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July  2019, 39(7): 4225-4257. doi: 10.3934/dcds.2019171

On the existence of invariant tori in non-conservative dynamical systems with degeneracy and finite differentiability

1. 

Key Laboratory of High Performance Computing and Stochastic Information Processing, Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China

2. 

HLM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author: Xuemei Li

Received  November 2018 Published  April 2019

Fund Project: This work is supported by the NNSF (11371132, 11671392) of China.

In this paper, we establish a KAM-theorem about the existenceof invariant tori in non-conservative dynamical systems with finitely differentiable vector fields and multiple degeneracies under the assumption that theintegrable part is finitely differentiable with respect to parameters, instead ofthe usual assumption of analyticity. We prove these results by constructingapproximation and inverse approximation lemmas in which all functions arefinitely differentiable in parameters.

Citation: Xuemei Li, Zaijiu Shang. On the existence of invariant tori in non-conservative dynamical systems with degeneracy and finite differentiability. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4225-4257. doi: 10.3934/dcds.2019171
References:
[1]

J. Albrecht, On the existence of invariant tori in nearly integrable Hamiltonian systems with finitely differentiable perturbations, Regul. Chaotic Dyn., 12 (2007), 281-320.  doi: 10.1134/S1560354707030033.

[2]

V. I. Arnol'd, Proof of a theorem by A. N. Kolmogorov on the invariance of quasi periodic motions under small perturbations of the Hamiltonian, Russian Math. Survey, 18 (1963), 9-36. 

[3]

V. I. Arnol'd, Small divisor problems in classical and celestial mechanics, Russian Math. Survey, 18 (1963), 85-191. 

[4]

D. BambusiM. Berti and E. Magistrelli, Degenerate KAM theory for partial differential equations, J. Differential Equations, 250 (2011), 3379-3397.  doi: 10.1016/j.jde.2010.11.002.

[5]

D. Bambusi and G. Gaeta, Invariant tori for non-conservative perturbations of integrable systems, NoDEA Nonlinear Differ. Equ. Appl., 8 (2001), 99-116.  doi: 10.1007/PL00001441.

[6]

N. N. Bogoljubov, Ju. A. Mitropolskii and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer, Berlin, 1976.

[7]

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., Springer, Berlin, 1996.

[8]

A. D. Bruno, On conditions for nondegeneracy in Kolmogorov's theorem, Soviet Math. Dokl., 45 (1992), 221-225. 

[9]

C.-Q. Cheng and Y. Sun, Existence of KAM tori in degenerate Hamiltonian systems, J. Differential Equations, 114 (1994), 288-335.  doi: 10.1006/jdeq.1994.1152.

[10]

C.-Q. Cheng and S. Wang, The surviving of lower dimensional tori from a resonant torus of Hamiltonian systems, J. Differential Equations, 155 (1999), 311-326.  doi: 10.1006/jdeq.1998.3586.

[11]

L. Chierchia, KAM Lectures, Dynamical Systems, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Sc. Norm. Sup., Pisa, Part Ⅰ: (2003), 1–55.

[12]

L. Chierchia and G. Pinzari, Properly degenerate KAM theory (following V. I. Arnold), Discrete. Contin. Dyn. Syst. S, 3 (2010), 545-578.  doi: 10.3934/dcdss.2010.3.545.

[13]

L. Chierchia and D. Qian, Moser's theorem for lower dimensional tori, J. Differential Equations, 206 (2004), 55-93.  doi: 10.1016/j.jde.2004.06.014.

[14]

J. Féjoz, Dèmonstration du `théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman), Ergodic Theory Dyn. Syst., 24 (2004), 1521-1582.  doi: 10.1017/S0143385704000410.

[15]

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

[16]

G. Gentile and G. Gallavotti, Degenerate elliptic resonances, Comm. Math. Phys., 257 (2005), 319-362.  doi: 10.1007/s00220-005-1325-6.

[17]

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.

[18]

Y. HanY. Li and Y. Yi, Invariant tori in Hamiltonian systems with high order proper degeneracy, Ann. Henri Poincaré, 10 (2010), 1419-1436.  doi: 10.1007/s00023-010-0026-7.

[19]

M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau, Vol. 1.Astérisque, 103 (1983), i+221pp.

[20]

X. Li, On the persistence of quasi-periodic invariant tori for double Hopf bifurcation of vector fields, J. Differential Equations, 260 (2016), 7320-7357.  doi: 10.1016/j.jde.2016.01.025.

[21]

X. Li and R. de la Llave, Convergence of differentiable functions on closed sets and remarks on the proofs of the "converse approximation lemmas", Discrete Contin. Dyn. Syst. S, 3 (2010), 623-641.  doi: 10.3934/dcdss.2010.3.623.

