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Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$
Convergence radius in the Poincaré-Siegel problem
1. | Dipartimento di Matematica, Universitµa degli Studi di Milano, via Saldini 50, 20133 | Milano, Italy |
2. | Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126, Pisa (PI) |
References:
[1] |
V. I. Arnold, Proof of a theorem of A. N. Kolmogorov on the persistence of quasi-periodic motions under small perturbations of the Hamiltonian,, Usp. Mat. Nauk, 18 (1963). Google Scholar |
[2] |
V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics,, Usp. Math. Nauk {\bf 18} (1963), 18 (1963).
doi: 10.1070/RM1963v018n06ABEH001143. |
[3] |
A. Berretti and G. Gentile, Scaling properties for the radius of convergence of a Lindstedt series: the standard map,, J. Math. Pures Appl., 78 (1999), 159. Google Scholar |
[4] |
A. Berretti and G. Gentile, Bryuno function and the standard map,, Comm. Math. Phys., 220 (2001), 623. Google Scholar |
[5] |
A. D. Bruno, Analytical form of differential equations,, Trans. Moscow Math. Soc., 25 (1971), 131. Google Scholar |
[6] |
X. Buff and A. Chéritat, The Brjuno function continuously estimates the size of quadratic Siegel disks,, Ann. of Math., 164 (2006), 265. Google Scholar |
[7] |
T. Carletti and S. Marmi, Linearization of analytic and non-analytic germs of diffeomorphisms of $C, 0)$,, Bull. Soc. Math. France, 128 (2000), 69. Google Scholar |
[8] |
A. M. Davie, The critical function for the semistandard map,, Nonlinearity, 7 (1994), 219. Google Scholar |
[9] |
A. Giorgilli, Quantitative methods in classical perturbation theory,, in Proceedings of the NATO ASI school, 336 (1995), 21. Google Scholar |
[10] |
A. Giorgilli and U. Locatelli, On classical series expansions for quasi-periodic motions,, MPEJ, 3 (1997). Google Scholar |
[11] |
A. Giorgilli and U. Locatelli, A classical self-contained proof of Kolmogorov's theorem on invariant tori,, in Proceedings of the NATO ASI school, 533 (1999), 72. Google Scholar |
[12] |
W. Gröbner, "Die Lie-Reihen und Ihre Anwendungen,", VEB Deutscher Verlag der Wissenschaften, (1967). Google Scholar |
[13] |
G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers,", Oxford University Press, (1979). Google Scholar |
[14] |
A. N. Kolmogorov, Preservation of conditionally periodic movements with small change in the Hamilton function,, Dokl. Akad. Nauk SSSR, 98 (1954), 71. Google Scholar |
[15] |
A. Ya. Khinchin, "Continued Fractions,", The University of Chicago Press, (1964). Google Scholar |
[16] |
S. Marmi, Critical functions for complex analytic maps,, J. Phys. A: Math. Gen., 23 (1990), 3447. Google Scholar |
[17] |
S. Marmi and J. Stark, On the standard map critical function,, Nonlinearity, 5 (1992), 743. Google Scholar |
[18] |
S. Marmi and J.-C. Yoccoz, Some open problems related to small divisors,, in, 1784 (2002), 175. Google Scholar |
[19] |
S. Marmi, P. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties,, Comm. Math. Phys., 186 (1997), 265. Google Scholar |
[20] |
J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. Gött., (1962), 1. Google Scholar |
[21] |
J. Moser, Convergent series expansions for quasi-periodic motions,, Math. Ann., 169 (1967), 136. Google Scholar |
[22] |
R. Pérez-Marco, Fixed points and circle maps,, Acta Math., 179 (1997), 243. Google Scholar |
[23] |
H. Poincaré, Mémoire sur les courbes définies par une équation différentielle,, Journal de Mathématiques, 7 (1881), 375. Google Scholar |
[24] |
H. Rüssmann, Über die iteration analytischer Funktionen,, J. Math. Mech., 17 (1967), 523. Google Scholar |
[25] |
E. Schröder, Über iterierte Funktionen,, Math. Ann., 3 (1871), 296. Google Scholar |
[26] |
C. L. Siegel, Iteration of analytic functions,, Annals of Math., 43 (1942), 607. Google Scholar |
[27] |
C. L. Siegel, Über die normalform analytischer differentialgleichungen in der Nähe einer Gleichgewichtslösung,, Nachr. Akad. Wiss. Göttingen, (1952), 21. Google Scholar |
[28] |
C. L. Siegel, and J. K. Moser, "Lectures on Celestial Mechanics,", Springer-Verlag, (1971). Google Scholar |
[29] |
J.-C. Yoccoz, Théeorème de Siegel, nombres de Bruno et polynômes quadratiques,, Astérisque, 231 (1995), 3. Google Scholar |
[30] |
J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms,, in, 1784 (2002), 125. Google Scholar |
show all references
References:
[1] |
V. I. Arnold, Proof of a theorem of A. N. Kolmogorov on the persistence of quasi-periodic motions under small perturbations of the Hamiltonian,, Usp. Mat. Nauk, 18 (1963). Google Scholar |
[2] |
V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics,, Usp. Math. Nauk {\bf 18} (1963), 18 (1963).
