November  2019, 12(7): 1851-1866. doi: 10.3934/dcdss.2019122

On good deformations of $ A_m $-singularities

1. 

Department of Mathematics, Xiangnan University, Chenzhou, Hunan 423000, China

2. 

Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, USA

3. 

Department of Mathematics, Bar-Ilan University, Ramat-Gan, 5290002, Israel

* Corresponding author: Zalman Balanov

Received  February 2018 Revised  April 2018 Published  December 2018

Fund Project: The first author is supported by NSF grant DMS-1413223. The first author is thankful for the support from the Gelbart Research Institute through Bar Ilan University (Israel)

The standard way to study a compound singularity is to decompose it into the simpler ones using either blow up techniques or appropriate deformations. Among deformations, one distinguishes between miniversal deformations (related to deformations of a basis of the local algebra of singularity) and good deformations (one-parameter deformations with simple singularities coalescing into a multiple one). In concrete settings, explicit construction of a good deformation is an art rather than a science. In this paper, we discuss some cases important from the application viewpoint when explicit good deformations can be constructed and effectively used. Our applications include: (a) an $ n $-dimensional Euler-Jacobi formula with simple and double roots, and (b) a simple approach to the known classification of phase portraits of planar differential systems around linearly non-zero equilibrium.

Citation: Zalman Balanov, Yakov Krasnov. On good deformations of $ A_m $-singularities. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1851-1866. doi: 10.3934/dcdss.2019122
References:
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M. J. ÁlvarezA. Gasull and R. Prohens, Configurations of critical points in complex polynomial differential equations, Nonlinear Anal., 71 (2009), 923-934.  doi: 10.1016/j.na.2008.11.018.  Google Scholar

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M. J. ÁlvarezA. Gasull and R. Prohens, Topological classification of polynomial complex differential equations with all the critical points of centre type, J. Difference Equ. Appl., 16 (2010), 411-423.  doi: 10.1080/10236190903232654.  Google Scholar

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F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006.  Google Scholar

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E. B. Dynkin, The structure of semi-simple algebras, Uspehi Matem. Nauk (N.S.), 2 (1947), 59-127.   Google Scholar

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M. Frommer, Die Integralkurven einer gewönlichen Differentialgleichungen erster Ordnung in der Umgebung rationaler Unbestimmtheitsstellen, Math. Annalen, 99 (1928), 222-272.  doi: 10.1007/BF01459096.  Google Scholar

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A. Gasull and J. Torregrosa, Euler-Jacobi formula for double points and applications to quadratic and cubic systems, Bulletin of the Belgian Mathematical Society Simon Stevin, 6 (1999), 337-346.   Google Scholar

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O. A. Gelfond and A. G. Khovanskii, Toric geometry and Grothendieck residues, Mosc. Math. J., 2 (2002), 99-112.   Google Scholar

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P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, New York, 1978.  Google Scholar

[19]

D. M. Grobman, Homeomorphisms of systems of differential equations, Dokl. Akad. Nauk SSSR, 128 (1959), 880-881.   Google Scholar

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J. Gross and J. Yellen, Graph Theory and Its Applications, Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar

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P. Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc., 11 (1960), 610-620.  doi: 10.1090/S0002-9939-1960-0121542-7.  Google Scholar

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Y. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations, Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008.  Google Scholar

[23]

M. A. Krasnoselsky and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Grundlehren Der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 263. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69409-7.  Google Scholar

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M. G. Soares, Lectures on Point Residues, Pontificia Universidad Católica del Perú, Lima, 2002.  Google Scholar

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H. Zoladek, Analytic ordinary differential equations and their local classification, in Handbook of Differential Equations: Ordinary Differential Equations, Elsevier/North-Holland, Amsterdam, 4 (2008), 593–687. doi: 10.1016/S1874-5725(08)80011-9.  Google Scholar

show all references

References:
[1]

L. A. Aizenberg and A. P. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis, Translations of Mathematical Monographs, 58. American Mathematical Society, Providence, RI, 1983.  Google Scholar

[2]

M. J. ÁlvarezA. Gasull and R. Prohens, Configurations of critical points in complex polynomial differential equations, Nonlinear Anal., 71 (2009), 923-934.  doi: 10.1016/j.na.2008.11.018.  Google Scholar

[3]

M. J. ÁlvarezA. Gasull and R. Prohens, Topological classification of polynomial complex differential equations with all the critical points of centre type, J. Difference Equ. Appl., 16 (2010), 411-423.  doi: 10.1080/10236190903232654.  Google Scholar

[4]

A. F. Andreev, Investigation of the behaviour of the integral curves of a system of two differential equations in the neighbourhood of a singular point, Trans. Amer. Math. Soc., 8 (1958), 183-207.  doi: 10.1090/trans2/008/07.  Google Scholar

[5]

V. I. Arnold, Normal forms for functions near degenerate critical points, the Weyl groups of $A_,D_k,E_k$ and Lagrangian singularities, Funktsional'nyi Analiz i Ego Prilozheniya, 6 (1972), 3-25.   Google Scholar

[6]

V. I. Arnold, Singularities, bifurcations, and catastrophes, Usp. Fiz. Nauk, 141 (1983), 569-590.  doi: 10.3367/UFNr.0141.198312a.0569.  Google Scholar

