American Institute of Mathematical Sciences

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

Response solutions for degenerate reversible harmonic oscillators

Wen Si ,

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

* Corresponding author: Wen Si

Received  July 2020 Revised  December 2020 Published  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.
Keywords: Degenerate harmonic oscillators, reversible system, KAM theory, weak non-resonant condition, response solution.
Mathematics Subject Classification: Primary: 34J40, 34C27; Secondary: 34E20.
Citation: Wen Si. Response solutions for degenerate reversible harmonic oscillators. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

G. Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory Dynam. Systems, 27 (2007), 427-457.  doi: 10.1017/S0143385706000757.  Google Scholar

[7]

G. Gentile, Quasi-periodic motions in strongly dissipative forced systems, Ergodic Theory Dynam. Systems, 30 (2010), 1457-1469.  doi: 10.1017/S0143385709000583.  Google Scholar

[8]

G. Gentile, Construction of quasi-periodic response solutions in forced strongly dissipative systems, Forum Math., 24 (2012), 791-808.   Google Scholar

[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.  Google Scholar

[10]

H. Hanßmann, Quasi-periodic bifurcations in reversible systems, Regul. Chaotic Dyn., 16 (2011), 51-60.  doi: 10.1134/S1560354710520059.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. Moser, Combination tones for Duffings equation, Comm. Pure Appl. Math., 18 (1965), 167-181.  doi: 10.1002/cpa.3160180116.  Google Scholar

[15]

J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publisher, New York, 1950.  Google Scholar

[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.  Google Scholar

[17]

W. Si and Y. Yi, Completely degenerate responsive tori in Hamiltonian systems, Nonlinearity, 33 (2020), 6072-6098.  doi: 10.1088/1361-6544/aba093.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

G. Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory Dynam. Systems, 27 (2007), 427-457.  doi: 10.1017/S0143385706000757.  Google Scholar

[7]

G. Gentile, Quasi-periodic motions in strongly dissipative forced systems, Ergodic Theory Dynam. Systems, 30 (2010), 1457-1469.  doi: 10.1017/S0143385709000583.  Google Scholar

[8]

G. Gentile, Construction of quasi-periodic response solutions in forced strongly dissipative systems, Forum Math., 24 (2012), 791-808.   Google Scholar

[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.  Google Scholar

[10]

H. Hanßmann, Quasi-periodic bifurcations in reversible systems, Regul. Chaotic Dyn., 16 (2011), 51-60.  doi: 10.1134/S1560354710520059.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. Moser, Combination tones for Duffings equation, Comm. Pure Appl. Math., 18 (1965), 167-181.  doi: 10.1002/cpa.3160180116.  Google Scholar

[15]

J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publisher, New York, 1950.  Google Scholar

[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.  Google Scholar

[17]

W. Si and Y. Yi, Completely degenerate responsive tori in Hamiltonian systems, Nonlinearity, 33 (2020), 6072-6098.  doi: 10.1088/1361-6544/aba093.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Virginie Bonnaillie-Noël. Harmonic oscillators with Neumann condition on the half-line. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2221-2237. doi: 10.3934/cpaa.2012.11.2221
[2]

Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57
[3]

Hongyu Cheng, Shimin Wang. Response solutions to harmonic oscillators beyond multi–dimensional Brjuno frequency. Communications on Pure & Applied Analysis, 2021, 20 (2) : 467-494. doi: 10.3934/cpaa.2020222
[4]

Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069
[5]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080
[6]

Shengqing Hu, Bin Liu. Degenerate lower dimensional invariant tori in reversible system. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3735-3763. doi: 10.3934/dcds.2018162
[7]

Luigi Chierchia, Gabriella Pinzari. Properly-degenerate KAM theory (following V. I. Arnold). Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 545-578. doi: 10.3934/dcdss.2010.3.545
[8]

Maxime Zavidovique. Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory. Journal of Modern Dynamics, 2010, 4 (4) : 693-714. doi: 10.3934/jmd.2010.4.693
[9]

Kazuyuki Yagasaki. Degenerate resonances in forced oscillators. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 423-438. doi: 10.3934/dcdsb.2003.3.423
[10]

Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191
[11]

Xiaocai Wang, Junxiang Xu. Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 701-718. doi: 10.3934/dcds.2009.25.701
[12]

Fabiana Maria Ferreira, Francisco Odair de Paiva. On a resonant and superlinear elliptic system. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5775-5784. doi: 10.3934/dcds.2019253
[13]

H. M. Yin. Optimal regularity of solution to a degenerate elliptic system arising in electromagnetic fields. Communications on Pure & Applied Analysis, 2002, 1 (1) : 127-134. doi: 10.3934/cpaa.2002.1.127
[14]

Michal Fečkan. Blue sky catastrophes in weakly coupled chains of reversible oscillators. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 193-200. doi: 10.3934/dcdsb.2003.3.193
[15]

Meng Wang, Wendong Wang, Zhifei Zhang. On the uniqueness of weak solution for the 2-D Ericksen--Leslie system. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 919-941. doi: 10.3934/dcdsb.2016.21.919
[16]

Yong Zeng. Existence and uniqueness of very weak solution of the MHD type system. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5617-5638. doi: 10.3934/dcds.2020240
[17]

Thi-Bich-Ngoc Mac. Existence of solution for a system of repulsion and alignment: Comparison between theory and simulation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3013-3027. doi: 10.3934/dcdsb.2015.20.3013
[18]

Jian Wu, Jiansheng Geng. Almost periodic solutions for a class of semilinear quantum harmonic oscillators. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 997-1015. doi: 10.3934/dcds.2011.31.997
[19]

Meina Gao, Jianjun Liu. A degenerate KAM theorem for partial differential equations with periodic boundary conditions. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5911-5928. doi: 10.3934/dcds.2020252
[20]

Roberto Livrea, Salvatore A. Marano. A min-max principle for non-differentiable functions with a weak compactness condition. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1019-1029. doi: 10.3934/cpaa.2009.8.1019

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (54)
  • HTML views (138)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences