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Persistence of Hölder continuity for non-local integro-differential equations
Formal Poincaré-Dulac renormalization for holomorphic germs
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 |
References:
[1] |
M. Abate, Holomorphic classification of $2$-dimensional quadratic maps tangent to the identity, Sūkikenkyūsho Kōkyūroku, 1447 (2005), 1-14. |
[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. |
[3] |
M. Abate and F. Tovena, Formal classification of holomorphic maps tangent to the identity, Discrete Contin. Dyn. Syst. Suppl (2005), 1-10. |
[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. |
[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. |
[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. |
[7] |
V. I. Arnold, "Geometrical Methods In The Theory Of Ordinary Differential Equations," Springer Verlag, New York, 1988. |
[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. |
[9] |
A. Baider and R. Churchill, Unique normal forms for planar vector fields, Math. Z., 199 (988), 303-310. |
[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. |
[11] |
G. R. Belitskii, Invariant normal forms of formal series, Functional Anal. Appl., 13 (1979), 46-67. |
[12] |
G. R. Belitskii, Normal forms relative to a filtering action of a group, Trans. Moscow Math. Soc., 40 (1979), 3-46. |
[13] |
F. Bracci and D. Zaitsev, Dynamics of one-resonant biholomorphisms, J. Eur. Math. Soc. arXiv:0912.0428v2. |
[14] |
A. D. Brjuno, Analytic form of differential equations. I, Trans. Moscow Math. Soc. 25 (1971), 131-288. |
[15] |
A. D. Brjuno, Analytic form of differential equations. II, Trans. Moscow Math. Soc. 26 (1972), 199-239. |
[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. |
[17] |
H. Cartan, "Cours de calcul différentiel," Hermann, Paris, 1977. |
[18] |
G. T. Chen and J. Della Dora, Normal forms for differentiable maps near a fixed point, Numer. Algorithms, 22 (1999), 213-230. |
[19] |
G. T. Chen and J. Della Dora, Further reductions of normal forms for dynamical systems, J. Differential Equations, 166 (2000), 79-106. |
[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. |
[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. |
[22] |
E. Fischer, Über die differentiationsprozesse der algebra, J. für Math., 148 (1917), 1-78. |
[23] |
G. Gaeta, Further reduction of Poincaré-Dulac normal forms in symmetric systems, Cubo, 9 (2007), 1-11. |
[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. |
[25] |
F. Ichikawa, On finite determinacy of formal vector fields, Invent. Math. 70 (1982/83), 45-52. |
[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. |
[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. |
[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. |
[29] |
D. Malonza and J. Murdock, An improved theory of asymptotic unfoldings, J. Differential Equations, 247 (2009), 685-709. |
[30] |
J. Murdock, "Normal Forms And Unfoldings For Local Dynamical Systems," Springer Verlag, Berlin, 2003. |
[31] |
J. Murdock, Hypernormal form theory: Foundations and algorithms, J. Differential Equations, 205 (2004), 424-465. |
[32] |
J. Murdock and J. A. Sanders, A new transvectant algorithm for nilpotent normal forms, J. Differential Equations, 238 (2007), 234-256. |
[33] |
J. Raissy, Torus actions in the normalization problem, J. Geom. Anal. 20 (2010), 472-524. |
[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. |
[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. |
[36] |
J. A. Sanders, Normal form theory and spectral sequences, J. Differential Equations, 192 (2003), 536-552. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
M. Abate, Holomorphic classification of $2$-dimensional quadratic maps tangent to the identity, Sūkikenkyūsho Kōkyūroku, 1447 (2005), 1-14. |
[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. |
[3] |
M. Abate and F. Tovena, Formal classification of holomorphic maps tangent to the identity, Discrete Contin. Dyn. Syst. Suppl (2005), 1-10. |
[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. |
[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. |
[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. |
[7] |
V. I. Arnold, "Geometrical Methods In The Theory Of Ordinary Differential Equations," Springer Verlag, New York, 1988. |
[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. |
[9] |
A. Baider and R. Churchill, Unique normal forms for planar vector fields, Math. Z., 199 (988), 303-310. |
[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. |
[11] |
G. R. Belitskii, Invariant normal forms of formal series, Functional Anal. Appl., 13 (1979), 46-67. |
[12] |
G. R. Belitskii, Normal forms relative to a filtering action of a group, Trans. Moscow Math. Soc., 40 (1979), 3-46. |
[13] |
F. Bracci and D. Zaitsev, Dynamics of one-resonant biholomorphisms, J. Eur. Math. Soc. arXiv:0912.0428v2. |
[14] |
A. D. Brjuno, Analytic form of differential equations. I, Trans. Moscow Math. Soc. 25 (1971), 131-288. |
[15] |
A. D. Brjuno, Analytic form of differential equations. II, Trans. Moscow Math. Soc. 26 (1972), 199-239. |
[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. |
[17] |
H. Cartan, "Cours de calcul différentiel," Hermann, Paris, 1977. |
[18] |
G. T. Chen and J. Della Dora, Normal forms for differentiable maps near a fixed point, Numer. Algorithms, 22 (1999), 213-230. |
[19] |
G. T. Chen and J. Della Dora, Further reductions of normal forms for dynamical systems, J. Differential Equations, 166 (2000), 79-106. |
[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. |
[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. |
[22] |
E. Fischer, Über die differentiationsprozesse der algebra, J. für Math., 148 (1917), 1-78. |
[23] |
G. Gaeta, Further reduction of Poincaré-Dulac normal forms in symmetric systems, Cubo, 9 (2007), 1-11. |
[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. |
[25] |
F. Ichikawa, On finite determinacy of formal vector fields, Invent. Math. 70 (1982/83), 45-52. |
[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. |
[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. |
[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. |
[29] |
D. Malonza and J. Murdock, An improved theory of asymptotic unfoldings, J. Differential Equations, 247 (2009), 685-709. |
[30] |
J. Murdock, "Normal Forms And Unfoldings For Local Dynamical Systems," Springer Verlag, Berlin, 2003. |
[31] |
J. Murdock, Hypernormal form theory: Foundations and algorithms, J. Differential Equations, 205 (2004), 424-465. |
[32] |
J. Murdock and J. A. Sanders, A new transvectant algorithm for nilpotent normal forms, J. Differential Equations, 238 (2007), 234-256. |
[33] |
J. Raissy, Torus actions in the normalization problem, J. Geom. Anal. 20 (2010), 472-524. |
[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. |
[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. |
[36] |
J. A. Sanders, Normal form theory and spectral sequences, J. Differential Equations, 192 (2003), 536-552. |
[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. |
[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. |
[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. |
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