems are classified in [4]. Several normal forms for quadratic Hamiltonian systems are provided, and some conditions are found for each one of them in order to distinguish different phase portraits.…”

Section: Applicationsmentioning

confidence: 99%

“…We notice that we shall apply Tame changes of variables to F λ,µ in some cases for which the degree of x or y is equal to one. If ade = 0, then the resultant of ∆ 2 (F λ,µ ) with respect to µ is ade(b 2 −4ad)(c 2 −4ae)(b 2 e− c 2 d) 4 . This case covers completely the families 1 , 2 , 3 and 4 of [7] and their phase portraits by distinguishing whether each factor is zero or not.…”

Section: Applicationsmentioning

confidence: 99%

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Detection of Special Curves Via the Double Resultant

Ferragut

García-Saldaña

Gasull

2015

Qual. Theory Dyn. Syst.

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Abstract. We introduce several applications of the use of the double resultant through some examples of computation of different nature: special level sets of rational first integrals for rational discrete dynamical systems; remarkable values of rational first integrals of polynomial vector fields; bifurcation values in phase portraits of polynomial vector fields; and the different topologies of the offset of curves.

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Quadratic Hamiltonian Vector Fields

Cited by 64 publications

References 0 publications

The Period Function for Hamiltonian Systems with Homogeneous Nonlinearities

The Period Function for Hamiltonian Systems with Homogeneous Nonlinearities

Geometry of quadratic differential systems in the neighborhood of infinity

Detection of Special Curves Via the Double Resultant

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