Duke Mathematical Journal

Polynomial normal forms for vector fields on ℝ3

Jiazhong Yang

Abstract

The present paper is devoted to studying a class of smoothly (C) finitely determined vector fields on ℝ3. Given any such generic local system of the form $\dot{x}$=Ax+⋯, where A is a 3×3 matrix, we find the minimal possible number i(A) such that the vector field is i(A)-jet determined, and we find the number μ(A) of moduli in the C classification. We also give a list of the simplest normal forms, that is, polynomials of degree i(A) containing exactly μ(A) parameters.

Article information

Source
Duke Math. J., Volume 106, Number 1 (2001), 1-18.

Dates
First available in Project Euclid: 13 August 2004

https://projecteuclid.org/euclid.dmj/1092403887

Digital Object Identifier
doi:10.1215/S0012-7094-01-10611-X

Mathematical Reviews number (MathSciNet)
MR1810364

Zentralblatt MATH identifier
1020.34033

Citation

Yang, Jiazhong. Polynomial normal forms for vector fields on ℝ 3. Duke Math. J. 106 (2001), no. 1, 1--18. doi:10.1215/S0012-7094-01-10611-X. https://projecteuclid.org/euclid.dmj/1092403887

References

• \lccV. Arnold and Yu. Ilyashenko, “Ordinary differential equations” in Dynamical Systems, Vol. 1, Encyclopaedia Math. Sci. 1, Springer, Berlin, 1988, 1–148.
• \lccG. Belitskii, Equivalence and normal forms of germs of smooth mappings, Russian Math. Surveys 33 (1978), 107–177.
• ––––, Smooth equivalence of germs of $C^\infty$ of vector fields with one zero or a pair of pure imaginary eigenvalues, Funct. Anal. Appl. 20 (1986), 253–259.
• \lccA. Bruno, Local Methods in Nonlinear Differential Equations, Springer Ser. Soviet Math., Springer, Berlin, 1989.
• \lccF. Ichikawa, Finitely determined singularities of formal vector fields, Invent. Math. 66 (1982), 199–214.
• ––––, On finite determinacy of formal vector fields, Invent. Math. 70 (1982), 45–52.
• \lccF. Takens, Normal forms for certain singularities of vector fields, Ann. Inst. Fourier (Grenoble) 23 (1973), 163–195.
• \lccJ. Yang, Polynomial Normal Forms of Vector Fields, thesis, Technion-Israel Institute of Technology, 1997.
• ––––, Normal forms of quasi-strongly $1$-resonant vector fields, in preparation.
• \lccM. Zhitomirskii, Local classification of differential $1$-forms and vector fields, thesis, Kharkov University, 1983.