# American Institute of Mathematical Sciences

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:
 [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. 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.  Google Scholar [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.  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

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. 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.  Google Scholar [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.  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
Geometric graphs for singularities of multiplicity four
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)
 [1] Chang-Shou Lin, Lei Zhang. Classification of radial solutions to Liouville systems with singularities. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2617-2637. doi: 10.3934/dcds.2014.34.2617 [2] F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605 [3] Jaume Llibre, Claudia Valls. Analytic integrability of a class of planar polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2657-2661. doi: 10.3934/dcdsb.2015.20.2657 [4] Armengol Gasull, Hector Giacomini. Upper bounds for the number of limit cycles of some planar polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 217-229. doi: 10.3934/dcds.2010.27.217 [5] Jaume Llibre, Roland Rabanal. Center conditions for a class of planar rigid polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1075-1090. doi: 10.3934/dcds.2015.35.1075 [6] Shimin Li, Jaume Llibre. On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5885-5901. doi: 10.3934/dcdsb.2019111 [7] Yilei Tang, Long Wang, Xiang Zhang. Center of planar quintic quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2177-2191. doi: 10.3934/dcds.2015.35.2177 [8] Igor E. Shparlinski. On some dynamical systems in finite fields and residue rings. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 901-917. doi: 10.3934/dcds.2007.17.901 [9] Dongsheng Kang, Fen Yang. Semilinear elliptic systems involving multiple critical exponents and singularities in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4247-4263. doi: 10.3934/dcds.2012.32.4247 [10] Robert Roussarie. A topological study of planar vector field singularities. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5217-5245. doi: 10.3934/dcds.2020226 [11] Jacek Banasiak, Aleksandra Puchalska. Generalized network transport and Euler-Hille formula. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 1873-1893. doi: 10.3934/dcdsb.2018185 [12] José Natário. An elementary derivation of the Montgomery phase formula for the Euler top. Journal of Geometric Mechanics, 2010, 2 (1) : 113-118. doi: 10.3934/jgm.2010.2.113 [13] C. Alonso-González, M. I. Camacho, F. Cano. Topological classification of multiple saddle connections. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 395-414. doi: 10.3934/dcds.2006.15.395 [14] Francisco Crespo, Sebastián Ferrer. On the extended Euler system and the Jacobi and Weierstrass elliptic functions. Journal of Geometric Mechanics, 2015, 7 (2) : 151-168. doi: 10.3934/jgm.2015.7.151 [15] David McCaffrey. A representational formula for variational solutions to Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1205-1215. doi: 10.3934/cpaa.2012.11.1205 [16] Ulrike Kant, Werner M. Seiler. Singularities in the geometric theory of differential equations. Conference Publications, 2011, 2011 (Special) : 784-793. doi: 10.3934/proc.2011.2011.784 [17] Piermarco Cannarsa, Marco Mazzola, Carlo Sinestrari. Global propagation of singularities for time dependent Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4225-4239. doi: 10.3934/dcds.2015.35.4225 [18] Yuan Xu, Xin Jin, Saiwei Wang, Yang Tang. Optimal synchronization control of multiple euler-lagrange systems via event-triggered reinforcement learning. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020377 [19] Yeping Li, Jie Liao. Stability and $L^{p}$ convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1281-1302. doi: 10.3934/cpaa.2019062 [20] Xingwu Chen, Jaume Llibre, Weinian Zhang. Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3953-3965. doi: 10.3934/dcdsb.2017203

2019 Impact Factor: 1.233