American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Convergence of differentiable functions on closed sets and remarks on the proofs of the "Converse Approximation Lemmas''
  • DCDS-S Home
  • This Issue
  • Next Article
    Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$
December  2010, 3(4): 601-621. doi: 10.3934/dcdss.2010.3.601

Convergence radius in the Poincaré-Siegel problem

Antonio Giorgilli 1, and  Stefano Marmi 2,

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)

Received  March 2009 Revised  May 2010 Published  August 2010

We reconsider the Poincaré-Siegel center problem, namely the problem of conjugating an analytic system of differential equations in the neighbourhood of an equilibrium to its linear part $\Lambda=\diag(\lambda_1,\ldots,\lambda_n)$. If the linear part is non--resonant we show that the convergence radius $r$ of the conjugating transformation satisfies $\log r(\Lambda )\ge -CB+C'$ with $C=1$ and a constant $C'$ not depending on $\Lambda$. The convergence condition is the same as the Bruno condition since $B = -\sum_{r\ge 1}2^{-r}\log\alpha_{2^r-1}$, where $\alpha_r = \min_{0\le s\le r} \beta_r$ for $r\ge 0$ and $\beta_r = \min_{j=1,\ldots,n}\ \ \ \min_{k\inZ_+^n,|k|=r+1}$ | < k, $\lambda$ > $- \lambda_j$|. Our lower bound improves the previous results for $n\gt 1$, where the known proofs give $C=2$. We also recall that $C=1$ is known to be the optimal value for the discrete time version of the center problem when $n=1$, namely the linearization problem for germs of holomorphic maps when the eigenvalue of the fixed point is on the unit circle.
Keywords: linearization, diophantine conditions, Small divisors, normal forms..
Mathematics Subject Classification: Primary: 37F50; Secondary: 37f2.
Citation: Antonio Giorgilli, Stefano Marmi. Convergence radius in the Poincaré-Siegel problem. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 601-621. doi: 10.3934/dcdss.2010.3.601
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), 13; Russ. Math. Surv., 18 (1963), 9.

[2]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Usp. Math. Nauk 18 (1963), 91; Russ. Math. Surv., 18 (1963), 85. 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), 159-176.

[4]

A. Berretti and G. Gentile, Bryuno function and the standard map, Comm. Math. Phys., 220 (2001), 623-656.

[5]

A. D. Bruno, Analytical form of differential equations, Trans. Moscow Math. Soc., 25 (1971), 131-288; 26 (1972), 199-239.

[6]

X. Buff and A. Chéritat, The Brjuno function continuously estimates the size of quadratic Siegel disks, Ann. of Math., 164 (2006), 265-312.

[7]

T. Carletti and S. Marmi, Linearization of analytic and non-analytic germs of diffeomorphisms of $C, 0)$, Bull. Soc. Math. France, 128 (2000), 69-85.

[8]

A. M. Davie, The critical function for the semistandard map, Nonlinearity, 7 (1994), 219-229.

[9]

A. Giorgilli, Quantitative methods in classical perturbation theory, in Proceedings of the NATO ASI school "From Newton to Chaos: Modern Techniques for Understanding and Coping with Chaos in $N$-body Dynamical Systems,'' A.E. Roy and B.A. Steves eds., NATO ASI Series B: Physics, Vol. 336, Plenum Press, New York, 21-37 (1995).

[10]

A. Giorgilli and U. Locatelli, On classical series expansions for quasi-periodic motions, MPEJ, 3 (1997).

[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 "Hamiltonian systems with three or more degrees of freedom,'' C. Simó ed., NATO ASI series C: Math. Phys. Sci., Vol. 533, Kluwer Academic Publishers, Dordrecht-Boston-London, (1999), 72-89.

[12]

W. Gröbner, "Die Lie-Reihen und Ihre Anwendungen," VEB Deutscher Verlag der Wissenschaften, Berlin, 1967.

[13]

G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers," Oxford University Press, New York, 1979.

[14]

A. N. Kolmogorov, Preservation of conditionally periodic movements with small change in the Hamilton function, Dokl. Akad. Nauk SSSR, 98 (1954), 527; English translation in: Los Alamos Scientific Laboratory translation LA-TR-71-67; reprinted in: G. Casati and J. Ford: Stochastic behavior in classical and quantum Hamiltonian systems, Lecture Notes in Physics, 93 (1979), 51-56.

[15]

A. Ya. Khinchin, "Continued Fractions," The University of Chicago Press, Chicago-London, 1964.

[16]

S. Marmi, Critical functions for complex analytic maps, J. Phys. A: Math. Gen., 23 (1990), 3447-3474.

[17]

S. Marmi and J. Stark, On the standard map critical function, Nonlinearity, 5 (1992), 743-761.

[18]

S. Marmi and J.-C. Yoccoz, Some open problems related to small divisors, in "Dynamical Systems and Small Divisors,'' Lecture Notes in Mathematics, 1784 (2002), 175-191.

