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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
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]

A. Bounemoura and S. Fischler, A Diophantine duality applied to the KAM and Nekhoroshev theorems, Math. Z., 275 (2014), 1135-1167. doi: 10.1007/s00209-013-1174-5.  Google Scholar

[3]

A. Bounemoura and S. Fischler, The classical KAM theorem for Hamiltonian systems via rational approximations, Regular and Chaotic Dynamics, 19 (2014), 251-265. doi: 10.1134/S1560354714020087.  Google Scholar

[4]

A. Delshams and P. Gutiérrez, Effective stability and KAM theory, J. Differential Equations, 128 (1996), 415-490. doi: 10.1006/jdeq.1996.0102.  Google Scholar

[5]

A. Fortunati and S. Wiggins, Normal form and Nekhoroshev stability for nearly integrable Hamiltonian systems with unconditionally slow aperiodic time dependence, Regular and Chaotic Dynamics, 19 (2014), 363-373. doi: 10.1134/S1560354714030071.  Google Scholar

[6]

M. Guzzo, L. Chierchia and G. Benettin, Mathematical analysis-The steep Nekhoroshev's theorem and optimal stability exponents, Rend. Lincel. Mat. Appl., 25 (2014), 293-299. doi: 10.4171/RLM/679.  Google Scholar

[7]

A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function(Russian), Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530.  Google Scholar

[8]

M. Kunze and David M. A. Stuart, Nekhoroshev type stability results for Hamiltonian systems with an additional transversal component, J. Math. Anal. Appl., 419 (2014), 1351-1386. doi: 10.1016/j.jmaa.2014.05.035.  Google Scholar

[9]

P. Lochak and A. Neishtadt, Estimate of stability for nearly integrable systems with quasi-convex Hamiltonian, Chaos, 2 (1992), 495-499. doi: 10.1063/1.165891.  Google Scholar

[10]

A. Morbidelli and A. Giorgilli, Quantitative perturbation theory by successive elimination of harmonics, Celest. Mech., 55 (1993), 131-159. doi: 10.1007/BF00692425.  Google Scholar

[11]

A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori, J. Statist. Phys., 78 (1995), 1607-1617. doi: 10.1007/BF02180145.  Google Scholar

[12]

A. Morbidelli and A. Giorgilli, On a connection between KAM and Nekhoroshev's theorems, Phys. D, 86 (1995), 514-516. doi: 10.1016/0167-2789(95)00199-E.  Google Scholar

[13]

J. Moser, On invariant curves of area preserving mappings of an annulus, Nachr. Akad. Wiss. Gött. Math. Phys. Kl., 2 (1962), 1-20.  Google Scholar

[14]

N. N. Nekhoroshev, An exponential estimate of the stability time of nearly integerable Hamiltonian systems, Russ. Math. Surveys, 32 (1977), 5-66.  Google Scholar

[15]

A. D. Perry and S. Wiggins, KAM tori very sticky: Rigorous lower bounds on the time to move away from an invariant lagrangian torus with linear flow, Phys. D, 71 (1994), 102-121. doi: 10.1016/0167-2789(94)90184-8.  Google Scholar

[16]

J. Pöschel, Über invariante tori in differenzierbaren Hamiltonschen systemen, Bonn. Math. Schr., 120 (1982), 1-103.  Google Scholar

[17]

J. Pöschel, Nekhoroshev estimate for quasi-convex Hamiltonian systems, Math. Z., 213 (1993), 187-216. doi: 10.1007/BF03025718.  Google Scholar

[18]

H. Rössmann, On optimal estimates for the solution of linear partial differential equations of first order with constant coefficients on the torus, Dynamical Systems, Theory and Applications, J. Moser (ed.), Lecture Notes in Physics, 38, Springer, 1975, 598-624.  Google Scholar

[19]

G. Schirinzi and M. Guzzo, On the formulation of new explicit conditions for steepness from a former result of N.N. Nekhoroshev, J. Math. Phys., 54 (2013), 072702, 22pp. doi: 10.1063/1.4813059.  Google Scholar

[20]

