A Parameterization Method for Lagrangian Tori of Exact Symplectic Maps of R2r

We are concerned with analytic exact symplectic maps of ${\mathbb R}^{2r}$ endowed with the standard symplectic form. We study the existence of a real analytic torus of dimension $r$, invariant by ...

[1]  H. Rüssmann.  On optimal estimates for the solutions of linear difference equations on the circle , 1976 .

[2]  J. Moser.  On invariant curves of area-preserving mappings of an anulus , 1962 .

[3]  A. Celletti,et al.  On the Stability of Realistic Three-Body Problems , 1997 .

[4]  R. Llave,et al.  A KAM theory for conformally symplectic systems: Efficient algorithms and their validation , 2013 .

[5]  Rafael de la Llave,et al.  A numerically accessible criterion for the breakdown of quasi-periodic solutions and its rigorous justification , 2010 .

[6]  J. Moser.  A rapidly convergent iteration method and non-linear partial differential equations - I , 1966 .

[7]  R. Llave,et al.  KAM theory without action-angle variables , 2005 .

[8]  H. Rüssmann.  On a new proof of Moser's twist mapping theorem , 1976 .

[9]  À. Haro.  An algorithm to generate canonical transformations: application to normal forms , 2002 .

[10]  Jürgen Pöschel,et al.  Integrability of Hamiltonian systems on cantor sets , 1982 .

[11]  V. I. Arnol'd,et al.  PROOF OF A THEOREM OF A.?N.?KOLMOGOROV ON THE INVARIANCE OF QUASI-PERIODIC MOTIONS UNDER SMALL PERTURBATIONS OF THE HAMILTONIAN , 1963 .

[12]  R. Llave,et al.  Construction of invariant whiskered tori by a parameterization method. Part I: Maps and flows in finite dimensions , 2009, 0903.0311.

[13]  R. Llave,et al.  Computation of whiskered invariant tori and their associated manifolds: new fast algorithms , 2010, 1004.5231.

[14]  H. Broer,et al.  KAM Theory : Quasi-periodicity in Dynamical Systems , 2010 .

[15]  Àlex Haro,et al.  A Parameterization Method for the Computation of Invariant Tori and Their Whiskers in Quasi-Periodic Maps: Explorations and Mechanisms for the Breakdown of Hyperbolicity , 2006, SIAM J. Appl. Dyn. Syst..

[16]  E. Zehnder,et al.  Generalized implicit function theorems with applications to some small divisor problems, I , 1976 .

[17]  R. Llave,et al.  Lindstedt series for lower dimensional tori , 1999 .

[18]  Eduard Zehnder,et al.  KAM theory in configuration space , 1989 .

[19]  Analytic smoothing of geometric maps with applications to KAM theory , 2008 .

[20]  J. Villanueva,et al.  A KAM theorem without action-angle variables for elliptic lower dimensional tori , 2011 .

[21]  A. Neishtadt.  Estimates in the kolmogorov theorem on conservation of conditionally periodic motions , 1981 .

[22]  J. Villanueva,et al.  A numerical method for computing initial conditions of Lagrangian invariant tori using the frequency map , 2016 .

[23]  Rafael de la Llave,et al.  A Tutorial on Kam Theory , 2003 .

[24]  A. Kolmogorov.  On conservation of conditionally periodic motions for a small change in Hamilton's function , 1954 .

[25]  Rafael de la Llave,et al.  A parameterization method for the computation of invariant tori andtheir whiskers in quasi-periodic maps: Numerical algorithms , 2006 .

[26]  Jürgen Moser,et al.  A rapidly convergent iteration method and non-linear differential equations = II , 1966 .

[27]  J. Villanueva.  Kolmogorov Theorem revisited , 2008 .

[28]  Jordi-Lluís Figueras,et al.  Rigorous Computer-Assisted Application of KAM Theory: A Modern Approach , 2016, Found. Comput. Math..

[29]  R. Llave,et al.  Singularity Theory for Non-Twist Kam Tori , 2014 .

[30]  Jordi Villanueva.  A New Approach to the Parameterization Method for Lagrangian Tori of Hamiltonian Systems , 2017, J. Nonlinear Sci..

[31]  Helmut Rüssmann,et al.  On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus , 1975 .