[22]

X. Li and X. Yuan, Quasi-periodic solutions for perturbed autonomous delay differential equations, J. Differential Equations, 252 (2012), 3752-3796.  doi: 10.1016/j.jde.2011.11.014.

[23]

Y. Li and Y. Yi, A quasi-periodic Poincare's theorem, Math. Ann., 326 (2003), 649-690.  doi: 10.1007/s00208-002-0399-0.

[24]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nach. Akad. Wiss. Göttingen, Ⅱ Math. Phys. KI, 1962 (1962), 1-20. 

[25]

J. Moser, A rapidly convergent iteration method and nonlinear partial differential equations Ⅰ and Ⅱ, Ann. Scuola Norm. Sup. Pisa(3), 20 (1966), 265-315. 

[26]

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

[27]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Commun. Pure Appl. Math., 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.

[28]

J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Sc. Norm. Sup. Pisa, 23 (1996), 119-148. 

[29]

J. Pöschel, A lecture on the classical KAM theorem, Proc. Symp. Pure Math., 69 (2001), 707-732.  doi: 10.1090/pspum/069/1858551.

[30]

H. Rüssmann, Kleine Nenner I: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nach. Akad. Wiss. Göttingen, Ⅱ Math. Phys. KI., 1970 (1970), 67-105. 

[31]

H. Rüssmann, On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, Lecture Notes in Phys., Springer, Berlin, 38 (1975), 598–624.

[32]

H. Rüssmann, On the existence of invariant curves of twist mappings of an annulus, Lecture Notes in Math., Springer, Berlin, 1007 (1983), 677–718. doi: 10.1007/BFb0061441.

[33]

H. Rüssmann, Nondegeneracy in the perturbation theory of integrable dynamical systems, London Math. Soc. Lecture Note Ser., 134 (1989), 5-18.  doi: 10.1017/CBO9780511661983.002.

[34]

H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn., 6 (2001), 119-204.  doi: 10.1070/RD2001v006n02ABEH000169.

[35]

Z. Shang, A note on the KAM theorem for symplectic mappings, J. Dyn. Diff. Eqs., 12 (2000), 357-383.  doi: 10.1023/A:1009068425415.

[36]

C. L. Siegel, Verlesungen über Himmelsmechanik, Springer, 1956.

[37] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Series, No. 30, Princeton University Press, 1970. 
[38]

F. Wagener, A parameterised version of Moser's modifying terms theorem, Discrete Contin. Dyn. Syst. S, 3 (2010), 719-768.  doi: 10.3934/dcdss.2010.3.719.

[39]

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.

[40]

E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems Ⅰ and Ⅱ, Comm. Pure Appl. Math., textbf29 (1975), 91–140 and 29 (1976), 49–111. doi: 10.1002/cpa.3160290104.

show all references

References:
[1]

J. Albrecht, On the existence of invariant tori in nearly integrable Hamiltonian systems with finitely differentiable perturbations, Regul. Chaotic Dyn., 12 (2007), 281-320.  doi: 10.1134/S1560354707030033.

[2]

V. I. Arnol'd, Proof of a theorem by A. N. Kolmogorov on the invariance of quasi periodic motions under small perturbations of the Hamiltonian, Russian Math. Survey, 18 (1963), 9-36. 

[3]

V. I. Arnol'd, Small divisor problems in classical and celestial mechanics, Russian Math. Survey, 18 (1963), 85-191. 

[4]

D. BambusiM. Berti and E. Magistrelli, Degenerate KAM theory for partial differential equations, J. Differential Equations, 250 (2011), 3379-3397.  doi: 10.1016/j.jde.2010.11.002.

[5]

D. Bambusi and G. Gaeta, Invariant tori for non-conservative perturbations of integrable systems, NoDEA Nonlinear Differ. Equ. Appl., 8 (2001), 99-116.  doi: 10.1007/PL00001441.

[6]

N. N. Bogoljubov, Ju. A. Mitropolskii and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer, Berlin, 1976.

[7]

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., Springer, Berlin, 1996.

[8]

A. D. Bruno, On conditions for nondegeneracy in Kolmogorov's theorem, Soviet Math. Dokl., 45 (1992), 221-225. 

[9]

C.-Q. Cheng and Y. Sun, Existence of KAM tori in degenerate Hamiltonian systems, J. Differential Equations, 114 (1994), 288-335.  doi: 10.1006/jdeq.1994.1152.

[10]

C.-Q. Cheng and S. Wang, The surviving of lower dimensional tori from a resonant torus of Hamiltonian systems, J. Differential Equations, 155 (1999), 311-326.  doi: 10.1006/jdeq.1998.3586.

