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January  2016, 21(1): 67-80. doi: 10.3934/dcdsb.2016.21.67

Quasi-effective stability for nearly integrable Hamiltonian systems

Fuzhong Cong 1, , Jialin Hong 2, and  Hongtian Li 3,

1. 

Fundamental Department, Aviation University of Air Force, Changchun 130022, China
2. 

State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190
3. 

School of Mathematics, Jilin University, Changchun, 130012

Received  June 2015 Revised  September 2015 Published  November 2015

This paper concerns with the stability of the orbits for nearly integrable Hamiltonian systems. Based on Nekehoroshev's original works in [14], we present the definition of quasi-effective stability and prove a theorem on quasi-effective stability under the Rüssmann's non-degeneracy. Our result gives a relation between KAM theorem and effective stability. A rapidly converging iteration procedure with two parameters is designed.
Keywords: quasi-effective stability, near-invariant torus, nearly integrable Hamiltonian system., Effective stability.
Mathematics Subject Classification: 37J25, 37J40, 70H0.
Citation: Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934/dcdsb.2016.21.67
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show all references

References:
[1]

Russ. Math. Surv., 18 (1963), 13-40.  Google Scholar

[2]

Math. Z., 275 (2014), 1135-1167. doi: 10.1007/s00209-013-1174-5.  Google Scholar

[3]

Regular and Chaotic Dynamics, 19 (2014), 251-265. doi: 10.1134/S1560354714020087.  Google Scholar

[4]

J. Differential Equations, 128 (1996), 415-490. doi: 10.1006/jdeq.1996.0102.  Google Scholar

[5]

Regular and Chaotic Dynamics, 19 (2014), 363-373. doi: 10.1134/S1560354714030071.  Google Scholar

[6]

Rend. Lincel. Mat. Appl., 25 (2014), 293-299. doi: 10.4171/RLM/679.  Google Scholar

[7]

Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530.  Google Scholar

[8]

J. Math. Anal. Appl., 419 (2014), 1351-1386. doi: 10.1016/j.jmaa.2014.05.035.  Google Scholar

[9]

Chaos, 2 (1992), 495-499. doi: 10.1063/1.165891.  Google Scholar

[10]

Celest. Mech., 55 (1993), 131-159. doi: 10.1007/BF00692425.  Google Scholar

[11]

J. Statist. Phys., 78 (1995), 1607-1617. doi: 10.1007/BF02180145.  Google Scholar

[12]

Phys. D, 86 (1995), 514-516. doi: 10.1016/0167-2789(95)00199-E.  Google Scholar

[13]

Nachr. Akad. Wiss. Gött. Math. Phys. Kl., 2 (1962), 1-20.  Google Scholar

[14]

Russ. Math. Surveys, 32 (1977), 5-66.  Google Scholar

[15]

Phys. D, 71 (1994), 102-121. doi: 10.1016/0167-2789(94)90184-8.  Google Scholar

[16]

Bonn. Math. Schr., 120 (1982), 1-103.  Google Scholar

[17]

Math. Z., 213 (1993), 187-216. doi: 10.1007/BF03025718.  Google Scholar

[18]

Dynamical Systems, Theory and Applications, J. Moser (ed.), Lecture Notes in Physics, 38, Springer, 1975, 598-624.  Google Scholar

[19]

J. Math. Phys., 54 (2013), 072702, 22pp. doi: 10.1063/1.4813059.  Google Scholar

[20]

Math. Z., 226 (1997), 375-387. doi: 10.1007/PL00004344.  Google Scholar

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