[7]

V. I. Arnold, V. Goryunov, O. Lyashko and V. Vasiliev, Singularities I. Local and global theory, Current Problems in Mathematics. Fundamental Directions, Vol. 6, 5–257, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988.  Google Scholar

[8]

V. I. Arnold, V. Goryunov, O. Lyashko and V. Vasiliev, Singularities Ⅱ. Classification and Applications, Encyclopedia of Math. Sciences, 39. Springer, 1992 (Russian edition 1989).  Google Scholar

[9]

V. I. Arnold, A. N. Varchenko and S. M. Gusein-Zade, Singularities of Differentiable Maps. Vol. Ⅰ, Monographs in Mathematics, 82. Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5154-5.  Google Scholar

[10]

I. Bendixson, Sur les courbes définies par des équations différentielles, Acta Math., 24 (1901), 1-88.  doi: 10.1007/BF02403068.  Google Scholar

[11]

A. N. Berlinskii, On the behavior of the integral curves of a differential equation, Izv. Vysh. Uchebn. Zaved. Matematika, 1960 (1960), 3-18.   Google Scholar

[12]

F. Dumortier, Singularities of vector fields on the plane, J. Differential Equations, 23 (1977), 53-106.  doi: 10.1016/0022-0396(77)90136-X.  Google Scholar

[13]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006.  Google Scholar

[14]

E. B. Dynkin, The structure of semi-simple algebras, Uspehi Matem. Nauk (N.S.), 2 (1947), 59-127.   Google Scholar

[15]

M. Frommer, Die Integralkurven einer gewönlichen Differentialgleichungen erster Ordnung in der Umgebung rationaler Unbestimmtheitsstellen, Math. Annalen, 99 (1928), 222-272.  doi: 10.1007/BF01459096.  Google Scholar

[16]

A. Gasull and J. Torregrosa, Euler-Jacobi formula for double points and applications to quadratic and cubic systems, Bulletin of the Belgian Mathematical Society Simon Stevin, 6 (1999), 337-346.   Google Scholar

[17]

O. A. Gelfond and A. G. Khovanskii, Toric geometry and Grothendieck residues, Mosc. Math. J., 2 (2002), 99-112.   Google Scholar

[18]

P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, New York, 1978.  Google Scholar

[19]

D. M. Grobman, Homeomorphisms of systems of differential equations, Dokl. Akad. Nauk SSSR, 128 (1959), 880-881.   Google Scholar

[20]

J. Gross and J. Yellen, Graph Theory and Its Applications, Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar

[21]

P. Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc., 11 (1960), 610-620.  doi: 10.1090/S0002-9939-1960-0121542-7.  Google Scholar

[22]

Y. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations, Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008.  Google Scholar

[23]

M. A. Krasnoselsky and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Grundlehren Der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 263. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69409-7.  Google Scholar

[24]

M. G. Soares, Lectures on Point Residues, Pontificia Universidad Católica del Perú, Lima, 2002.  Google Scholar

[25]

H. Zoladek, Analytic ordinary differential equations and their local classification, in Handbook of Differential Equations: Ordinary Differential Equations, Elsevier/North-Holland, Amsterdam, 4 (2008), 593–687. doi: 10.1016/S1874-5725(08)80011-9.  Google Scholar

Figure 1.  Geometric graphs for singularities of multiplicity four
Figure 2.  Singularity of type $A_m$
Table 1.   
$m < 2n+1$ $m > 2n+1$
$m=2k$ $\mathrm{saddle}+\mathrm{focus}=\mathrm{cusp}$ $\mathrm{saddle}+\mathrm{node}=\textrm{saddle-node}$
$m=2k+1$ $a < 0$ $\mathrm{focus}+\mathrm{focus}=\mathrm{focus/center}$ $a > 0$ $\mathrm{saddle}+\mathrm{saddle}=\mathrm{saddle}$
$a > 0$ $\mathrm{saddle}+\mathrm{saddle}=\mathrm{saddle}$ $a < 0$ n - even
$\mathrm{node}+\mathrm{node}=\mathrm{node}$
(both nodes have the same stability)
n - odd
$\mathrm{node}+\mathrm{node}=\mathrm{hyperbolic}+\mathrm{elliptic}$
(the nodes have different stability)
$m < 2n+1$ $m > 2n+1$
$m=2k$ $\mathrm{saddle}+\mathrm{focus}=\mathrm{cusp}$ $\mathrm{saddle}+\mathrm{node}=\textrm{saddle-node}$
$m=2k+1$ $a < 0$ $\mathrm{focus}+\mathrm{focus}=\mathrm{focus/center}$ $a > 0$ $\mathrm{saddle}+\mathrm{saddle}=\mathrm{saddle}$
$a > 0$ $\mathrm{saddle}+\mathrm{saddle}=\mathrm{saddle}$ $a < 0$ n - even
$\mathrm{node}+\mathrm{node}=\mathrm{node}$
(both nodes have the same stability)
n - odd
$\mathrm{node}+\mathrm{node}=\mathrm{hyperbolic}+\mathrm{elliptic}$
(the nodes have different stability)
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