[11]

L. Chierchia, KAM Lectures, Dynamical Systems, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Sc. Norm. Sup., Pisa, Part Ⅰ: (2003), 1–55.

[12]

L. Chierchia and G. Pinzari, Properly degenerate KAM theory (following V. I. Arnold), Discrete. Contin. Dyn. Syst. S, 3 (2010), 545-578.  doi: 10.3934/dcdss.2010.3.545.

[13]

L. Chierchia and D. Qian, Moser's theorem for lower dimensional tori, J. Differential Equations, 206 (2004), 55-93.  doi: 10.1016/j.jde.2004.06.014.

[14]

J. Féjoz, Dèmonstration du `théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman), Ergodic Theory Dyn. Syst., 24 (2004), 1521-1582.  doi: 10.1017/S0143385704000410.

[15]

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

[16]

G. Gentile and G. Gallavotti, Degenerate elliptic resonances, Comm. Math. Phys., 257 (2005), 319-362.  doi: 10.1007/s00220-005-1325-6.

[17]

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.

[18]

Y. HanY. Li and Y. Yi, Invariant tori in Hamiltonian systems with high order proper degeneracy, Ann. Henri Poincaré, 10 (2010), 1419-1436.  doi: 10.1007/s00023-010-0026-7.

[19]

M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau, Vol. 1.Astérisque, 103 (1983), i+221pp.

[20]

X. Li, On the persistence of quasi-periodic invariant tori for double Hopf bifurcation of vector fields, J. Differential Equations, 260 (2016), 7320-7357.  doi: 10.1016/j.jde.2016.01.025.

[21]

X. Li and R. de la Llave, Convergence of differentiable functions on closed sets and remarks on the proofs of the "converse approximation lemmas", Discrete Contin. Dyn. Syst. S, 3 (2010), 623-641.  doi: 10.3934/dcdss.2010.3.623.

[22]

X. Li and X. Yuan, Quasi-periodic solutions for perturbed autonomous delay differential equations, J. Differential Equations, 252 (2012), 3752-3796.  doi: 10.1016/j.jde.2011.11.014.

[23]

Y. Li and Y. Yi, A quasi-periodic Poincare's theorem, Math. Ann., 326 (2003), 649-690.  doi: 10.1007/s00208-002-0399-0.

[24]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nach. Akad. Wiss. Göttingen, Ⅱ Math. Phys. KI, 1962 (1962), 1-20. 

[25]

J. Moser, A rapidly convergent iteration method and nonlinear partial differential equations Ⅰ and Ⅱ, Ann. Scuola Norm. Sup. Pisa(3), 20 (1966), 265-315. 

[26]

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

[27]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Commun. Pure Appl. Math., 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.

[28]

J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Sc. Norm. Sup. Pisa, 23 (1996), 119-148. 

[29]

J. Pöschel, A lecture on the classical KAM theorem, Proc. Symp. Pure Math., 69 (2001), 707-732.  doi: 10.1090/pspum/069/1858551.

[30]

H. Rüssmann, Kleine Nenner I: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nach. Akad. Wiss. Göttingen, Ⅱ Math. Phys. KI., 1970 (1970), 67-105. 

[31]

H. Rüssmann, On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, Lecture Notes in Phys., Springer, Berlin, 38 (1975), 598–624.

[32]

H. Rüssmann, On the existence of invariant curves of twist mappings of an annulus, Lecture Notes in Math., Springer, Berlin, 1007 (1983), 677–718. doi: 10.1007/BFb0061441.

[33]

H. Rüssmann, Nondegeneracy in the perturbation theory of integrable dynamical systems, London Math. Soc. Lecture Note Ser., 134 (1989), 5-18.  doi: 10.1017/CBO9780511661983.002.

[34]

H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn., 6 (2001), 119-204.  doi: 10.1070/RD2001v006n02ABEH000169.

[35]

Z. Shang, A note on the KAM theorem for symplectic mappings, J. Dyn. Diff. Eqs., 12 (2000), 357-383.  doi: 10.1023/A:1009068425415.

[36]

C. L. Siegel, Verlesungen über Himmelsmechanik, Springer, 1956.

[37] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Series, No. 30, Princeton University Press, 1970. 
[38]

F. Wagener, A parameterised version of Moser's modifying terms theorem, Discrete Contin. Dyn. Syst. S, 3 (2010), 719-768.  doi: 10.3934/dcdss.2010.3.719.

[39]

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.

[40]

E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems Ⅰ and Ⅱ, Comm. Pure Appl. Math., textbf29 (1975), 91–140 and 29 (1976), 49–111. doi: 10.1002/cpa.3160290104.

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