We investigate the existence of whiskered tori in some dissipative systems, called conformally symplectic systems, having the property that they transform the symplectic form into a multiple of itself. We consider a family ${f}_{\mu}$ of conformally symplectic maps which depends on a drift parameter $\mu $.

We fix a Diophantine frequency of the torus and we assume to have a drift ${\mu}_{0}$ and an embedding of the torus ${K}_{0}$, which satisfy approximately the invariance equation ${f}_{{\mu}_{0}}\circ {K}_{0}={K}_{0}\circ {T}_{\omega}$ (where ${T}_{\omega}$ denotes the shift by $\omega $). We also assume to have a splitting of the tangent space at the range of ${K}_{0}$ into three bundles. We assume that the bundles are approximately invariant under $D{f}_{{\mu}_{0}}$ and the derivative satisfies some rate conditions.

Under suitable nondegeneracy conditions, we prove that there exist ${\mu}_{\infty}$, ${K}_{\infty}$ invariant under ${f}_{{\mu}_{\infty}}$, close to the original ones, and a splitting which is invariant under $D{f}_{{\mu}_{\infty}}$. The proof provides an efficient algorithm to construct whiskered tori. Full details of the statements and proofs are given in [10].

Received: 2019-01-23

Accepted: 2019-05-21

Published online: 2020-06-30

DOI:
https://doi.org/10.5802/mrr.4

Classification: 70K43, 70K20, 34D30

Keywords: Whiskered tori, Conformally symplectic systems, KAM theory.

Accepted: 2019-05-21

Published online: 2020-06-30

Classification: 70K43, 70K20, 34D30

Keywords: Whiskered tori, Conformally symplectic systems, KAM theory.

@article{MRR_2020__1__15_0, author = {Renato C. Calleja and Alessandra Celletti and Rafael de la Llave}, title = {Whiskered KAM tori of conformally symplectic systems}, journal = {Mathematics Research Reports}, publisher = {MathOA foundation}, volume = {1}, year = {2020}, pages = {15-29}, doi = {10.5802/mrr.4}, language = {en}, url = {mrr.centre-mersenne.org/item/MRR_2020__1__15_0/} }

Renato C. Calleja; Alessandra Celletti; Rafael de la Llave. Whiskered KAM tori of conformally symplectic systems. Mathematics Research Reports, Volume 1 (2020) , pp. 15-29. doi : 10.5802/mrr.4. https://mrr.centre-mersenne.org/item/MRR_2020__1__15_0/

[1] Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Russian Math. Surveys, Volume 18 (1963) no. 5, pp. 9-36 | Article | MR 0163025

[2] Instability of dynamical systems with many degrees of freedom, Sov. Math. Doklady, Volume 156 (1964), pp. 581-585 | MR 0163026

[3] Some properties of locally conformal symplectic structures, Comment. Math. Helv., Volume 77 (2002) no. 2, pp. 383-398 | Article | MR 1915047 | Zbl 1020.53050

[4] Unfoldings and bifurcations of quasi-periodic tori, Mem. Amer. Math. Soc., Volume 83 (1990) no. 421, viii+175 pages | Article | MR 1041003 | Zbl 0717.58043

[5] Quasi-periodic motions in families of dynamical systems. Order amidst chaos, Lecture Notes in Mathematics, Volume 1645, Springer-Verlag, Berlin, 1996, xii+196 pages | MR 1484969 | Zbl 0870.58087

[6] Computation of domains of analyticity for the dissipative standard map in the limit of small dissipation, Phys. D, Volume 395 (2019), pp. 15-23 | Article | MR 3958298

[7] Response solutions for quasi-periodically forced, dissipative wave equations, SIAM J. Math. Anal., Volume 49 (2017) no. 4, pp. 3161-3207 | Article | MR 3686798 | Zbl 06764375

[8] A KAM theory for conformally symplectic systems: efficient algorithms and their validation, J. Differential Equations, Volume 255 (2013) no. 5, pp. 978-1049 | Article | MR 3062760 | Zbl 1345.37078

[9] Domains of analyticity and Lindstedt expansions of KAM tori in some dissipative perturbations of Hamiltonian systems, Nonlinearity, Volume 30 (2017) no. 8, pp. 3151-3202 | Article | MR 3685665 | Zbl 1425.70037

[10] Existence of whiskered KAM tori of conformally symplectic systems, Nonlinearity, Volume 33 (2020) no. 1, pp. 538-597 | Article | MR 4039781 | Zbl 07141430

[11] Computation of quasiperiodic normally hyperbolic invariant tori: rigorous results, J. Nonlinear Sci., Volume 27 (2017) no. 6, pp. 1869-1904 | Article | MR 3713933 | Zbl 1380.37061

[12] Stability and chaos in celestial mechanics, Springer-Verlag, Berlin; published in association with Praxis Publishing, Chichester, 2010, xvi+261 pages | Article | MR 2571993 | Zbl 1203.70001

[13] Dichotomies in stability theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978, ii+98 pages | Article | MR 0481196 | Zbl 0376.34001

[14] Convergence of the solutions of the discounted equation: the discrete case, Math. Z., Volume 284 (2016) no. 3-4, pp. 1021-1034 | Article | MR 3563265 | Zbl 1352.37170

[15] Convergence of the solutions of the discounted Hamilton-Jacobi equation: convergence of the discounted solutions, Invent. Math., Volume 206 (2016) no. 1, pp. 29-55 | Article | MR 3556524 | Zbl 1362.35094

[16] Proof of Lyapunov exponent pairing for systems at constant kinetic energy, Phys. Rev. E, Volume 53 (1996) no. 6, p. R5545-R5548 | Article

[17] Construction of invariant whiskered tori by a parameterization method. Part I. Maps and flows in finite dimensions, J. Differential Equations, Volume 246 (2009) no. 8, pp. 3136-3213 | Article | MR 2507954 | Zbl 1209.37066

[18] Construction of invariant whiskered tori by a parameterization method. Part II: Quasi-periodic and almost periodic breathers in coupled map lattices, J. Differential Equations, Volume 259 (2015) no. 6, pp. 2180-2279 | Article | MR 3353644 | Zbl 1351.37258

[19] Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977, ii+149 pages | MR 0501173 | Zbl 0355.58009

[20] KAM theory without action-angle variables, Nonlinearity, Volume 18 (2005) no. 2, pp. 855-895 | Article | MR 2122688 | Zbl 1067.37081

[21] A tutorial on KAM theory, Smooth ergodic theory and its applications (Seattle, WA, 1999) (Proc. Sympos. Pure Math.) Volume 69, Amer. Math. Soc., Providence, RI, 2001, pp. 175-292 | Article | MR 1858536 | Zbl 1055.37064

[22] An a posteriori KAM theorem for whiskered tori in Hamiltonian partial differential equations with applications to some ill-posed equations, Arch. Ration. Mech. Anal., Volume 231 (2019) no. 2, pp. 971-1044 | Article | MR 3900818 | Zbl 1407.37108

[23] Quasi-periodic motions in a special class of dynamical equations with dissipative effects: a pair of detection methods, Discrete Contin. Dyn. Syst. Ser. B, Volume 20 (2015) no. 4, pp. 1155-1187 | Article | MR 3315491 | Zbl 1307.70010

[24] Convergent series expansions for quasi-periodic motions, Math. Ann., Volume 169 (1967), pp. 136-176 | Article | MR 208078 | Zbl 0149.29903

[25] Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, Volume 15 (1974), pp. 429-458 | Article | MR 341458 | Zbl 0294.58008

[26] Kolmogorov’s normal form for equations of motion with dissipative effects, Discrete Contin. Dyn. Syst. Ser. B, Volume 17 (2012) no. 7, pp. 2561-2593 | Article | MR 2946318 | Zbl 1287.37040

[27] Convergence of discrete Aubry-Mather model in the continuous limit, Nonlinearity, Volume 31 (2018) no. 5, pp. 2126-2155 | Article | MR 3816667 | Zbl 1394.37097

[28] Conformally symplectic dynamics and symmetry of the Lyapunov spectrum, Comm. Math. Phys., Volume 194 (1998) no. 1, pp. 47-60 | Article | MR 1628286 | Zbl 0951.37016