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September  2015, 20(7): 1959-1970. doi: 10.3934/dcdsb.2015.20.1959

Quasi-effective stability for a nearly integrable volume-preserving mapping

Fuzhong Cong 1, and  Hongtian Li 1,

1. 

School of Mathematics, Jilin University, Changchun, 130012, China, China

Received  September 2014 Revised  December 2014 Published  July 2015

This paper is concerned with the stability of the orbits for a nearly integrable volume-preserving mapping. We prove that the nearly integrable volume-preserving mapping possesses quasi-effective stability under the classical KAM-type nondegeneracy, that is, there is an open subset of the phase space whose measure is nearly full, such that the considered mapping is effective stable on this subset. This announces a connection between the Nekhoroshev theory and KAM theory.
Keywords: near-invariant torus, Quasi-effective stability, nearly integrable volume-preserving mapping., effective stability.
Mathematics Subject Classification: Primary: 37J40, 37E40, 37F50; Secondary: 65L2.
Citation: Fuzhong Cong, Hongtian Li. Quasi-effective stability for a nearly integrable volume-preserving mapping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1959-1970. doi: 10.3934/dcdsb.2015.20.1959
References:
[1]

V. I. Arnol'd, Proof of a theorem by A. N. Komolgorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Russ. Math. Surv., 18 (1963), 13-40.  Google Scholar

[2]

V. Arnol'd, Sur une propriététopologique des applications globalement canoniques de la mécanique classique, C. R. Acad. Sci. Paris, 261 (1965), 3719-3722.  Google Scholar

[3]

J. E. Cartwright, M. Feingold and O. Piro, Passive scalar and three-dimensional liouvillian mappings, Phys. D, 76 (1994), 22-23. doi: 10.1016/0167-2789(94)90247-X.  Google Scholar

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C. Cheng and Y. Sun, Existence of invariant tori in three-dimensional measure-preserving mappings, Celestial Mech. Dynam. Astronom., 47 (1990), 275-292. doi: 10.1007/BF00053456.  Google Scholar

[5]

F. Cong, The approximate decomposition of exponential order of slow-fast motions in multifrequency systems, J. Differential Equations, 196 (2004), 466-480. doi: 10.1016/j.jde.2003.09.003.  Google Scholar

[6]

F. Cong, J. Hong and Y. Han, Near-invariant tori on exponentially long time for Poisson systems, J. Math. Anal. Appl., 334 (2007), 59-68. Google Scholar

[7]

F. Cong, Y. Li and M. Huang, Invariant tori for nearly twist mappings with intersection property, Northest. Math. J., 12 (1996), 280-298.  Google Scholar

[8]

M. Feingold, L. P. Kadanoff and O. Piro, Passive Scalars, three-dimensional volume preserving maps, and Chaos, J. Stat. Phys., 50 (1988), 529-565. doi: 10.1007/BF01026490.  Google Scholar

[9]

M. Feingold, L. P. Kadanoff and O. Piro, Transport of passive scalars: KAM surface and diffusion in three-dimensional Liouvillian maps, in Instabilities and nonequilbrium structures II (eds. E. Tirapegui and D. Villarroel), Kluwer Academic Publishers Group, 50 (1989), 37-51.  Google Scholar

[10]

M. Guzzo, A direct proof of the Nekhoroshev theorem for nearly integrable symplectic mappings, Ann. Henri Poincaré, 5 (2004), 1013-1039. doi: 10.1007/s00023-004-0188-2.  Google Scholar

[11]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations, Springer, 2002. doi: 10.1007/978-3-662-05018-7.  Google Scholar

[12]

Y. Han and F. Cong, Effective stablity for nearly integrable mappings with intersection property, Ann. Diff. Eqs., 21 (2005), 294-299.  Google Scholar

[13]

I. Mezić, Break-up of invariant surfaces in action-angle-angle mappings and flows, Phys. D, 154 (2001), 51-67. doi: 10.1016/S0167-2789(01)00226-3.  Google Scholar

[14]

N. N. Nekhoroshev, Exponential estimate of the stability time of nearly integerable Hamiltonian systems, Russ. Math. Surv., 32 (1977), 1-65.  Google Scholar

[15]

O. Piro and M. Feingold, Diffusion in three-dimensional liouvillian maps, Phys. Rev. Lett., 61 (1988), 1799-1802. doi: 10.1103/PhysRevLett.61.1799.  Google Scholar

