# American Institute of Mathematical Sciences

August  2021, 41(8): 3951-3972. doi: 10.3934/dcds.2021023

## Response solutions for degenerate reversible harmonic oscillators

 School of Mathematics, Shandong University, Jinan, Shandong 250100, China

* Corresponding author: Wen Si

Received  July 2020 Revised  December 2020 Published  August 2021 Early access  January 2021

Fund Project: W. Si was partially supported by the National Natural Science Foundation of China (Grant Nos. 12001315); Shandong Provincial Natural Science Foundation, China (Grant Nos. ZR2020MA015); China Postdoctoral Science Foundation (Grant Nos. 2020M680089) and the Fundamental Research Funds of Shandong University (Grant Nos. 2019GN077). This paper is also supported by the National Natural Science Foundation of China (Grant Nos. 11971261, 11571201)

We consider the existence of response solutions for the quasi-periodic perturbation of degenerate reversible harmonic oscillators
 $\ddot{x}-\lambda x^n = \epsilon f(\omega t, x, \dot x, \epsilon), \; \; x\in \mathbb{R},$
where
 $\lambda = \pm 1$
,
 $n>1$
is an integer and
 $f(-\omega t, x, -\dot x, \epsilon) = f(\omega t, x, \dot x, \epsilon)$
. With
 $f$
satisfying certain non-degenerate conditions, we obtain the following results: (1) For
 $\lambda = 1$
and
 $\epsilon$
sufficiently small, response solutions exist for each
 $\omega$
satisfying a weak non-resonant condition; (2) For
 $\lambda = -1$
and
 $\epsilon_*$
sufficiently small, there exists a Cantor set
 $\mathcal{E}\in(0, \epsilon_*)$
with almost full Lebesgue measure such that response solutions exist for each
 $\epsilon\in\mathcal{E}$
if
 $\omega$
satisfies a Diophantine condition. Non-existence of response solutions is also discussed when
 $f$
fails to satisfy the non-degenerate conditions.
Citation: Wen Si. Response solutions for degenerate reversible harmonic oscillators. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3951-3972. doi: 10.3934/dcds.2021023
##### References:
 [1] B. L. J. Braaksma and H. W. Broer, On a quasi-periodic Hopf bifurcation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 115-168.  doi: 10.1016/S0294-1449(16)30370-5. [2] H. W. Broer, M. C. Ciocci and H. Hanßmann, The quasi-periodic reversible Hopf bifurcation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2605-2623.  doi: 10.1142/S021812740701866X. [3] L. Corsi and G. Gentile, Oscillator synchronisation under arbitrary quasi-periodic forcing, Comm. Math. Phys., 316 (2012), 489-529.  doi: 10.1007/s00220-012-1548-2. [4] L. Corsi and G. Gentile, Resonant tori of arbitrary codimension for quasi-periodically forced systems, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Paper No. 3, 21 pp. doi: 10.1007/s00030-016-0425-7. [5] 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. [6] G. Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory Dynam. Systems, 27 (2007), 427-457.  doi: 10.1017/S0143385706000757. [7] G. Gentile, Quasi-periodic motions in strongly dissipative forced systems, Ergodic Theory Dynam. Systems, 30 (2010), 1457-1469.  doi: 10.1017/S0143385709000583. [8] G. Gentile, Construction of quasi-periodic response solutions in forced strongly dissipative systems, Forum Math., 24 (2012), 791-808. [9] Y. Han, Y. Li and Y. Yi, Degenerate lower-dimensional tori in Hamiltonian systems, J. Differential Equations, 227 (2006), 670-691.  doi: 10.1016/j.jde.2006.02.006. [10] H. Hanßmann, Quasi-periodic bifurcations in reversible systems, Regul. Chaotic Dyn., 16 (2011), 51-60.  doi: 10.1134/S1560354710520059. [11] S. Hu and B. Liu, Degenerate lower dimensional invariant tori in reversible system, Discrete Contin. Dyn. Syst., 38 (2018), 3735-3763.  doi: 10.3934/dcds.2018162. [12] S. Hu and B. Liu, Completely degenerate lower-dimensional invariant tori for Hamiltonian system, J. Differential Equations, 266 (2019), 7459-7480.  doi: 10.1016/j.jde.2018.12.001. [13] Z. Lou and J. Geng, Quasi-periodic response solutions in forced reversible systems with Liouvillean frequencies, J. Differential Equations, 263 (2017), 3894-3927.  doi: 10.1016/j.jde.2017.05.007. [14] J. Moser, Combination tones for Duffings equation, Comm. Pure Appl. Math., 18 (1965), 167-181.  doi: 10.1002/cpa.3160180116. [15] J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publisher, New York, 1950. [16] W. Si and J. Si, Construction of response solutions for two classes of quasi-periodically forced four-dimensional nonlinear systems with degenerate equilibrium point under small perturbations, J. Differential Equations, 262 (2017), 4771-4822.  doi: 10.1016/j.jde.2016.12.019. [17] W. Si and Y. Yi, Completely degenerate responsive tori in Hamiltonian systems, Nonlinearity, 33 (2020), 6072-6098.  doi: 10.1088/1361-6544/aba093. [18] J. Wang, J. You and Q. Zhou, Response solutions for quasi-periodically forced harmonic oscillators, Trans. Amer. Math. Soc., 369 (2017), 4251-4274.  doi: 10.1090/tran/6800. [19] J. You, A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems, Commun. Math. Phys., 192 (1998), 145-168.  doi: 10.1007/s002200050294. [20] X. Wang, J. Xu and D. Zhang, Degenerate lower dimensional tori in reversible systems, J. Math. Anal. Appl., 387 (2012), 776-790.  doi: 10.1016/j.jmaa.2011.09.030. [21] 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.  doi: 10.1017/etds.2014.34.

