American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold
  • DCDS Home
  • This Issue
  • Next Article
    Global attractor for weakly damped gKdV equations in higher sobolev spaces
August  2015, 35(8): 3827-3855. doi: 10.3934/dcds.2015.35.3827

Continuous averaging proof of the Nekhoroshev theorem

Jinxin Xue 1,

1. 

Department of mathematics, the University of Chicago, Chicago, IL, 60637, United States

Received  August 2013 Revised  December 2014 Published  February 2015

In this paper we develop the continuous averaging method of Treschev to work on the simultaneous Diophantine approximation and apply the result to give a new proof of the Nekhoroshev theorem. We obtain a sharp normal form theorem and explicit estimates of the stability constants appearing in the Nekhoroshev theorem.
Keywords: Hamiltonian systems., Continuous averaging, Nekhoroshev theorem.
Mathematics Subject Classification: Primary: 37J40, 37J25; Secondary: 34D1.
Citation: Jinxin Xue. Continuous averaging proof of the Nekhoroshev theorem. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3827-3855. doi: 10.3934/dcds.2015.35.3827
References:
[1]

A. Bounemoura and J.-P. Marco, Improved exponential stability for near-integrable quasi-convex Hamiltonians, Nonlinearity, 24 (2011), 97-112. doi: 10.1088/0951-7715/24/1/005.  Google Scholar

[2]

A. Córdoba, D. Córdoba and M. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Annals of Mathematics, 162 (2005), 1377-1389. doi: 10.4007/annals.2005.162.1377.  Google Scholar

[3]

J. Féjoz, M. Guardia, V. Kaloshin and P. Raldan, Kirkwood gaps and diffusion along mean motion resonance in the restricted planar three-body problem,, , ().   Google Scholar

[4]

A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Inventiones Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3.  Google Scholar

[5]

P. Lochak, Canonical perturbation theory via simultaneous approximation, Russian Mathematical Surveys, 47 (1992), 57-133. doi: 10.1070/RM1992v047n06ABEH000965.  Google Scholar

[6]

P. Lochak, Simultaneous Diophantine approximation in classical perturbation theory: Why and what for? Progress in nonlinear science., 1 RAS, Inst. Appl. Phys., Nizhnii (Novgorod, (2002), 116-138.  Google Scholar

[7]

P. Lochak and A. I. Neishtadt, Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian, Chaos, 2 (1992), 495-499. doi: 10.1063/1.165891.  Google Scholar

[8]

P. Lochak, A. I. Neishtadt and L. Niederman, Stability of nearly integrable convex Hamiltonian systems over exponentially long times. Kuksin, S. (ed.) et al., Seminar on dynamical systems. Basel: Birkhäuser. Prog. Nonlinear Diff. Equ. Appl. 12 (1994), 15-34.  Google Scholar

[9]

L. Niederman, Stability over exponentially long times in the planetary problem, Nonlinearity, 9 (1996), 1703-1751. doi: 10.1088/0951-7715/9/6/017.  Google Scholar

[10]

N. Nekhorochev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Russ. Math. Surv., 32 (1977), 5-66, 287.  Google Scholar

[11]

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

[12]

A. Pronin and D. Treschev, Continuous averaging in multi-frequency slow-fast systems, Regular and Chaotic Dynamics, 5 (2000), 157-170. doi: 10.1070/rd2000v005n02ABEH000138.  Google Scholar

[13]

D. Treschev and O. Zubelevich, Introduction to the Perturbation Theory of Hamiltonian Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-03028-4.  Google Scholar

[14]

D. V. Treschev, The continuous averaging method in the problem of separation of fast and slow motions, Regular and Chaotic Dynamics, 2 (1997), 9-20.  Google Scholar

[15]

D. Treschev, Separatrix splitting for a pendulum with rapidly oscillating suspension point, Russian J. Math. Phys., 5 (1997), 63-98.  Google Scholar

show all references

References:
[1]

A. Bounemoura and J.-P. Marco, Improved exponential stability for near-integrable quasi-convex Hamiltonians, Nonlinearity, 24 (2011), 97-112. doi: 10.1088/0951-7715/24/1/005.  Google Scholar

[2]

A. Córdoba, D. Córdoba and M. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Annals of Mathematics, 162 (2005), 1377-1389. doi: 10.4007/annals.2005.162.1377.  Google Scholar

[3]

