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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 |
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.
Keywords:
Good deformations,
classification of multiple singularities,
Euler-Jacobi formula,
Grothendieck residue,
planar differential systems.
Mathematics Subject Classification:
Primary: 34C05, 13H15, 14B07; Secondary: 37C10, 37C15.
Citation:
Zalman Balanov, Yakov Krasnov. On good deformations of $ A_m $-singularities. Discrete and Continuous Dynamical Systems - S,
2019, 12
(7)
: 1851-1866.
doi: 10.3934/dcdss.2019122
![]()
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. |
| [2] |
M. J. Álvarez, A. 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. |
| [3] |
M. J. Álvarez, A. 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. |
| [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. |
| [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.
|
| [6] |
V. I. Arnold,
Singularities, bifurcations, and catastrophes, Usp. Fiz. Nauk, 141 (1983), 569-590.
doi: 10.3367/UFNr.0141.198312a.0569. |
| [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. |
| [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). |
| [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. |
| [10] |
I. Bendixson,
Sur les courbes définies par des équations différentielles, Acta Math., 24 (1901), 1-88.
doi: 10.1007/BF02403068. |
| [11] |
A. N. Berlinskii,
On the behavior of the integral curves of a differential equation, Izv. Vysh. Uchebn. Zaved. Matematika, 1960 (1960), 3-18.
|
| [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. |
| [13] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006. |
| [14] |
E. B. Dynkin,
The structure of semi-simple algebras, Uspehi Matem. Nauk (N.S.), 2 (1947), 59-127.
|
| [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. |
| [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.
|
| [17] |
O. A. Gelfond and A. G. Khovanskii,
Toric geometry and Grothendieck residues, Mosc. Math. J., 2 (2002), 99-112.
|
| [18] |
P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, New York, 1978. |
| [19] |
D. M. Grobman,
Homeomorphisms of systems of differential equations, Dokl. Akad. Nauk SSSR, 128 (1959), 880-881.
|
| [20] |
J. Gross and J. Yellen, Graph Theory and Its Applications, Chapman & Hall/CRC, Boca Raton, FL, 2006. |
| [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. |
| [22] |
Y. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations, Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008. |
| [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. |
| [24] |
M. G. Soares, Lectures on Point Residues, Pontificia Universidad Católica del Perú, Lima, 2002. |
| [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. |
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. |
| [2] |
M. J. Álvarez, A. 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. |
| [3] |
M. J. Álvarez, A. 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. |
| [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. |
| [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.
|
| [6] |
V. I. Arnold,
Singularities, bifurcations, and catastrophes, Usp. Fiz. Nauk, 141 (1983), 569-590.
doi: 10.3367/UFNr.0141.198312a.0569. |
| [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. |
| [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). |
| [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. |
| [10] |
I. Bendixson,
Sur les courbes définies par des équations différentielles, Acta Math., 24 (1901), 1-88.
doi: 10.1007/BF02403068. |
| [11] |
A. N. Berlinskii,
On the behavior of the integral curves of a differential equation, Izv. Vysh. Uchebn. Zaved. Matematika, 1960 (1960), 3-18.
|
| [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. |
| [13] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006. |
| [14] |
E. B. Dynkin,
The structure of semi-simple algebras, Uspehi Matem. Nauk (N.S.), 2 (1947), 59-127.
|
| [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. |
| [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.
|
| [17] |
O. A. Gelfond and A. G. Khovanskii,
Toric geometry and Grothendieck residues, Mosc. Math. J., 2 (2002), 99-112.
|
| [18] |
P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, New York, 1978. |
| [19] |
D. M. Grobman,
Homeomorphisms of systems of differential equations, Dokl. Akad. Nauk SSSR, 128 (1959), 880-881.
|
| [20] |
J. Gross and J. Yellen, Graph Theory and Its Applications, Chapman & Hall/CRC, Boca Raton, FL, 2006. |
| [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. |
| [22] |
Y. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations, Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008. |
| [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. |
| [24] |
M. G. Soares, Lectures on Point Residues, Pontificia Universidad Católica del Perú, Lima, 2002. |
| [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. |
Figure 2.
Singularity of type $A_m$
| $m < 2n+1$ | $m > 2n+1$ | |||
| $m=2k$ | $\mathrm{saddle}+\mathrm{focus}=\mathrm{cusp}$ | $\mathrm{saddle}+\mathrm{node}=\textrm{saddle-node}$ | ||
| $m=2k+1$ | $a < 0$ | $\mathrm{focus}+\mathrm{focus}=\mathrm{focus/center}$ | $a > 0$ | $\mathrm{saddle}+\mathrm{saddle}=\mathrm{saddle}$ |
| $a > 0$ | $\mathrm{saddle}+\mathrm{saddle}=\mathrm{saddle}$ | $a < 0$ | 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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