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##### References:
 [1] B. L. J. Braaksma and H. W. Broer, On a quasi-periodic Hopf bifurcation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 115-168.  doi: 10.1016/S0294-1449(16)30370-5. [2] H. W. Broer, M. C. Ciocci and H. Hanßmann, The quasi-periodic reversible Hopf bifurcation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2605-2623.  doi: 10.1142/S021812740701866X. [3] L. Corsi and G. Gentile, Oscillator synchronisation under arbitrary quasi-periodic forcing, Comm. Math. Phys., 316 (2012), 489-529.  doi: 10.1007/s00220-012-1548-2. [4] L. Corsi and G. Gentile, Resonant tori of arbitrary codimension for quasi-periodically forced systems, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Paper No. 3, 21 pp. doi: 10.1007/s00030-016-0425-7. [5] 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. [6] G. Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory Dynam. Systems, 27 (2007), 427-457.  doi: 10.1017/S0143385706000757. [7] G. Gentile, Quasi-periodic motions in strongly dissipative forced systems, Ergodic Theory Dynam. Systems, 30 (2010), 1457-1469.  doi: 10.1017/S0143385709000583. [8] G. Gentile, Construction of quasi-periodic response solutions in forced strongly dissipative systems, Forum Math., 24 (2012), 791-808. [9] Y. Han, Y. Li and Y. Yi, Degenerate lower-dimensional tori in Hamiltonian systems, J. Differential Equations, 227 (2006), 670-691.  doi: 10.1016/j.jde.2006.02.006. [10] H. Hanßmann, Quasi-periodic bifurcations in reversible systems, Regul. Chaotic Dyn., 16 (2011), 51-60.  doi: 10.1134/S1560354710520059. [11] S. Hu and B. Liu, Degenerate lower dimensional invariant tori in reversible system, Discrete Contin. Dyn. Syst., 38 (2018), 3735-3763.  doi: 10.3934/dcds.2018162. [12] S. Hu and B. Liu, Completely degenerate lower-dimensional invariant tori for Hamiltonian system, J. Differential Equations, 266 (2019), 7459-7480.  doi: 10.1016/j.jde.2018.12.001. [13] Z. Lou and J. Geng, Quasi-periodic response solutions in forced reversible systems with Liouvillean frequencies, J. Differential Equations, 263 (2017), 3894-3927.  doi: 10.1016/j.jde.2017.05.007. [14] J. Moser, Combination tones for Duffings equation, Comm. Pure Appl. Math., 18 (1965), 167-181.  doi: 10.1002/cpa.3160180116. [15] J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publisher, New York, 1950. [16] W. Si and J. Si, Construction of response solutions for two classes of quasi-periodically forced four-dimensional nonlinear systems with degenerate equilibrium point under small perturbations, J. Differential Equations, 262 (2017), 4771-4822.  doi: 10.1016/j.jde.2016.12.019. [17] W. Si and Y. Yi, Completely degenerate responsive tori in Hamiltonian systems, Nonlinearity, 33 (2020), 6072-6098.  doi: 10.1088/1361-6544/aba093. [18] J. Wang, J. You and Q. Zhou, Response solutions for quasi-periodically forced harmonic oscillators, Trans. Amer. Math. Soc., 369 (2017), 4251-4274.  doi: 10.1090/tran/6800. [19] J. You, A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems, Commun. Math. Phys., 192 (1998), 145-168.  doi: 10.1007/s002200050294. [20] X. Wang, J. Xu and D. Zhang, Degenerate lower dimensional tori in reversible systems, J. Math. Anal. Appl., 387 (2012), 776-790.  doi: 10.1016/j.jmaa.2011.09.030. [21] 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.  doi: 10.1017/etds.2014.34.
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