# American Institute of Mathematical Sciences

February  2021, 20(2): 467-494. doi: 10.3934/cpaa.2020222

## Response solutions to harmonic oscillators beyond multi–dimensional Brjuno frequency

 1 School of Mathematical Sciences, Tiangong University, Tianjin 300387, China 2 School of Mathematics, Shandong University, Jinan 250100, China

* Corresponding author

Received  January 2020 Revised  May 2020 Published  February 2021 Early access  December 2020

Fund Project: The first author was supported by NSFC (No. 12001294). The second author was partially supported by CSC (No. 201706220147) and NSFC (No. 12001397)

This paper focuses on the quasi–periodically forced nonlinear harmonic oscillators
 $\begin{equation*} \ddot{x}+\lambda^{2}x = \epsilon f(\omega t,x), \end{equation*}$
where
 $\lambda \in \mathcal{O}$
, a closed interval not containing zero, the forcing term
 $f$
is real analytic, and the frequency vector
 $\omega \in \mathbb{R}^d \, (d \geq 2)$
is beyond Brjuno frequency, which we call as Liouvillean frequency. For the given class of the frequency
 $\omega\in\mathbb{R}^{d},$
which will be given later, we prove the existence of real analytic response solutions (the response solution is the quasi–periodic solution with the same frequency as the forcing) for the above equation. The proof is based on a modified KAM (Kolmogorov–Arnold–Moser) theorem for finite–dimensional harmonic oscillator systems with Liouvillean frequency.
Citation: Hongyu Cheng, Shimin Wang. Response solutions to harmonic oscillators beyond multi–dimensional Brjuno frequency. Communications on Pure and Applied Analysis, 2021, 20 (2) : 467-494. doi: 10.3934/cpaa.2020222
##### References:
 [1] A. Avila, B. Fayad and R. Krikorian, A KAM scheme for $\mathrm{SL}(2, \mathbb{R})$ cocycles with Liouvillean frequencies, Geom. Funct. Anal., 21 (2011), 1001-1019.  doi: 10.1007/s00039-011-0135-6. [2] A. Avila, J. You and Q. Zhou, Sharp phase transitions for the almost Mathieu operator, Duke Math. J., 166 (2017), 2697-2718.  doi: 10.1215/00127094-2017-0013. [3] M. Berti, KAM theory for partial differential equations, Anal. Theory Appl., 35 (2019), 235-267.  doi: 10.4208/ata.oa-0013. [4] B. L. J. Braaksma and H. W. Broer, On a quasiperiodic Hopf bifurcation, Ann. Inst. Henri Poincare Anal. Non Lineaire, 4 (1987), 115-168. [5] H. Cheng, W. Si and J. Si, Whiskered tori for forced beam equations with multi-dimensional liouvillean frequency, J. Dyn. Differ. Equ., 32 (2020), 705-739.  doi: 10.1007/s10884-019-09754-1. [6] Y. Cheung, Hausdorff dimension of the set of singular pairs, Ann. Math., 173 (2011), 127-167.  doi: 10.4007/annals.2011.173.1.4. [7] L. H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 15 (1988), 115-147. [8] L. H. Eliasson, B. Grébert and S. B. Kuksin, KAM for the nonlinear beam equation, Geom. Funct. Anal., 26 (2016), 1588-1715.  doi: 10.1007/s00039-016-0390-7. [9] L. H. Eliasson and S. B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. Math., 172 (2010), 371-435.  doi: 10.4007/annals.2010.172.371. [10] M. Friedman, Quasi-periodic solutions of nonlinear ordinary differential equations with small damping, Bull. Amer. Math. Soc., 73 (1967), 460-464.  doi: 10.1090/S0002-9904-1967-11783-X. [11] J. Geng and X. Ren, Lower dimensional invariant tori with prescribed frequency for nonlinear wave equation, J. Differ. Equ., 249 (2010), 2796-2821.  doi: 10.1016/j.jde.2010.04.003. [12] J. Geng, X. Xu and J. You, An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation, Adv. Math., 226 (2011), 5361-5402.  