[19]

S. Marmi, P. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties, Comm. Math. Phys., 186 (1997), 265-293.

[20]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gött., II Math.-Phys. Kl., (1962), 1-20.

[21]

J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176.

[22]

R. Pérez-Marco, Fixed points and circle maps, Acta Math., 179 (1997), 243-294.

[23]

H. Poincaré, Mémoire sur les courbes définies par une équation différentielle, Journal de Mathématiques, 7 (1881), 375-422 and 8 (1882), 251-296.

[24]

H. Rüssmann, Über die iteration analytischer Funktionen, J. Math. Mech., 17 (1967), 523-532.

[25]

E. Schröder, Über iterierte Funktionen, Math. Ann., 3 (1871), 296-322.

[26]

C. L. Siegel, Iteration of analytic functions, Annals of Math., 43 (1942), 607-612.

[27]

C. L. Siegel, Über die normalform analytischer differentialgleichungen in der Nähe einer Gleichgewichtslösung, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. Math.-Phys.-Chem. Abt., (1952), 21-30.

[28]

C. L. Siegel, and J. K. Moser, "Lectures on Celestial Mechanics," Springer-Verlag, Berlin-Heidelberg-New York 1971.

[29]

J.-C. Yoccoz, Théeorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque, 231 (1995), 3-88.

[30]

J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms, in "Dynamical Systems and Small Divisors,'' Lecture Notes in Mathematics, 1784 (2002), 125-173.

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), 13; Russ. Math. Surv., 18 (1963), 9.

[2]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Usp. Math. Nauk 18 (1963), 91; Russ. Math. Surv., 18 (1963), 85. 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), 159-176.

[4]

A. Berretti and G. Gentile, Bryuno function and the standard map, Comm. Math. Phys., 220 (2001), 623-656.

[5]

A. D. Bruno, Analytical form of differential equations, Trans. Moscow Math. Soc., 25 (1971), 131-288; 26 (1972), 199-239.

[6]

X. Buff and A. Chéritat, The Brjuno function continuously estimates the size of quadratic Siegel disks, Ann. of Math., 164 (2006), 265-312.

[7]

T. Carletti and S. Marmi, Linearization of analytic and non-analytic germs of diffeomorphisms of $C, 0)$, Bull. Soc. Math. France, 128 (2000), 69-85.

[8]

A. M. Davie, The critical function for the semistandard map, Nonlinearity, 7 (1994), 219-229.

[9]

A. Giorgilli, Quantitative methods in classical perturbation theory, in Proceedings of the NATO ASI school "From Newton to Chaos: Modern Techniques for Understanding and Coping with Chaos in $N$-body Dynamical Systems,'' A.E. Roy and B.A. Steves eds., NATO ASI Series B: Physics, Vol. 336, Plenum Press, New York, 21-37 (1995).

[10]

A. Giorgilli and U. Locatelli, On classical series expansions for quasi-periodic motions, MPEJ, 3 (1997).

[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 "Hamiltonian systems with three or more degrees of freedom,'' C. Simó ed., NATO ASI series C: Math. Phys. Sci., Vol. 533, Kluwer Academic Publishers, Dordrecht-Boston-London, (1999), 72-89.

[12]

W. Gröbner, "Die Lie-Reihen und Ihre Anwendungen," VEB Deutscher Verlag der Wissenschaften, Berlin, 1967.

[13]

G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers," Oxford University Press, New York, 1979.

[14]

A. N. Kolmogorov, Preservation of conditionally periodic movements with small change in the Hamilton function, Dokl. Akad. Nauk SSSR, 98 (1954), 527; English translation in: Los Alamos Scientific Laboratory translation LA-TR-71-67; reprinted in: G. Casati and J. Ford: Stochastic behavior in classical and quantum Hamiltonian systems, Lecture Notes in Physics, 93 (1979), 51-56.

[15]

A. Ya. Khinchin, "Continued Fractions," The University of Chicago Press, Chicago-London, 1964.

[16]

S. Marmi, Critical functions for complex analytic maps, J. Phys. A: Math. Gen., 23 (1990), 3447-3474.

[17]

S. Marmi and J. Stark, On the standard map critical function, Nonlinearity, 5 (1992), 743-761.

[18]

S. Marmi and J.-C. Yoccoz, Some open problems related to small divisors, in "Dynamical Systems and Small Divisors,'' Lecture Notes in Mathematics, 1784 (2002), 175-191.

[19]

S. Marmi, P. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties, Comm. Math. Phys., 186 (1997), 265-293.

[20]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gött., II Math.-Phys. Kl., (1962), 1-20.

[21]

J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176.

