×

zbMATH — the first resource for mathematics

Algebraic and topological classification of the homogeneous cubic vector fields in the plane. (English) Zbl 0711.34061
By using the classification of fourth order binary forms in the real domain, an algebraic classification of systems \(x'=P(x,y)\), \(y'=Q(x,y)\), P, Q homogeneous polynomials of degree 3 is obtained. After a study of phase portraits for the general case when P and Q have the same degree and no common factor, all possible phase portraits are obtained for degree three.
Reviewer: A.Halanay

MSC:
37G05 Normal forms for dynamical systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Argemi, J., Sur LES points singuliers multiples de systèmes dynamiques dans R2, Ann. mat. pura appl., 4, 35-70, (1968) · Zbl 0203.39802
[2] Andronov, A.A.; Leontovich, E.A.; Gordon, I.I.; Maier, A.L., Qualitative theory of second order dynamic systems, (1973), Wiley New York/Toronto · Zbl 0282.34022
[3] Cima, A., Indices of polynomial vector fields with applications, ()
[4] \scA. Cima and J. Llibre, Bounded polynomial vector fields, Universitat Autònoma de Barcelona, Trans. Amer. Math. Soc., to appear. · Zbl 0695.34028
[5] Date, T., Classification and analysis of two-dimensional real homogeneous quadratic differential equation systems, J. differential equations, 32, 311-334, (1979) · Zbl 0378.34014
[6] Date, T.; Iri, M., Canonical forms of real homogeneous quadratic transformations, J. math. anal. appl., 56, 650-682, (1976) · Zbl 0342.15009
[7] Dieudonné, J.A.; Carrell, J.B., Invariant theory, old and new, (1971), Academic Press New York/London · Zbl 0258.14011
[8] Gonzales, E.A., Generic properties of polynomial vector fields at infinity, Trans. amer. math. soc., 143, 201-222, (1969) · Zbl 0187.34401
[9] Gurevich, G., Foundations of the theory of algebraic invariants, (1964), Noordhoff Groningen · Zbl 0128.24601
[10] Sotomayor, J., Curvas definidas por equacoes diferenciais no plano, (1981), Instituto de Matematica Pura e Aplicada Rio de Janeiro
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.