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]

A. Bounemoura and S. Fischler, A Diophantine duality applied to the KAM and Nekhoroshev theorems, Math. Z., 275 (2014), 1135-1167. doi: 10.1007/s00209-013-1174-5.  Google Scholar

[3]

A. Bounemoura and S. Fischler, The classical KAM theorem for Hamiltonian systems via rational approximations, Regular and Chaotic Dynamics, 19 (2014), 251-265. doi: 10.1134/S1560354714020087.  Google Scholar

[4]

A. Delshams and P. Gutiérrez, Effective stability and KAM theory, J. Differential Equations, 128 (1996), 415-490. doi: 10.1006/jdeq.1996.0102.  Google Scholar

[5]

A. Fortunati and S. Wiggins, Normal form and Nekhoroshev stability for nearly integrable Hamiltonian systems with unconditionally slow aperiodic time dependence, Regular and Chaotic Dynamics, 19 (2014), 363-373. doi: 10.1134/S1560354714030071.  Google Scholar

[6]

M. Guzzo, L. Chierchia and G. Benettin, Mathematical analysis-The steep Nekhoroshev's theorem and optimal stability exponents, Rend. Lincel. Mat. Appl., 25 (2014), 293-299. doi: 10.4171/RLM/679.  Google Scholar

[7]

A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function(Russian), Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530.  Google Scholar

[8]

M. Kunze and David M. A. Stuart, Nekhoroshev type stability results for Hamiltonian systems with an additional transversal component, J. Math. Anal. Appl., 419 (2014), 1351-1386. doi: 10.1016/j.jmaa.2014.05.035.  Google Scholar

[9]

P. Lochak and A. Neishtadt, Estimate of stability for nearly integrable systems with quasi-convex Hamiltonian, Chaos, 2 (1992), 495-499. doi: 10.1063/1.165891.  Google Scholar

[10]

A. Morbidelli and A. Giorgilli, Quantitative perturbation theory by successive elimination of harmonics, Celest. Mech., 55 (1993), 131-159. doi: 10.1007/BF00692425.  Google Scholar

[11]

A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori, J. Statist. Phys., 78 (1995), 1607-1617. doi: 10.1007/BF02180145.  Google Scholar

[12]

A. Morbidelli and A. Giorgilli, On a connection between KAM and Nekhoroshev's theorems, Phys. D, 86 (1995), 514-516. doi: 10.1016/0167-2789(95)00199-E.  Google Scholar

[13]

J. Moser, On invariant curves of area preserving mappings of an annulus, Nachr. Akad. Wiss. Gött. Math. Phys. Kl., 2 (1962), 1-20.  Google Scholar

[14]

N. N. Nekhoroshev, An exponential estimate of the stability time of nearly integerable Hamiltonian systems, Russ. Math. Surveys, 32 (1977), 5-66.  Google Scholar

[15]

A. D. Perry and S. Wiggins, KAM tori very sticky: Rigorous lower bounds on the time to move away from an invariant lagrangian torus with linear flow, Phys. D, 71 (1994), 102-121. doi: 10.1016/0167-2789(94)90184-8.  Google Scholar

[16]

J. Pöschel, Über invariante tori in differenzierbaren Hamiltonschen systemen, Bonn. Math. Schr., 120 (1982), 1-103.  Google Scholar

[17]

J. Pöschel, Nekhoroshev estimate for quasi-convex Hamiltonian systems, Math. Z., 213 (1993), 187-216. doi: 10.1007/BF03025718.  Google Scholar

[18]

H. Rössmann, On optimal estimates for the solution of linear partial differential equations of first order with constant coefficients on the torus, Dynamical Systems, Theory and Applications, J. Moser (ed.), Lecture Notes in Physics, 38, Springer, 1975, 598-624.  Google Scholar

[19]

G. Schirinzi and M. Guzzo, On the formulation of new explicit conditions for steepness from a former result of N.N. Nekhoroshev, J. Math. Phys., 54 (2013), 072702, 22pp. doi: 10.1063/1.4813059.  Google Scholar

[20]

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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