doi: 10.3934/dcdsb.2022004
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On the persistence of lower-dimensional tori in reversible systems with high dimensional degenerate equilibrium under small perturbations

Faculty of mathematics and physics, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, China

Received  April 2021 Revised  October 2021 Early access January 2022

Fund Project: This work is supported by National Natural Science Foundation of China (11501234, 11871146, 117030006)

This paper focuses on the persistence of lower-dimensional tori in reversible systems with high dimensional degenerate equilibrium under small perturbations. By an improved KAM iteration and Topological degree theory, we prove that the invariant torus with given frequency persists under small perturbations. Our result is a generalization of X. Wang et al [On the persistence of degenerate lower-dimensional tori in reversible systems, Ergodic Theory Dynam. Systems, 35(2015), 2311-2333].

Citation: Xiaocai Wang, Xiaofei Cao, Xuqing Liu. On the persistence of lower-dimensional tori in reversible systems with high dimensional degenerate equilibrium under small perturbations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022004
References:
[1]

V. I. Arnol'd, Reversible systems, in: Nonlinear and Turbulent Processes in Physics, Vol. 3 (Kiev, 1983), Harwood Academic Publ., Chur, (1984), 1161–1174.

[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]

H. W. BroerM. C. CiocciH. Hanßmann and A. Vanderbauwhede, Quasi-periodic stability of normally resonant tori, Physica D, 238 (2009), 309-318.  doi: 10.1016/j.physd.2008.10.004.

[4]

H. W. BroerJ. Hoo and V. Naudot, Normal linear stability of quasi-periodic tori, J. Differ. Equations, 232 (2007), 355-418.  doi: 10.1016/j.jde.2006.08.022.

[5]

H. W. Broer and G. B. Huitema, Unfoldings of quasi-periodic tori in reversible systems, J. Dynam. Differ. Equations, 7 (1995), 191-212.  doi: 10.1007/BF02218818.

[6]

H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems, Order Amidst Chaos, Lecture Notes in Math., 1645, Springer-Verlag, Berlin, 1996.

[7]

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

[8]

B. Liu, On lower dimensional invariant tori in reversible systems, J. Differ. Equations, 176 (2001), 158-194.  doi: 10.1006/jdeq.2000.3960.

[9]

Z. Lou and J. Geng, Quasi-periodic response solutions in forced reversible systems with Liouvillean frequencies, J. Differ. Equations, 263 (2017), 3894-3927.  doi: 10.1016/j.jde.2017.05.007.

[10]

J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176.  doi: 10.1007/BF01399536.

[11]

I. O. Parasyuk, Conservation of quasiperiodic motions in reversible multifrequency systems, Dokl. Akad. Nauk Ukrain. SSR. Ser. A, (1982), 19–22, 85.

[12]

J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z., 202 (1989), 559-608.  doi: 10.1007/BF01221590.

[13]

M. B. Sevryuk, Reversible Systems, Lecture Notes in Math., 1211, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0075877.

[14]

M. B. Sevryuk, Invariant $m$-dimensional tori of reversible systems with phase space of dimension greater than $2m$, J. Soviet. Math., 51 (1990), 2374-2386.  doi: 10.1007/BF01094996.

[15]

M. B. Sevryuk, New results in the reversible KAM theory, in: Seminar on Dynamical Systems, eds. S. B. Kuksin, V. F. Lazutkin and J. Pöschel, Birkhäuser, Basel, (1994), 184–199. doi: 10.1007/978-3-0348-7515-8_14.

[16]

M. B. Sevryuk, The iteration-approximation decoupling in the reversible KAM theory, Chaos, 5 (1995), 552-565.  doi: 10.1063/1.166125.

[17]

M. B. Sevryuk, Partial preservation of frequency in KAM theory, Nonlinearity, 19 (2006), 1099-1140.  doi: 10.1088/0951-7715/19/5/005.

[18]

X. Wang, Non-Floquet invariant tori in reversible systems, Discrete Contin. Dyn. Syst., 38 (2018), 3439-3457.  doi: 10.3934/dcds.2018147.

