• Previous Article
    Equality of Kolmogorov-Sinai and permutation entropy for one-dimensional maps consisting of countably many monotone parts
  • DCDS Home
  • This Issue
  • Next Article
    A polygonal scheme and the lower bound on density for the isentropic gas dynamics
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 & Continuous Dynamical Systems - A, 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.

[1]

Fuzhong Cong, Yong Li. Invariant hyperbolic tori for Hamiltonian systems with degeneracy. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 371-382. doi: 10.3934/dcds.1997.3.371

[2]

Yuusuke Sugiyama. Degeneracy in finite time of 1D quasilinear wave equations Ⅱ. Evolution Equations & Control Theory, 2017, 6 (4) : 615-628. doi: 10.3934/eect.2017031

[3]

Deconinck Bernard, Olga Trichtchenko. High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1323-1358. doi: 10.3934/dcds.2017055

[4]

C. Chandre. Renormalization for cubic frequency invariant tori in Hamiltonian systems with two degrees of freedom. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 457-465. doi: 10.3934/dcdsb.2002.2.457

[5]

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

[6]

Stefania Fanali, Massimo Giulietti, Irene Platoni. On maximal curves over finite fields of small order. Advances in Mathematics of Communications, 2012, 6 (1) : 107-120. doi: 10.3934/amc.2012.6.107

[7]

Robert Granger, Thorsten Kleinjung, Jens Zumbrägel. Indiscreet logarithms in finite fields of small characteristic. Advances in Mathematics of Communications, 2018, 12 (2) : 263-286. doi: 10.3934/amc.2018017

[8]

Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443

[9]

Lei Wang, Quan Yuan, Jia Li. Persistence of the hyperbolic lower dimensional non-twist invariant torus in a class of Hamiltonian systems. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1233-1250. doi: 10.3934/cpaa.2016.15.1233

[10]

Nils Ackermann, Thomas Bartsch, Petr Kaplický. An invariant set generated by the domain topology for parabolic semiflows with small diffusion. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 613-626. doi: 10.3934/dcds.2007.18.613

[11]

Kazuo Aoki, Ansgar Jüngel, Peter A. Markowich. Small velocity and finite temperature variations in kinetic relaxation models. Kinetic & Related Models, 2010, 3 (1) : 1-15. doi: 10.3934/krm.2010.3.1

[12]

Sergey Popov, Volker Reitmann. Frequency domain conditions for finite-dimensional projectors and determining observations for the set of amenable solutions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 249-267. doi: 10.3934/dcds.2014.34.249

[13]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085

[14]

Harald Fripertinger. The number of invariant subspaces under a linear operator on finite vector spaces. Advances in Mathematics of Communications, 2011, 5 (2) : 407-416. doi: 10.3934/amc.2011.5.407

[15]

Martin Brokate, Pavel Krejčí. Weak differentiability of scalar hysteresis operators. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2405-2421. doi: 10.3934/dcds.2015.35.2405

[16]

Misha Guysinsky, Boris Hasselblatt, Victoria Rayskin. Differentiability of the Hartman--Grobman linearization. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 979-984. doi: 10.3934/dcds.2003.9.979

[17]

Rémi Carles, Erwan Faou. Energy cascades for NLS on the torus. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2063-2077. doi: 10.3934/dcds.2012.32.2063

[18]

Simon Lloyd. On the Closing Lemma problem for the torus. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 951-962. doi: 10.3934/dcds.2009.25.951

[19]

Peter Seibt. A period formula for torus automorphisms. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1029-1048. doi: 10.3934/dcds.2003.9.1029

[20]

Aaron W. Brown. Smooth stabilizers for measures on the torus. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 43-58. doi: 10.3934/dcds.2015.35.43

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (43)
  • HTML views (138)
  • Cited by (0)

Other articles
by authors

[Back to Top]