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. BroerM. 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. HanY. 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. WangJ. 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. WangJ. 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. WangJ. 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.

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.

[2]

H. W. BroerM. 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. HanY. 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. WangJ. 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. WangJ. 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. WangJ. 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.

[1]

Virginie Bonnaillie-Noël. Harmonic oscillators with Neumann condition on the half-line. Communications on Pure and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 2019, 39 (10) : 5775-5784. doi: 10.3934/dcds.2019253

[13]

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

[14]

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

[15]

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

[16]

Ahmad El Hajj, Aya Oussaily. Continuous solution for a non-linear eikonal system. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3795-3823. doi: 10.3934/cpaa.2021131

[17]

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

[18]

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

[19]

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

[20]

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

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (184)
  • HTML views (178)
  • Cited by (0)

Other articles
by authors

[Back to Top]