On the linearization of vector fields on a torus with prescribed frequency

Dongfebg Zhang, Xindong Xu

DOI: http://dx.doi.org/10.12775/TMNA.2019.058

Abstract


In this paper we are mainly concerned with the linearization of the flow with prescribed frequency for analytic perturbation of constant vector fields on a torus under weaker non-degeneracy condition and non-resonant condition. As is well known the perturbation of constant vector fields may induce a shift of frequency, when Kolmogorov's non-degeneracy condition is violated. By introducing external parameters and using the polynomial structure to truncate, we prove that if the frequency mapping has the nonzero Brouwer's topological degree at some non-resonant frequency, then the conjugated vector fields will have a linear flow with this frequency.

Keywords


KAM theory; linear flow; non-degeneracy condition; non-resonant condition

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References


V.I. Arnold, Proof of a theorem of A.N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Math. Nauk 18 (1963), 13–40; Russian Math. Surveys 18 (1963), 9–36.

A. Avila, B. Fayad and R. Krikorian, A KAM scheme for SL(2, R) cocycles with Liouvillean frequencies, Geom. Funct. Anal. 21 (2011), 1001–1019.

A. Bounemoura and S. Fischler, A Diophantine duality applied to the KAM and Nekhoroshev theorems, Math. Z. 275 (2013), 1135–1167.

A. Bounemoura and S. Fischler, The classical KAM theorem for Hamiltonian systems via rational approximations, Regul. Chaotic Dyn. 19 (2014), 251–265.

H.W. Broer, G.B. Huitema, F. Takens and B.L.J. Braaksma, Unfoldings of quasiperiodic tori, Mem. Amer. Math. Soc. 83 (1990), no. 421, viii+175 pp.

H.W. Broer, G.B. Huitema, Unfoldings of quasi-periodic tori in reversible systems, J. Dynam. Differential Equations 7 (1995), no. 1, 191–212.

H.W. Broer, G.B. Huitema and M.B. Sevryuk, Families of quasi-periodic motions in dynamical systems depending on parameters, Nonlinear Dynamical Systems and Chaos (Gröningen, 1995), 171–211; Progr. Nonlinear Differential Equations Appl., vol. 19, Birkhäuser, Basel, 1996.

M. Herman, Dynamics connected with indefinite normal torsion, (English summary) Twist mappings and their applications (English summary), IMA Vol. Math. Appl., vol. 44, Springer, New York, 1992, 153–182.

X. Hou and J. You, Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems, Invent. Math. 190 (2012), 209–260.

A.N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk SSSR 98 (1954), 527–530.

Yu.V. Loveiki and I.O. Parasyuk, Invariant tori of locally Hamiltonian systems that are close to conditionally integrable systems, Ukrainian Math. 59 (2007), 70–99.

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

J. Pöschel, A Lecture on the classical KAM theorem, Proc. Symp. Pure Math. 69 (2001), 707–732.

J. Pöschel, KAM à la R, Regul. Chaotic Dyn. 16 (2011), 17–23.

H. Rüssmann, Nondegeneracy in the perturbation theory of integrable dynamical systems, Stochastics, algebra and analysis in classical and quantum dynamics (Marseille, 1988), Math. Appl., 59, Kluwer Acad. Publ., Dordrecht, 1990, 211–223.

H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn. 6 (2001), 119–204.

M.B. Sevryuk, KAM stable Hamiltonians, J. Dynamics Control Systems 1 (1995), 351–366.

J. Xu, J. You and Q. Qiu, Invariant tori of nearly integrable Hamiltonian systems with degeneracy, Math. Z. 226 (1997), 375–386.

J. Xu and J. You, Persistence of the non-twist torus in nearly integrable Hamiltonian systems, Proc. Amer. Math. Soc. 138 (2010), 2385–2395.

J.-C. Yoccoz, Travaux de Herman sur les tores invariants, Séminaire Bourbaki, Vol. 1991/92. Astérisque No. 206 (1992), Exp. No. 754, 311–344.

D. Zhang and J. Xu, Invariant curves of analytic reversible mappings under Brjuno–Rüssmann’s non-resonant condition, J. Dynam. Differential Equations 26 (2014), no. 4, 989–1005.

D. Zhang and J. Xu, On invariant tori of vector field under weaker non-degeneracy condition, Nonlinear Differential Equations and Applications 22 (2015), 1381–1394.

X. Wang, J. Xu and D. Zhang, On the persistence of degenerate lower-dimensional tori in reversible systems, Ergodic Theory Dynam. Systems, 35 (2015), 2311–2333.

X. Wang, J. Xu and D. Zhang, On the persistence of lower-dimensional elliptic tori with prescribed frequencies in reversible systems, Discrete Contin. Dyn. Syst. Ser. A 36 (2016), 1677–1692.


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