doi: 10.1070/RM1963v018n06ABEH001143. |
[3] |
A. Berretti and G. Gentile, Scaling properties for the radius of convergence of a Lindstedt series: the standard map,, J. Math. Pures Appl., 78 (1999), 159. Google Scholar |
[4] |
A. Berretti and G. Gentile, Bryuno function and the standard map,, Comm. Math. Phys., 220 (2001), 623. Google Scholar |
[5] |
A. D. Bruno, Analytical form of differential equations,, Trans. Moscow Math. Soc., 25 (1971), 131. Google Scholar |
[6] |
X. Buff and A. Chéritat, The Brjuno function continuously estimates the size of quadratic Siegel disks,, Ann. of Math., 164 (2006), 265. Google Scholar |
[7] |
T. Carletti and S. Marmi, Linearization of analytic and non-analytic germs of diffeomorphisms of $C, 0)$,, Bull. Soc. Math. France, 128 (2000), 69. Google Scholar |
[8] |
A. M. Davie, The critical function for the semistandard map,, Nonlinearity, 7 (1994), 219. Google Scholar |
[9] |
A. Giorgilli, Quantitative methods in classical perturbation theory,, in Proceedings of the NATO ASI school, 336 (1995), 21. Google Scholar |
[10] |
A. Giorgilli and U. Locatelli, On classical series expansions for quasi-periodic motions,, MPEJ, 3 (1997). Google Scholar |
[11] |
A. Giorgilli and U. Locatelli, A classical self-contained proof of Kolmogorov's theorem on invariant tori,, in Proceedings of the NATO ASI school, 533 (1999), 72. Google Scholar |
[12] |
W. Gröbner, "Die Lie-Reihen und Ihre Anwendungen,", VEB Deutscher Verlag der Wissenschaften, (1967). Google Scholar |
[13] |
G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers,", Oxford University Press, (1979). Google Scholar |
[14] |
A. N. Kolmogorov, Preservation of conditionally periodic movements with small change in the Hamilton function,, Dokl. Akad. Nauk SSSR, 98 (1954), 71. Google Scholar |
[15] |
A. Ya. Khinchin, "Continued Fractions,", The University of Chicago Press, (1964). Google Scholar |
[16] |
S. Marmi, Critical functions for complex analytic maps,, J. Phys. A: Math. Gen., 23 (1990), 3447. Google Scholar |
[17] |
S. Marmi and J. Stark, On the standard map critical function,, Nonlinearity, 5 (1992), 743. Google Scholar |
[18] |
S. Marmi and J.-C. Yoccoz, Some open problems related to small divisors,, in, 1784 (2002), 175. Google Scholar |
[19] |
S. Marmi, P. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties,, Comm. Math. Phys., 186 (1997), 265. Google Scholar |
[20] |
J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. Gött., (1962), 1. Google Scholar |
[21] |
J. Moser, Convergent series expansions for quasi-periodic motions,, Math. Ann., 169 (1967), 136. Google Scholar |
[22] |
R. Pérez-Marco, Fixed points and circle maps,, Acta Math., 179 (1997), 243. Google Scholar |
[23] |
H. Poincaré, Mémoire sur les courbes définies par une équation différentielle,, Journal de Mathématiques, 7 (1881), 375. Google Scholar |
[24] |
H. Rüssmann, Über die iteration analytischer Funktionen,, J. Math. Mech., 17 (1967), 523. Google Scholar |
[25] |
E. Schröder, Über iterierte Funktionen,, Math. Ann., 3 (1871), 296. Google Scholar |
[26] |
C. L. Siegel, Iteration of analytic functions,, Annals of Math., 43 (1942), 607. Google Scholar |
[27] |
C. L. Siegel, Über die normalform analytischer differentialgleichungen in der Nähe einer Gleichgewichtslösung,, Nachr. Akad. Wiss. Göttingen, (1952), 21. Google Scholar |
[28] |
C. L. Siegel, and J. K. Moser, "Lectures on Celestial Mechanics,", Springer-Verlag, (1971). Google Scholar |
[29] |
J.-C. Yoccoz, Théeorème de Siegel, nombres de Bruno et polynômes quadratiques,, Astérisque, 231 (1995), 3. Google Scholar |
[30] |
J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms,, in, 1784 (2002), 125. Google Scholar |
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