[16]

M. B. Sevryuk, The classical KAM theory at the dawn of the twenty-first century, Mosc. Math. J., 3 (2003), 1113-1144, 1201-1202.  Google Scholar

[17]

Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms, Ergodic Theory Dynamical Systems, 12 (1992), 621-631. doi: 10.1017/S0143385700006969.  Google Scholar

[18]

J. Xu, J. You and Q. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z., 226 (1997), 375-387. doi: 10.1007/PL00004344.  Google Scholar

show all references

References:
[1]

V. I. Arnol'd, Proof of a theorem by A. N. Komolgorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Russ. Math. Surv., 18 (1963), 13-40.  Google Scholar

[2]

V. Arnol'd, Sur une propriététopologique des applications globalement canoniques de la mécanique classique, C. R. Acad. Sci. Paris, 261 (1965), 3719-3722.  Google Scholar

[3]

J. E. Cartwright, M. Feingold and O. Piro, Passive scalar and three-dimensional liouvillian mappings, Phys. D, 76 (1994), 22-23. doi: 10.1016/0167-2789(94)90247-X.  Google Scholar

[4]

C. Cheng and Y. Sun, Existence of invariant tori in three-dimensional measure-preserving mappings, Celestial Mech. Dynam. Astronom., 47 (1990), 275-292. doi: 10.1007/BF00053456.  Google Scholar

[5]

F. Cong, The approximate decomposition of exponential order of slow-fast motions in multifrequency systems, J. Differential Equations, 196 (2004), 466-480. doi: 10.1016/j.jde.2003.09.003.  Google Scholar

[6]

F. Cong, J. Hong and Y. Han, Near-invariant tori on exponentially long time for Poisson systems, J. Math. Anal. Appl., 334 (2007), 59-68. Google Scholar

[7]

F. Cong, Y. Li and M. Huang, Invariant tori for nearly twist mappings with intersection property, Northest. Math. J., 12 (1996), 280-298.  Google Scholar

[8]

M. Feingold, L. P. Kadanoff and O. Piro, Passive Scalars, three-dimensional volume preserving maps, and Chaos, J. Stat. Phys., 50 (1988), 529-565. doi: 10.1007/BF01026490.  Google Scholar

[9]

M. Feingold, L. P. Kadanoff and O. Piro, Transport of passive scalars: KAM surface and diffusion in three-dimensional Liouvillian maps, in Instabilities and nonequilbrium structures II (eds. E. Tirapegui and D. Villarroel), Kluwer Academic Publishers Group, 50 (1989), 37-51.  Google Scholar

[10]

M. Guzzo, A direct proof of the Nekhoroshev theorem for nearly integrable symplectic mappings, Ann. Henri Poincaré, 5 (2004), 1013-1039. doi: 10.1007/s00023-004-0188-2.  Google Scholar

[11]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations, Springer, 2002. doi: 10.1007/978-3-662-05018-7.  Google Scholar

[12]

Y. Han and F. Cong, Effective stablity for nearly integrable mappings with intersection property, Ann. Diff. Eqs., 21 (2005), 294-299.  Google Scholar

[13]

I. Mezić, Break-up of invariant surfaces in action-angle-angle mappings and flows, Phys. D, 154 (2001), 51-67. doi: 10.1016/S0167-2789(01)00226-3.  Google Scholar

[14]

N. N. Nekhoroshev, Exponential estimate of the stability time of nearly integerable Hamiltonian systems, Russ. Math. Surv., 32 (1977), 1-65.  Google Scholar

[15]

O. Piro and M. Feingold, Diffusion in three-dimensional liouvillian maps, Phys. Rev. Lett., 61 (1988), 1799-1802. doi: 10.1103/PhysRevLett.61.1799.  Google Scholar

[16]

M. B. Sevryuk, The classical KAM theory at the dawn of the twenty-first century, Mosc. Math. J., 3 (2003), 1113-1144, 1201-1202.  Google Scholar

[17]

Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms, Ergodic Theory Dynamical Systems, 12 (1992), 621-631. doi: 10.1017/S0143385700006969.  Google Scholar

[18]

J. Xu, J. You and Q. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z., 226 (1997), 375-387. doi: 10.1007/PL00004344.  Google Scholar

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