[19]

X. Wang and J. Xu, Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition, Discrete Contin. Dyn. Syst., 25 (2009), 701-718.  doi: 10.3934/dcds.2009.25.701.

[20]

X. WangJ. Xu and D. Zhang, Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1237-1249.  doi: 10.3934/dcdsb.2010.14.1237.

[21]

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.

[22]

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.

[23]

X. WangJ. Xu and D. Zhang, A new KAM theorem for the hyperbolic lower dimensional tori in reversible systems, Acta Appl. Math., 143 (2016), 45-61.  doi: 10.1007/s10440-015-0027-0.

[24]

X. WangJ. Xu and D. Zhang, On the persistence of lower-dimensional elliptic tori with prescribed frequencies in reversible systems, Discrete Contin. Dyn. Syst., 36 (2016), 1677-1692.  doi: 10.3934/dcds.2016.36.1677.

[25]

X. WangJ. Xu and D. Zhang, A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems, Discrete Contin. Dyn. Syst., 37 (2017), 2141-2160.  doi: 10.3934/dcds.2017092.

[26]

X. WangD. Zhang and J. Xu, Persistence of lower dimensional tori for a class of nearly integrable reversible systems, Acta Appl. Math., 115 (2011), 193-207.  doi: 10.1007/s10440-011-9615-9.

[27]

H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89.  doi: 10.1090/S0002-9947-1934-1501735-3.

[28]

B. Wei, Perturbations of lower dimensional tori in the resonant zone for reversible systems, J. Math. Anal. Appl., 253 (2001), 558-577.  doi: 10.1006/jmaa.2000.7165.

[29]

J. Xu, Normal form of reversible systems and persistence of lower dimensional tori under weaker nonresonance conditions, SIAM J. Math. Anal., 36 (2004), 233-255.  doi: 10.1137/S0036141003421923.

[30]

D. Zhang and J. Xu, Reducibility of a class of nonlinear quasi-periodic systems with Liouvillean basic frequencies, Ergodic Theory Dynam. Systems, 41 (2021), 1883-1920.  doi: 10.1017/etds.2020.23.

[31]

D. ZhangJ. XuH. Wu and X. Xu, On the reducibility of linear quasi-periodic systems with Liouvillean basic frequencies and multiple eigenvalues, J. Differential Equations, 269 (2020), 10670-10716.  doi: 10.1016/j.jde.2020.07.025.

show all references

References:
[1]

V. I. Arnol'd, Reversible systems, in: Nonlinear and Turbulent Processes in Physics, Vol. 3 (Kiev, 1983), Harwood Academic Publ., Chur, (1984), 1161–1174.

[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]

H. W. BroerM. C. CiocciH. Hanßmann and A. Vanderbauwhede, Quasi-periodic stability of normally resonant tori, Physica D, 238 (2009), 309-318.  doi: 10.1016/j.physd.2008.10.004.

[4]

H. W. BroerJ. Hoo and V. Naudot, Normal linear stability of quasi-periodic tori, J. Differ. Equations, 232 (2007), 355-418.  doi: 10.1016/j.jde.2006.08.022.

[5]

H. W. Broer and G. B. Huitema, Unfoldings of quasi-periodic tori in reversible systems, J. Dynam. Differ. Equations, 7 (1995), 191-212.  doi: 10.1007/BF02218818.

[6]

H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems, Order Amidst Chaos, Lecture Notes in Math., 1645, Springer-Verlag, Berlin, 1996.

[7]

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

[8]

B. Liu, On lower dimensional invariant tori in reversible systems, J. Differ. Equations, 176 (2001), 158-194.  doi: 10.1006/jdeq.2000.3960.

[9]

Z. Lou and J. Geng, Quasi-periodic response solutions in forced reversible systems with Liouvillean frequencies, J. Differ. Equations, 263 (2017), 3894-3927.  doi: 10.1016/j.jde.2017.05.007.

[10]

J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176.  doi: 10.1007/BF01399536.

[11]

I. O. Parasyuk, Conservation of quasiperiodic motions in reversible multifrequency systems, Dokl. Akad. Nauk Ukrain. SSR. Ser. A, (1982), 19–22, 85.

[12]

J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z., 202 (1989), 559-608.  doi: 10.1007/BF01221590.

[13]

M. B. Sevryuk, Reversible Systems, Lecture Notes in Math., 1211, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0075877.

[14]

M. B. Sevryuk, Invariant $m$-dimensional tori of reversible systems with phase space of dimension greater than $2m$, J. Soviet. Math., 51 (1990), 2374-2386.  doi: 10.1007/BF01094996.

[15]

M. B. Sevryuk, New results in the reversible KAM theory, in: Seminar on Dynamical Systems, eds. S. B. Kuksin, V. F. Lazutkin and J. Pöschel, Birkhäuser, Basel, (1994), 184–199. doi: 10.1007/978-3-0348-7515-8_14.

[16]

M. B. Sevryuk, The iteration-approximation decoupling in the reversible KAM theory, Chaos, 5 (1995), 552-565.  doi: 10.1063/1.166125.

[17]

M. B. Sevryuk, Partial preservation of frequency in KAM theory, Nonlinearity, 19 (2006), 1099-1140.  doi: 10.1088/0951-7715/19/5/005.

[18]

X. Wang, Non-Floquet invariant tori in reversible systems, Discrete Contin. Dyn. Syst., 38 (2018), 3439-3457.  doi: 10.3934/dcds.2018147.

[19]

X. Wang and J. Xu, Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition, Discrete Contin. Dyn. Syst., 25 (2009), 701-718.  doi: 10.3934/dcds.2009.25.701.

[20]

X. WangJ. Xu and D. Zhang, Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1237-1249.  doi: 10.3934/dcdsb.2010.14.1237.

[21]

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.

[22]

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.

[23]

X. WangJ. Xu and D. Zhang, A new KAM theorem for the hyperbolic lower dimensional tori in reversible systems, Acta Appl. Math., 143 (2016), 45-61.  doi: 10.1007/s10440-015-0027-0.

[24]

X. WangJ. Xu and D. Zhang, On the persistence of lower-dimensional elliptic tori with prescribed frequencies in reversible systems, Discrete Contin. Dyn. Syst., 36 (2016), 1677-1692.  doi: 10.3934/dcds.2016.36.1677.

[25]

X. WangJ. Xu and D. Zhang, A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems, Discrete Contin. Dyn. Syst., 37 (2017), 2141-2160.  doi: 10.3934/dcds.2017092.

[26]

X. WangD. Zhang and J. Xu, Persistence of lower dimensional tori for a class of nearly integrable reversible systems, Acta Appl. Math., 115 (2011), 193-207.  doi: 10.1007/s10440-011-9615-9.

[27]

H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89.  doi: 10.1090/S0002-9947-1934-1501735-3.

[28]

B. Wei, Perturbations of lower dimensional tori in the resonant zone for reversible systems, J. Math. Anal. Appl., 253 (2001), 558-577.  doi: 10.1006/jmaa.2000.7165.

[29]

J. Xu, Normal form of reversible systems and persistence of lower dimensional tori under weaker nonresonance conditions, SIAM J. Math. Anal., 36 (2004), 233-255.  doi: 10.1137/S0036141003421923.

[30]

D. Zhang and J. Xu, Reducibility of a class of nonlinear quasi-periodic systems with Liouvillean basic frequencies, Ergodic Theory Dynam. Systems, 41 (2021), 1883-1920.  doi: 10.1017/etds.2020.23.

[31]

D. ZhangJ. XuH. Wu and X. Xu, On the reducibility of linear quasi-periodic systems with Liouvillean basic frequencies and multiple eigenvalues, J. Differential Equations, 269 (2020), 10670-10716.  doi: 10.1016/j.jde.2020.07.025.

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