Quasi-effective stability for nearly integrable Hamiltonian systems

Previous Article
Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity
DCDS-B Home
This Issue
Next Article
Stochastic PDE model for spatial population growth in random environments

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

Abstract
Related Papers

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 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.

[1]	Fuzhong Cong, Hongtian Li. Quasi-effective stability for a nearly integrable volume-preserving mapping. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1959-1970. doi: 10.3934/dcdsb.2015.20.1959
[2]	Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643
[3]	Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443
[4]	Carles Simó, Dmitry Treschev. Stability islands in the vicinity of separatrices of near-integrable symplectic maps. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 681-698. doi: 10.3934/dcdsb.2008.10.681
[5]	Michael Kastner, Jacques-Alexandre Sepulchre. Effective Hamiltonian for traveling discrete breathers in the FPU chain. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 719-734. doi: 10.3934/dcdsb.2005.5.719
[6]	Hartmut Schwetlick, Daniel C. Sutton, Johannes Zimmer. Effective Hamiltonian dynamics via the Maupertuis principle. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1395-1410. doi: 10.3934/dcdss.2020078
[7]	Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81
[8]	Marcel Guardia. Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2829-2859. doi: 10.3934/dcds.2013.33.2829
[9]	Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1237-1249. doi: 10.3934/dcdsb.2010.14.1237
[10]	Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227
[11]	Carlos A. Berenstein and Alain Yger. Residues and effective Nullstellensatz. Electronic Research Announcements, 1996, 2: 82-91.
[12]	Luci H. Fatori, Marcio A. Jorge Silva, Vando Narciso. Quasi-stability property and attractors for a semilinear Timoshenko system. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6117-6132. doi: 10.3934/dcds.2016067
[13]	Baowei Feng. On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4729-4751. doi: 10.3934/dcds.2017203
[14]	Siniša Slijepčević. Stability of invariant measures. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1345-1363. doi: 10.3934/dcds.2009.24.1345
[15]	Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109
[16]	Zhi Lin, Katarína Boďová, Charles R. Doering. Models & measures of mixing & effective diffusion. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 259-274. doi: 10.3934/dcds.2010.28.259
[17]	Lei Wang, Quan Yuan, Jia Li. Persistence of the hyperbolic lower dimensional non-twist invariant torus in a class of Hamiltonian systems. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1233-1250. doi: 10.3934/cpaa.2016.15.1233
[18]	Dong Chen. Positive metric entropy in nondegenerate nearly integrable systems. Journal of Modern Dynamics, 2017, 11: 43-56. doi: 10.3934/jmd.2017003
[19]	Junmin Yang, Maoan Han. On the number of limit cycles of a cubic Near-Hamiltonian system. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 827-840. doi: 10.3934/dcds.2009.24.827
[20]	Paolo Luzzini, Paolo Musolino. Perturbation analysis of the effective conductivity of a periodic composite. Networks and Heterogeneous Media, 2020, 15 (4) : 581-603. doi: 10.3934/nhm.2020015

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Fuzhong Cong Jialin Hong Hongtian Li
on Google Scholar
Fuzhong Cong Jialin Hong Hongtian Li

[Back to Top]