J. Féjoz, M. Guardia, V. Kaloshin and P. Raldan, Kirkwood gaps and diffusion along mean motion resonance in the restricted planar three-body problem,, , ().   Google Scholar

[4]

A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Inventiones Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3.  Google Scholar

[5]

P. Lochak, Canonical perturbation theory via simultaneous approximation, Russian Mathematical Surveys, 47 (1992), 57-133. doi: 10.1070/RM1992v047n06ABEH000965.  Google Scholar

[6]

P. Lochak, Simultaneous Diophantine approximation in classical perturbation theory: Why and what for? Progress in nonlinear science., 1 RAS, Inst. Appl. Phys., Nizhnii (Novgorod, (2002), 116-138.  Google Scholar

[7]

P. Lochak and A. I. Neishtadt, Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian, Chaos, 2 (1992), 495-499. doi: 10.1063/1.165891.  Google Scholar

[8]

P. Lochak, A. I. Neishtadt and L. Niederman, Stability of nearly integrable convex Hamiltonian systems over exponentially long times. Kuksin, S. (ed.) et al., Seminar on dynamical systems. Basel: Birkhäuser. Prog. Nonlinear Diff. Equ. Appl. 12 (1994), 15-34.  Google Scholar

[9]

L. Niederman, Stability over exponentially long times in the planetary problem, Nonlinearity, 9 (1996), 1703-1751. doi: 10.1088/0951-7715/9/6/017.  Google Scholar

[10]

N. Nekhorochev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Russ. Math. Surv., 32 (1977), 5-66, 287.  Google Scholar

[11]

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

[12]

A. Pronin and D. Treschev, Continuous averaging in multi-frequency slow-fast systems, Regular and Chaotic Dynamics, 5 (2000), 157-170. doi: 10.1070/rd2000v005n02ABEH000138.  Google Scholar

[13]

D. Treschev and O. Zubelevich, Introduction to the Perturbation Theory of Hamiltonian Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-03028-4.  Google Scholar

[14]

D. V. Treschev, The continuous averaging method in the problem of separation of fast and slow motions, Regular and Chaotic Dynamics, 2 (1997), 9-20.  Google Scholar

[15]

D. Treschev, Separatrix splitting for a pendulum with rapidly oscillating suspension point, Russian J. Math. Phys., 5 (1997), 63-98.  Google Scholar

[1]

Paolo Perfetti. A Nekhoroshev theorem for some infinite--dimensional systems. Communications on Pure & Applied Analysis, 2006, 5 (1) : 125-146. doi: 10.3934/cpaa.2006.5.125
[2]

Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020
[3]

Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623
[4]

Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215
[5]

Massimiliano Guzzo, Giancarlo Benettin. A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 1-28. doi: 10.3934/dcdsb.2001.1.1
[6]

Armengol Gasull, Víctor Mañosa. Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 651-670. doi: 10.3934/dcdsb.2019259
[7]

Claudio A. Buzzi, Jeroen S.W. Lamb. Reversible Hamiltonian Liapunov center theorem. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 51-66. doi: 10.3934/dcdsb.2005.5.51
[8]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1
[9]

Janusz Mierczyński. Averaging in random systems of nonnegative matrices. Conference Publications, 2015, 2015 (special) : 835-840. doi: 10.3934/proc.2015.0835
[10]

Guan Huang. An averaging theorem for nonlinear Schrödinger equations with small nonlinearities. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3555-3574. doi: 10.3934/dcds.2014.34.3555
[11]

Mickael Chekroun, Michael Ghil, Jean Roux, Ferenc Varadi. Averaging of time - periodic systems without a small parameter. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 753-782. doi: 10.3934/dcds.2006.14.753
[12]

Jacques Demongeot, Dan Istrate, Hajer Khlaifi, Lucile Mégret, Carla Taramasco, René Thomas. From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (8) : 2121-2134. doi: 10.3934/dcdss.2020181
[13]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785
[14]

Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i
[15]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116
[16]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017
[17]

Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201
[18]

Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855
[19]

Miguel Rodríguez-Olmos. Continuous singularities in hamiltonian relative equilibria with abelian momentum isotropy. Journal of Geometric Mechanics, 2020, 12 (3) : 525-540. doi: 10.3934/jgm.2020019
[20]

D. P. Demuner, M. Federson, C. Gutierrez. The Poincaré-Bendixson Theorem on the Klein bottle for continuous vector fields. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 495-509. doi: 10.3934/dcds.2009.25.495

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy