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May  2013, 33(5): 1773-1807. doi: 10.3934/dcds.2013.33.1773

Formal Poincaré-Dulac renormalization for holomorphic germs

Marco Abate 1, and  Jasmin Raissy 2,

1. 

Dipartimento di Matematica, Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, Italy
2. 

Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via Cozzi 53, 20125 Milano, Italy

Received  April 2012 Revised  July 2012 Published  December 2012

We shall describe an alternative approach to a general renormalization procedure for formal self-maps, originally suggested by Chen-Della Dora and Wang-Zheng-Peng, giving formal normal forms simpler than the classical Poincaré-Dulac normal form. As example of application we shall compute a complete list of normal forms for bi-dimensional superattracting germs with non-vanishing quadratic term; in most cases, our normal forms will be the simplest possible ones (in the sense of Wang-Zheng-Peng). We shall also discuss a few examples of renormalization of germs tangent to the identity, revealing interesting second-order resonance phenomena.
Keywords: Poincaré-Dulac normal form, superattracting germs, renormalization, tangent to the identity maps., formal transformation.
Mathematics Subject Classification: Primary: 37G05, 37F99, 32H5.
Citation: Marco Abate, Jasmin Raissy. Formal Poincaré-Dulac renormalization for holomorphic germs. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1773-1807. doi: 10.3934/dcds.2013.33.1773
References:
[1]

M. Abate, Holomorphic classification of $2$-dimensional quadratic maps tangent to the identity, Sūkikenkyūsho Kōkyūroku, 1447 (2005), 1-14. Google Scholar

[2]

M. Abate, Discrete holomorphic local dynamical systems, in "Holomorphic Dynamical Systems" (G.Gentili, J. Guénot and G. Patrizio, eds.), Lect. Notes in Math. 1998, Springer, Berlin, 2010, pp. 1-55.  Google Scholar

[3]

M. Abate and F. Tovena, Formal classification of holomorphic maps tangent to the identity, Discrete Contin. Dyn. Syst. Suppl (2005), 1-10.  Google Scholar

[4]

M. Abate and F. Tovena, Poincaré-Bendixson theorems for meromorphic connections and holomorphic homogeneous vector fields, J. Differential Equations, 251 (2011), 2612-2684. doi: 10.1016/j.jde.2011.05.031.  Google Scholar

[5]

A. Algaba, E. Freire and E. Gamero, Hypernormal forms for equilibria of vector fields. Codimension one linear degeneracies, Rocky Mountain J. Math. 29 (1999), 13-45. doi: 10.1216/rmjm/1181071677.  Google Scholar

[6]

A. Algaba, E. Freire, E. Gamero and C. Garcia, Quasi-homogeneous normal forms, J. Comput. Appl. Math. 150 (2003), 193-216. doi: 10.1016/S0377-0427(02)00660-X.  Google Scholar

[7]

V. I. Arnold, "Geometrical Methods In The Theory Of Ordinary Differential Equations," Springer Verlag, New York, 1988.  Google Scholar

[8]

A. Baider, Unique normal forms for vector fields and Hamiltonians, J. Differential Equations, 78 (1989), 33-52. doi: 10.1016/0022-0396(89)90074-0.  Google Scholar

[9]

A. Baider and R. Churchill, Unique normal forms for planar vector fields,, Math. Z., 199 (): 303.   Google Scholar

[10]

A. Baider and J. Sanders, Further reduction of the Takens-Bogdanov normal form, J. Differential Equations, 99 (1992), 205-244. doi: 10.1016/0022-0396(92)90022-F.  Google Scholar

[11]

G. R. Belitskii, Invariant normal forms of formal series, Functional Anal. Appl., 13 (1979), 46-67.  Google Scholar

[12]

G. R. Belitskii, Normal forms relative to a filtering action of a group, Trans. Moscow Math. Soc., 40 (1979), 3-46.  Google Scholar

[13]

F. Bracci and D. Zaitsev, Dynamics of one-resonant biholomorphisms,, J. Eur. Math. Soc. , ().   Google Scholar

[14]

A. D. Brjuno, Analytic form of differential equations. I, Trans. Moscow Math. Soc. 25 (1971), 131-288.  Google Scholar

[15]

A. D. Brjuno, Analytic form of differential equations. II, Trans. Moscow Math. Soc. 26 (1972), 199-239.  Google Scholar

[16]

H. Broer, Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case, in "Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980)", Lecture Notes in Math. 898, Springer, Berlin, 1981, pp. 54-74.  Google Scholar

[17]

H. Cartan, "Cours de calcul différentiel," Hermann, Paris, 1977. Google Scholar

[18]

G. T. Chen and J. Della Dora, Normal forms for differentiable maps near a fixed point, Numer. Algorithms, 22 (1999), 213-230.  Google Scholar

[19]

G. T. Chen and J. Della Dora, Further reductions of normal forms for dynamical systems, J. Differential Equations, 166 (2000), 79-106.  Google Scholar

[20]

J. Écalle, "Les Fonctions Résurgentes. Tome III: L'Équation Du Pont Et La Classification Analytique Des Objects Locaux," Publ. Math. Orsay, 85-05, Université de Paris-Sud, Orsay, 1985.  Google Scholar

[21]

J. Écalle, Iteration and analytic classification of local diffeomorphisms of $\mathbbC^v$, in "Iteration Theory And Its Functional Equations (Lochau, 1984)", Lect. Notes in Math., 1163, Springer-Verlag, Berlin, 1985, pp. 41-48.  Google Scholar

[22]

E. Fischer, Über die differentiationsprozesse der algebra, J. für Math., 148 (1917), 1-78. Google Scholar

[23]

G. Gaeta, Further reduction of Poincaré-Dulac normal forms in symmetric systems, Cubo, 9 (2007), 1-11.  Google Scholar

[24]

A. Giorgilli and A. Posilicano, Estimates for normal forms of differential equations near an equilibrium point, Z. Angew. Math. Phys., 39 (1988), 713-732. doi: 10.1007/BF00948732.  Google Scholar

[25]

F. Ichikawa, On finite determinacy of formal vector fields,, Invent. Math. 70 (1982/83), 70 (): 45.   Google Scholar

[26]

F. Ichikawa, Classification of finitely determined singularities of formal vector fields on the plane, Tokyo J. Math. 8 (1985), 463-472. doi: 10.3836/tjm/1270151227.  Google Scholar

[27]

H. Kokubu, H. Oka and D. Wang, Linear grading function and further reduction of normal forms, J. Differential Equations, 132 (1996), 293-318. doi: 10.1006/jdeq.1996.0181.  Google Scholar

[28]

E. Lombardi and L. Stolovitch, Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation, Ann. Sci. Éc. Norm. Supér. 43 (2010), 659-718.  Google Scholar

[29]

D. Malonza and J. Murdock, An improved theory of asymptotic unfoldings, J. Differential Equations, 247 (2009), 685-709.  Google Scholar

[30]

J. Murdock, "Normal Forms And Unfoldings For Local Dynamical Systems," Springer Verlag, Berlin, 2003. Google Scholar

[31]

J. Murdock, Hypernormal form theory: Foundations and algorithms, J. Differential Equations, 205 (2004), 424-465.  Google Scholar

[32]

J. Murdock and J. A. Sanders, A new transvectant algorithm for nilpotent normal forms, J. Differential Equations, 238 (2007), 234-256.  Google Scholar

[33]

J. Raissy, Torus actions in the normalization problem, J. Geom. Anal. 20 (2010), 472-524.  Google Scholar

[34]

J. Raissy, Brjuno conditions for linearization in presence of resonances, in "Asymptotics In Dynamics, Geometry And PDE's; Generalized Borel Summation, Vol. I" (O. Costin, F. Fauvet, F. Menous and D. Sauzin, eds.), Edizioni Della Normale, Pisa, 2010, pp. 201-218.  Google Scholar

[35]

H. Rüssmann, Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition, Ergodic Theory Dynam. Systems, 22 (2002), 1551-1573.  Google Scholar

[36]

J. A. Sanders, Normal form theory and spectral sequences, J. Differential Equations, 192 (2003), 536-552.  Google Scholar

[37]

D. Wang, M. Zheng and J. Peng, Further reduction of normal forms of formal maps, C. R. Math. Acad. Sci. Paris, 343 (2006), 657-660.  Google Scholar

[38]

D. Wang, M. Zheng and J. Peng, Further reduction of normal forms and unique normal forms of smooth maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 18 (2008), 803-825. doi: 10.1142/S0218127408020665.  Google Scholar

[39]

P. Yu and Y. Yuan, The simplest normal form for the singularity of a pure imaginary pair and a zero eigenvalue, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 8 (2001), 219-249.  Google Scholar

show all references

References:
[1]

M. Abate, Holomorphic classification of $2$-dimensional quadratic maps tangent to the identity, Sūkikenkyūsho Kōkyūroku, 1447 (2005), 1-14. Google Scholar

[2]

M. Abate, Discrete holomorphic local dynamical systems, in "Holomorphic Dynamical Systems" (G.Gentili, J. Guénot and G. Patrizio, eds.), Lect. Notes in Math. 1998, Springer, Berlin, 2010, pp. 1-55.  Google Scholar

[3]

M. Abate and F. Tovena, Formal classification of holomorphic maps tangent to the identity, Discrete Contin. Dyn. Syst. Suppl (2005), 1-10.  Google Scholar

[4]

M. Abate and F. Tovena, Poincaré-Bendixson theorems for meromorphic connections and holomorphic homogeneous vector fields, J. Differential Equations, 251 (2011), 2612-2684. doi: 10.1016/j.jde.2011.05.031.  Google Scholar

[5]

A. Algaba, E. Freire and E. Gamero, Hypernormal forms for equilibria of vector fields. Codimension one linear degeneracies, Rocky Mountain J. Math. 29 (1999), 13-45. doi: 10.1216/rmjm/1181071677.  Google Scholar

[6]

A. Algaba, E. Freire, E. Gamero and C. Garcia, Quasi-homogeneous normal forms, J. Comput. Appl. Math. 150 (2003), 193-216. doi: 10.1016/S0377-0427(02)00660-X.  Google Scholar

[7]

V. I. Arnold, "Geometrical Methods In The Theory Of Ordinary Differential Equations," Springer Verlag, New York, 1988.  Google Scholar

[8]

A. Baider, Unique normal forms for vector fields and Hamiltonians, J. Differential Equations, 78 (1989), 33-52. doi: 10.1016/0022-0396(89)90074-0.  Google Scholar

[9]

A. Baider and R. Churchill, Unique normal forms for planar vector fields,, Math. Z., 199 (): 303.   Google Scholar

[10]

A. Baider and J. Sanders, Further reduction of the Takens-Bogdanov normal form, J. Differential Equations, 99 (1992), 205-244. doi: 10.1016/0022-0396(92)90022-F.  Google Scholar

[11]

G. R. Belitskii, Invariant normal forms of formal series, Functional Anal. Appl., 13 (1979), 46-67.  Google Scholar

[12]

G. R. Belitskii, Normal forms relative to a filtering action of a group, Trans. Moscow Math. Soc., 40 (1979), 3-46.  Google Scholar

[13]

F. Bracci and D. Zaitsev, Dynamics of one-resonant biholomorphisms,, J. Eur. Math. Soc. , ().   Google Scholar

[14]

A. D. Brjuno, Analytic form of differential equations. I, Trans. Moscow Math. Soc. 25 (1971), 131-288.  Google Scholar

[15]

A. D. Brjuno, Analytic form of differential equations. II, Trans. Moscow Math. Soc. 26 (1972), 199-239.  Google Scholar

[16]

H. Broer, Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case, in "Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980)", Lecture Notes in Math. 898, Springer, Berlin, 1981, pp. 54-74.  Google Scholar

[17]

H. Cartan, "Cours de calcul différentiel," Hermann, Paris, 1977. Google Scholar

[18]

G. T. Chen and J. Della Dora, Normal forms for differentiable maps near a fixed point, Numer. Algorithms, 22 (1999), 213-230.  Google Scholar

