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Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period
Quasi-effective stability for a nearly integrable volume-preserving mapping
1. | School of Mathematics, Jilin University, Changchun, 130012, China, China |
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. |
[2] |
V. Arnol'd, Sur une propriététopologique des applications globalement canoniques de la mécanique classique, C. R. Acad. Sci. Paris, 261 (1965), 3719-3722. |
[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. |
[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. |
[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. |
[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. |
[7] |
F. Cong, Y. Li and M. Huang, Invariant tori for nearly twist mappings with intersection property, Northest. Math. J., 12 (1996), 280-298. |
[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. |
[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. |
[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. |
[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. |
[12] |
Y. Han and F. Cong, Effective stablity for nearly integrable mappings with intersection property, Ann. Diff. Eqs., 21 (2005), 294-299. |
[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. |
[14] |
N. N. Nekhoroshev, Exponential estimate of the stability time of nearly integerable Hamiltonian systems, Russ. Math. Surv., 32 (1977), 1-65. |
[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. |
[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. |
[17] |
Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms, Ergodic Theory Dynamical Systems, 12 (1992), 621-631.
doi: 10.1017/S0143385700006969. |
[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. |
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. |
[2] |
V. Arnol'd, Sur une propriététopologique des applications globalement canoniques de la mécanique classique, C. R. Acad. Sci. Paris, 261 (1965), 3719-3722. |
[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. |
[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. |
[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. |
[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. |
[7] |
F. Cong, Y. Li and M. Huang, Invariant tori for nearly twist mappings with intersection property, Northest. Math. J., 12 (1996), 280-298. |
[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. |
[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. |
[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. |
[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. |
[12] |
Y. Han and F. Cong, Effective stablity for nearly integrable mappings with intersection property, Ann. Diff. Eqs., 21 (2005), 294-299. |
[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. |
[14] |
N. N. Nekhoroshev, Exponential estimate of the stability time of nearly integerable Hamiltonian systems, Russ. Math. Surv., 32 (1977), 1-65. |
[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. |
[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. |
[17] |
Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms, Ergodic Theory Dynamical Systems, 12 (1992), 621-631.
doi: 10.1017/S0143385700006969. |
[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. |
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