doi: 10.1016/j.aim.2011.01.013. [13] J. Geng and J. You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Commun. Math. Phys., 262 (2006), 343-372.  doi: 10.1007/s00220-005-1497-0. [14] Y. Han, Y. Li and Y. Yi, Degenerate lower-dimensional tori in Hamiltonian systems, J. Differ. Equ., 227 (2006), 670-691.  doi: 10.1016/j.jde.2006.02.006. [15] X. Hou and J. You, Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems, Invent. Math., 190 (2012), 209-260.  doi: 10.1007/s00222-012-0379-2. [16] T. Kappeler and J. Pöschel, KdV & KAM, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2. [17] R. Krikorian, J. Wang, J. You and Q. Zhou, Linearization of quasiperiodically forced circle flows beyond brjuno condition, Commun. Math. Phys., 358 (2018), 81-100.  doi: 10.1007/s00220-017-3021-8. [18] S. B. Kuksin, A KAM-theorem for equations of the Korteweg-de Vries type, Rev. Math. Math. Phys., 10 (1998), 1-64. [19] S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford University Press, Oxford, 2000. [20] S. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. Math., 143 (1996), 149-179.  doi: 10.2307/2118656. [21] Y. Li and Y. Yi, Persistence of lower dimensional tori of general types in Hamiltonian systems, T. Am. Math. Soc., 357 (2005), 1565-1600.  doi: 10.1090/S0002-9947-04-03564-0. [22] J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large-variable coefficient, Commun. Pure Appl. Math., 63 (2010), 1145-1172.  doi: 10.1002/cpa.20314. [23] J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Commun. Math. Phys., 307 (2011), 629-673.  doi: 10.1007/s00220-011-1353-3. [24] Z. Lou and J. Geng, Quasi-periodic response solutions in forced reversible systems with liouvillean frequencies, J. Differ. Equ., 263 (2017), 3894-3927.  doi: 10.1016/j.jde.2017.05.007. [25] J. Moser, Combination tones for Duffing's equation, Commun. Pure Appl. Math., 18 (1965), 167-181.  doi: 10.1002/cpa.3160180116. [26] J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z., 202 (1989), 559-608.  doi: 10.1007/BF01221590. [27] W. Si and J. Si, Response solutions and quasi-periodic degenerate bifurcations for quasi-periodically forced systems, Nonlinearity, 31 (2018), 2361-2418.  doi: 10.1088/1361-6544/aaa7b9. [28] J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publishers, Inc., New York, N.Y., 1950. [29] J. Wang and J. You, Boundedness of solutions for non-linear quasi-periodic differential equations with Liouvillean frequency, J. Differ. Equ., 261 (2016), 1068-1098.  doi: 10.1016/j.jde.2016.03.038. [30] J. Wang, J. You and Q. Zhou, Response solutions for quasi-periodically forced harmonic oscillators, T. Am. Math. Soc., 369 (2017), 4251-4274.  doi: 10.1090/tran/6800. [31] J. Xu, J. You and Q. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z., 226 (1997), 375-387.  doi: 10.1007/PL00004344. [32] X. Xu, J. You and Q. Zhou, Quasi-periodic solutions of NLS with Liouvillean frequency, preprint, arXiv: 1707.04048. [33] J. You and Q. Zhou, Phase transition and semi-global reducibility, Commun. Math. Phys., 330 (2014), 1095-1113.  doi: 10.1007/s00220-014-2012-2. [34] D. Zhang, J. Xu and X. Xu, Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies, Discrete Contin. Dyn. Syst., 38 (2018), 2851-2877.  doi: 10.3934/dcds.2018123. [35] Q. Zhou and J. Wang, Reducibility results for quasiperiodic cocycles with liouvillean frequency, J. Dyn. Differ. Equ., 24 (2012), 61-83.  doi: 10.1007/s10884-011-9235-0.

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##### References:
 [1] A. Avila, B. Fayad and R. Krikorian, A KAM scheme for $\mathrm{SL}(2, \mathbb{R})$ cocycles with Liouvillean frequencies, Geom. Funct. Anal., 21 (2011), 1001-1019.  doi: 10.1007/s00039-011-0135-6. [2] A. Avila, J. You and Q. Zhou, Sharp phase transitions for the almost Mathieu operator, Duke Math. J., 166 (2017), 2697-2718.  doi: 10.1215/00127094-2017-0013. [3] M. Berti, KAM theory for partial differential equations, Anal. Theory Appl., 35 (2019), 235-267.  doi: 10.4208/ata.oa-0013. [4] B. L. J. Braaksma and H. W. Broer, On a quasiperiodic Hopf bifurcation, Ann. Inst. Henri Poincare Anal. Non Lineaire, 4 (1987), 115-168. [5] H. Cheng, W. Si and J. Si, Whiskered tori for forced beam equations with multi-dimensional liouvillean frequency, J. Dyn. Differ. Equ., 32 (2020), 705-739.  doi: 10.1007/s10884-019-09754-1. [6] Y. Cheung, Hausdorff dimension of the set of singular pairs, Ann. Math., 173 (2011), 127-167.  doi: 10.4007/annals.2011.173.1.4. [7] L. H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 15 (1988), 115-147. [8] L. H. Eliasson, B. Grébert and S. B. Kuksin, KAM for the nonlinear beam equation, Geom. Funct. Anal., 26 (2016), 1588-1715.  doi: 10.1007/s00039-016-0390-7. [9] L. H. Eliasson and S. B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. Math., 172 (2010), 371-435.  doi: 10.4007/annals.2010.172.371. [10] M. Friedman, Quasi-periodic solutions of nonlinear ordinary differential equations with small damping, Bull. Amer. Math. Soc., 73 (1967), 460-464.  doi: 10.1090/S0002-9904-1967-11783-X. [11] J. Geng and X. Ren, Lower dimensional invariant tori with prescribed frequency for nonlinear wave equation, J. Differ. Equ., 249 (2010), 2796-2821.  doi: 10.1016/j.jde.2010.04.003. [12] J. Geng, X. Xu and J. You, An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation, Adv. Math., 226 (2011), 5361-5402.  doi: 10.1016/j.aim.2011.01.013. [13] J. Geng and J. You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Commun. Math. Phys., 262 (2006), 343-372.  doi: 10.1007/s00220-005-1497-0. [14] Y. Han, Y. Li and Y. Yi, Degenerate lower-dimensional tori in Hamiltonian systems, J. Differ. Equ., 227 (2006), 670-691.  doi: 10.1016/j.jde.2006.02.006. [15] X. Hou and J. You, Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems, Invent. Math., 190 (2012), 209-260.  doi: 10.1007/s00222-012-0379-2. [16] T. Kappeler and J. Pöschel, KdV & KAM, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2. [17] R. Krikorian, J. Wang, J. You and Q. Zhou, Linearization of quasiperiodically forced circle flows beyond brjuno condition, Commun. Math. Phys., 358 (2018), 81-100.  doi: 10.1007/s00220-017-3021-8. [18] S. B. Kuksin, A KAM-theorem for equations of the Korteweg-de Vries type, Rev. Math. Math. Phys., 10 (1998), 1-64. [19] S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford University Press, Oxford, 2000. [20] S. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. Math., 143 (1996), 149-179.  doi: 10.2307/2118656. [21] Y. Li and Y. Yi, Persistence of lower dimensional tori of general types in Hamiltonian systems, T. Am. Math. Soc., 357 (2005), 1565-1600.  doi: 10.1090/S0002-9947-04-03564-0. [22] J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large-variable coefficient, Commun. Pure Appl. Math., 63 (2010), 1145-1172.  doi: 10.1002/cpa.20314. [23] J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Commun. Math. Phys., 307 (2011), 629-673.  doi: 10.1007/s00220-011-1353-3. [24] Z. Lou and J. Geng, Quasi-periodic response solutions in forced reversible systems with liouvillean frequencies, J. Differ. Equ., 263 (2017), 3894-3927.  doi: 10.1016/j.jde.2017.05.007. [25] J. Moser, Combination tones for Duffing's equation, Commun. Pure Appl. Math., 18 (1965), 167-181.  doi: 10.1002/cpa.3160180116. [26] J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z., 202 (1989), 559-608.  doi: 10.1007/BF01221590. [27] W. Si and J. Si, Response solutions and quasi-periodic degenerate bifurcations for quasi-periodically forced systems, Nonlinearity, 31 (2018), 2361-2418.  doi: 10.1088/1361-6544/aaa7b9. [28] J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publishers, Inc., New York, N.Y., 1950. [29] J. Wang and J. You, Boundedness of solutions for non-linear quasi-periodic differential equations with Liouvillean frequency, J. Differ. Equ., 261 (2016), 1068-1098.  doi: 10.1016/j.jde.2016.03.038. [30] J. Wang, J. You and Q. Zhou, Response solutions for quasi-periodically forced harmonic oscillators, T. Am. Math. Soc., 369 (2017), 4251-4274.  doi: 10.1090/tran/6800. [31] J. Xu, J. You and Q. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z., 226 (1997), 375-387.  doi: 10.1007/PL00004344. [32] X. Xu, J. You and Q. Zhou, Quasi-periodic solutions of NLS with Liouvillean frequency, preprint, arXiv: 1707.04048. [33] J. You and Q. Zhou, Phase transition and semi-global reducibility, Commun. Math. Phys., 330 (2014), 1095-1113.  doi: 10.1007/s00220-014-2012-2. [34] D. Zhang, J. Xu and X. Xu, Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies, Discrete Contin. Dyn. Syst., 38 (2018), 2851-2877.  doi: 10.3934/dcds.2018123. [35] Q. Zhou and J. Wang, Reducibility results for quasiperiodic cocycles with liouvillean frequency, J. Dyn. Differ. Equ., 24 (2012), 61-83.  doi: 10.1007/s10884-011-9235-0.
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