[19]

G. T. Chen and J. Della Dora, Further reductions of normal forms for dynamical systems, J. Differential Equations, 166 (2000), 79-106.  Google Scholar

[20]

J. Écalle, "Les Fonctions Résurgentes. Tome III: L'Équation Du Pont Et La Classification Analytique Des Objects Locaux," Publ. Math. Orsay, 85-05, Université de Paris-Sud, Orsay, 1985.  Google Scholar

[21]

J. Écalle, Iteration and analytic classification of local diffeomorphisms of $\mathbbC^v$, in "Iteration Theory And Its Functional Equations (Lochau, 1984)", Lect. Notes in Math., 1163, Springer-Verlag, Berlin, 1985, pp. 41-48.  Google Scholar

[22]

E. Fischer, Über die differentiationsprozesse der algebra, J. für Math., 148 (1917), 1-78. Google Scholar

[23]

G. Gaeta, Further reduction of Poincaré-Dulac normal forms in symmetric systems, Cubo, 9 (2007), 1-11.  Google Scholar

[24]

A. Giorgilli and A. Posilicano, Estimates for normal forms of differential equations near an equilibrium point, Z. Angew. Math. Phys., 39 (1988), 713-732. doi: 10.1007/BF00948732.  Google Scholar

[25]

F. Ichikawa, On finite determinacy of formal vector fields,, Invent. Math. 70 (1982/83), 70 (): 45.   Google Scholar

[26]

F. Ichikawa, Classification of finitely determined singularities of formal vector fields on the plane, Tokyo J. Math. 8 (1985), 463-472. doi: 10.3836/tjm/1270151227.  Google Scholar

[27]

H. Kokubu, H. Oka and D. Wang, Linear grading function and further reduction of normal forms, J. Differential Equations, 132 (1996), 293-318. doi: 10.1006/jdeq.1996.0181.  Google Scholar

[28]

E. Lombardi and L. Stolovitch, Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation, Ann. Sci. Éc. Norm. Supér. 43 (2010), 659-718.  Google Scholar

[29]

D. Malonza and J. Murdock, An improved theory of asymptotic unfoldings, J. Differential Equations, 247 (2009), 685-709.  Google Scholar

[30]

J. Murdock, "Normal Forms And Unfoldings For Local Dynamical Systems," Springer Verlag, Berlin, 2003. Google Scholar

[31]

J. Murdock, Hypernormal form theory: Foundations and algorithms, J. Differential Equations, 205 (2004), 424-465.  Google Scholar

[32]

J. Murdock and J. A. Sanders, A new transvectant algorithm for nilpotent normal forms, J. Differential Equations, 238 (2007), 234-256.  Google Scholar

[33]

J. Raissy, Torus actions in the normalization problem, J. Geom. Anal. 20 (2010), 472-524.  Google Scholar

[34]

J. Raissy, Brjuno conditions for linearization in presence of resonances, in "Asymptotics In Dynamics, Geometry And PDE's; Generalized Borel Summation, Vol. I" (O. Costin, F. Fauvet, F. Menous and D. Sauzin, eds.), Edizioni Della Normale, Pisa, 2010, pp. 201-218.  Google Scholar

[35]

H. Rüssmann, Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition, Ergodic Theory Dynam. Systems, 22 (2002), 1551-1573.  Google Scholar

[36]

J. A. Sanders, Normal form theory and spectral sequences, J. Differential Equations, 192 (2003), 536-552.  Google Scholar

[37]

D. Wang, M. Zheng and J. Peng, Further reduction of normal forms of formal maps, C. R. Math. Acad. Sci. Paris, 343 (2006), 657-660.  Google Scholar

[38]

D. Wang, M. Zheng and J. Peng, Further reduction of normal forms and unique normal forms of smooth maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 18 (2008), 803-825. doi: 10.1142/S0218127408020665.  Google Scholar

[39]

P. Yu and Y. Yuan, The simplest normal form for the singularity of a pure imaginary pair and a zero eigenvalue, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 8 (2001), 219-249.  Google Scholar

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