[22]

R. Pérez-Marco, Fixed points and circle maps, Acta Math., 179 (1997), 243-294.

[23]

H. Poincaré, Mémoire sur les courbes définies par une équation différentielle, Journal de Mathématiques, 7 (1881), 375-422 and 8 (1882), 251-296.

[24]

H. Rüssmann, Über die iteration analytischer Funktionen, J. Math. Mech., 17 (1967), 523-532.

[25]

E. Schröder, Über iterierte Funktionen, Math. Ann., 3 (1871), 296-322.

[26]

C. L. Siegel, Iteration of analytic functions, Annals of Math., 43 (1942), 607-612.

[27]

C. L. Siegel, Über die normalform analytischer differentialgleichungen in der Nähe einer Gleichgewichtslösung, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. Math.-Phys.-Chem. Abt., (1952), 21-30.

[28]

C. L. Siegel, and J. K. Moser, "Lectures on Celestial Mechanics," Springer-Verlag, Berlin-Heidelberg-New York 1971.

[29]

J.-C. Yoccoz, Théeorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque, 231 (1995), 3-88.

[30]

J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms, in "Dynamical Systems and Small Divisors,'' Lecture Notes in Mathematics, 1784 (2002), 125-173.

[1]

E. Muñoz Garcia, R. Pérez-Marco. Diophantine conditions in small divisors and transcendental number theory. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1401-1409. doi: 10.3934/dcds.2003.9.1401
[2]

David DeLatte. Diophantine conditions for the linearization of commuting holomorphic functions. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 317-332. doi: 10.3934/dcds.1997.3.317
[3]

Luigi Chierchia, Gabriella Pinzari. Planetary Birkhoff normal forms. Journal of Modern Dynamics, 2011, 5 (4) : 623-664. doi: 10.3934/jmd.2011.5.623
[4]

Shrikrishna G. Dani. Simultaneous diophantine approximation with quadratic and linear forms. Journal of Modern Dynamics, 2008, 2 (1) : 129-138. doi: 10.3934/jmd.2008.2.129
[5]

Andreas Henrici. Symmetries of the periodic Toda lattice, with an application to normal forms and perturbations of the lattice with Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2949-2977. doi: 10.3934/dcds.2015.35.2949
[6]

Shui-Nee Chow, Kening Lu, Yun-Qiu Shen. Normal forms for quasiperiodic evolutionary equations. Discrete and Continuous Dynamical Systems, 1996, 2 (1) : 65-94. doi: 10.3934/dcds.1996.2.65
[7]

Xingwu Chen, Weinian Zhang. Normal forms of planar switching systems. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6715-6736. doi: 10.3934/dcds.2016092
[8]

A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133.

[9]

Marco Abate, Francesca Tovena. Formal normal forms for holomorphic maps tangent to the identity. Conference Publications, 2005, 2005 (Special) : 1-10. doi: 10.3934/proc.2005.2005.1
[10]

Boris Kalinin, Victoria Sadovskaya. Normal forms for non-uniform contractions. Journal of Modern Dynamics, 2017, 11: 341-368. doi: 10.3934/jmd.2017014
[11]

Chiara Caracciolo, Ugo Locatelli. Computer-assisted estimates for Birkhoff normal forms. Journal of Computational Dynamics, 2020, 7 (2) : 425-460. doi: 10.3934/jcd.2020017
[12]

Luchezar Stoyanov. Pinching conditions, linearization and regularity of Axiom A flows. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 391-412. doi: 10.3934/dcds.2013.33.391
[13]

P. De Maesschalck. Gevrey normal forms for nilpotent contact points of order two. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 677-688. doi: 10.3934/dcds.2014.34.677
[14]

Weigu Li, Jaume Llibre, Hao Wu. Polynomial and linearized normal forms for almost periodic differential systems. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 345-360. doi: 10.3934/dcds.2016.36.345
[15]

Ricardo Miranda Martins. Formal equivalence between normal forms of reversible and hamiltonian dynamical systems. Communications on Pure and Applied Analysis, 2014, 13 (2) : 703-713. doi: 10.3934/cpaa.2014.13.703
[16]

Vincent Naudot, Jiazhong Yang. Finite smooth normal forms and integrability of local families of vector fields. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 667-682. doi: 10.3934/dcdss.2010.3.667
[17]

Tomas Johnson, Warwick Tucker. Automated computation of robust normal forms of planar analytic vector fields. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 769-782. doi: 10.3934/dcdsb.2009.12.769
[18]

Gladston Duarte, Àngel Jorba. Using normal forms to study Oterma's transition in the Planar RTBP. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022073
[19]

Koichiro Naito. Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations II: Gaps of dimensions and Diophantine conditions. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 449-488. doi: 10.3934/dcds.2004.11.449
[20]

Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205

2021 Impact Factor: 1.865

Tools

Metrics

  • PDF downloads (109)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy