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Invariants of binary differential equations

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Abstract

In this paper, we study binary differential equations a(x, y)dy 2 + 2b(x, y) dx dy + c(x, y)dx 2 = 0, where a, b, and c are real analytic functions. Following the geometric approach of Bruce and Tari in their work on multiplicity of implicit differential equations, we introduce a definition of the index for this class of equations that coincides with the classical Hopf’s definition for positive binary differential equations. Our results also apply to implicit differential equations F(x, y, p) = 0, where F is an analytic function, p = dy/dx, F p  = 0, and F pp  ≠ 0 at the singular point. For these equations, we relate the index of the equation at the singular point with the index of the gradient of F and index of the 1-form ω = dy − pdx defined on the singular surface F = 0.

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References

  1. M. A. Aguilar, J. A. Seade, and A. Verjovsky, Indices of vector fields and topological invariants of real analytic singularities. J. Reine Angew. Math. 504 (1998), 159–176.

    MATH  MathSciNet  Google Scholar 

  2. J. Bochnak, M. Coste, and M.-F. Roy, Real algebraic geometry. Springer-Verlag, Berlin (1998).

    MATH  Google Scholar 

  3. J. W. Bruce, On families of symmetric matrices. Moscow Math. J. 3 (2003), 335–360.

    MATH  MathSciNet  Google Scholar 

  4. J. W. Bruce and F. Tari, On binary differential equations. Nonlinearity 8 (1995), 255–271.

    Article  MATH  MathSciNet  Google Scholar 

  5. _____, Implicit differential equations from the singularity theory viewpoint. Singularities and differential equations. Banach Center Publ. 33 (1996), 23–38.

    MathSciNet  Google Scholar 

  6. _____, On the multiplicity of implicit differential equations. J. Differential Equations 148 (1998), 122–147.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. W. Bruce and N. P. Kirk, Generic projections of stable mappings. Bull. London Math. Soc. 32 (2000), No. 6, 718–728.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Cibrario, Sulla reduzione a forma delle equationi lineari alle derviate parziale di secondo ordine di tipo misto. Accad. Sci. Lettere, Inst. Lombardo Redicconti 65 (1932), 889–906.

    MATH  Google Scholar 

  9. A. Cima, A. Gasull, and J. Torregrosa, On the relation between index and multiplicity. J. London Math. Soc. (2), 57 (1998), 757–768.

    Article  MathSciNet  Google Scholar 

  10. M. Coste, An introduction to o-minimal geometry. Dottorato di Ricerca in Matermatica, Dip. Mat. Univ. Pisa, Instituti Editoriali e Poligrafici Internazionali (2000).

  11. L. Dara, Singularités génériques des équations différentielles multiformes. Bol. Soc. Brasil. Math. 6 (1975), 95–128.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. A. Davydov, Normal forms of differential equations unresolved with respect to derivatives in a neighborhood of its singular point. Funct. Anal. Appl. 19 (1985), 1–10.

    Article  MathSciNet  Google Scholar 

  13. _____, Qualitative control theory. Transl. Math. Monogr. 142 (1994).

  14. A. A. Davydov and L. Ortiz-Bobadilla, Smooth normal forms of folded elementary singular points. J. Dynam. Control Systems 1 (1995), No. 4, 463–482.

    Article  MATH  MathSciNet  Google Scholar 

  15. J. N. Damon, The unfolding and determinacy theorems for subgroups of \(\mathcal{A}\) and \(\mathcal{K}\). Mem. Amer. Math. Soc. 50 (1984), No. 306.

    Google Scholar 

  16. W. Ebeling and S. M. Gusein-Zade, Indices of 1-forms on an isolated complete intersection singularity. Mosc. Math. J. 3 (2003), 439–455.

    MATH  MathSciNet  Google Scholar 

  17. D. Eisenbud and H. I. Levine, An algebraic formula for the degree of a C ∞ map-germs. Ann. Math. 106 (1977), 19–44.

    Article  MathSciNet  Google Scholar 

  18. T. Gaffney, The structure of TA(f), classification and an application to differential geometry. Proc. Symp. Pure Math. 40 (1983), 409–427.

    MathSciNet  Google Scholar 

  19. R. Garcia, C. Gutierrez, and J. Sotomayor, Structural stability of asymptotic lines on surfaces immersed in \( \mathbb{R}^3 \). Bull. Sci. Math. 123 (1999), 599–622.

    Article  MathSciNet  Google Scholar 

  20. V. Guíıñez, Positive quadratic differential forms and foliations with singularities on surfaces. Trans. Amer. Math. Soc. 309 (1988), 477–502.

    Article  MathSciNet  Google Scholar 

  21. C. Gutierrez and J. Sotomayor, Lines of curvature, umbilic points and Carathéodory conjecture. Resenhas 3 (1998), 291–322.

    MATH  MathSciNet  Google Scholar 

  22. H. Hopf, Differential geometry in the large. Lect. Notes Math. 1000 (1971).

  23. B. L. Keyfitz, Classification of one-state-variable bifurcation problems up to codimension seven. Dynam. Stability Systems 1 (1986), 1–41.

    MATH  MathSciNet  Google Scholar 

  24. G. M. Khimshiashvili, On the local degree of a smooth mapping. Tr. Mat. Inst. Razmadze Gruz. Akad. Nauk 64 (1980), 105–124.

    MathSciNet  Google Scholar 

  25. A. G. Kuzmin, On the behavior of the characteristics of equations of mixed type near the line of degeneracy. Differ. Uravn. 17 (1981), No. 11, 2052–2063.

    MathSciNet  Google Scholar 

  26. _____, Behavior of integral curve of the equation of first order near a branch of discriminant curve. Bull. Leningrad Univ. 7 (1981).

  27. _____, Nonclassical equations of mixed type and their applications in gas dynamics. Int. Ser. Numer. Math. 109 (1992).

  28. S. Lojasiewicz, Ensemble semi-analytiques. Notes I.H.E.S. (1965).

  29. J. Milnor, Topology from the differentiable viewpoint. Univ. Press Virginia (1965).

  30. _____, Singular points of complex hypersurfaces. Ann. Math. Stud. 61 (1968).

  31. A. Phakadze and A. A. Shestakov, On the classification of singular points of a first-order differential equation implicit in the derivative. Mat. Sb. 49 (1959), 3–12.

    MathSciNet  Google Scholar 

  32. M. H. Schwartz, Champs radiaux et préradiaux associés à une stratification. C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), 239–241.

    MATH  Google Scholar 

  33. F. Tari, Two-parameter families of implicit differential equations. Discrete Contin. Dyn. Syst. 13 (2005), 139–162.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos SP, Brazil

    L. S. Challapa

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  1. L. S. Challapa
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Correspondence to L. S. Challapa.

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This work was partially supported by Fapesp grant No. 02/09157-5.

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Challapa, L.S. Invariants of binary differential equations. J Dyn Control Syst 15, 157–176 (2009). https://doi.org/10.1007/s10883-009-9066-z

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  • DOI: https://doi.org/10.1007/s10883-009-9066-z

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