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

Quasi-effective stability for nearly integrable Hamiltonian systems

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.
Citation: Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete and 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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[13]

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

[14]

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

[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.

[16]

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

[17]

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

[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.

[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.

[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.

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]

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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[13]

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

[14]

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

[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.

[16]

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

[17]

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

[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.

[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.

[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.

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