American Institute of Mathematical Sciences

Advanced
July  2002, 8(3): 633-646. doi: 10.3934/dcds.2002.8.633

On the renormalization of Hamiltonian flows, and critical invariant tori

Hans Koch 1,

1. 

Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, United States

Received  August 2001 Revised  November 2001 Published  April 2002

We analyze a renormalization group transformation $\mathcal R$ for partially analytic Hamiltonians, with emphasis on what seems to be needed for the construction of non-integrable fixed points. Under certain assumptions, which are supported by numerical data in the golden mean case, we prove that such a fixed point has a critical invariant torus. The proof is constructive and can be used for numerical computations. We also relate $\mathcal R$ to a renormalization group transformation for commuting maps.
Keywords: fixed point., Hamiltonian flows.
Mathematics Subject Classification: 58F22, 58F2.
Citation: Hans Koch. On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 633-646. doi: 10.3934/dcds.2002.8.633
[1]

Hans Koch, Héctor E. Lomelí. On Hamiltonian flows whose orbits are straight lines. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2091-2104. doi: 10.3934/dcds.2014.34.2091
[2]

Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271
[3]

Gernot Greschonig. Real cocycles of point-distal minimal flows. Conference Publications, 2015, 2015 (special) : 540-548. doi: 10.3934/proc.2015.0540
[4]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807
[5]

Denis G. Gaidashev. Renormalization of isoenergetically degenerate hamiltonian flows and associated bifurcations of invariant tori. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 63-102. doi: 10.3934/dcds.2005.13.63
[6]

César J. Niche. Non-contractible periodic orbits of Hamiltonian flows on twisted cotangent bundles. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 617-630. doi: 10.3934/dcds.2006.14.617
[7]

Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273
[8]

Gianluca Crippa, Milton C. Lopes Filho, Evelyne Miot, Helena J. Nussenzveig Lopes. Flows of vector fields with point singularities and the vortex-wave system. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2405-2417. doi: 10.3934/dcds.2016.36.2405
[9]

Xavier Perrot, Xavier Carton. Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 971-995. doi: 10.3934/dcdsb.2009.11.971
[10]

Tiphaine Jézéquel, Patrick Bernard, Eric Lombardi. Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3153-3225. doi: 10.3934/dcds.2016.36.3153
[11]

Yuri B. Suris. Variational formulation of commuting Hamiltonian flows: Multi-time Lagrangian 1-forms. Journal of Geometric Mechanics, 2013, 5 (3) : 365-379. doi: 10.3934/jgm.2013.5.365
[12]

Calin Iulian Martin. A Hamiltonian approach for nonlinear rotational capillary-gravity water waves in stratified flows. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 387-404. doi: 10.3934/dcds.2017016
[13]

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
[14]

Hassan Najafi Alishah, Pedro Duarte. Hamiltonian evolutionary games. Journal of Dynamics & Games, 2015, 2 (1) : 33-49. doi: 10.3934/jdg.2015.2.33
[15]

Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855
[16]

G. A. Swarup. On the cut point conjecture. Electronic Research Announcements, 1996, 2: 98-100.

[17]

Marek Rychlik. The Equichordal Point Problem. Electronic Research Announcements, 1996, 2: 108-123.

[18]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024
[19]

V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277
[20]

P. Balseiro, M. de León, Juan Carlos Marrero, D. Martín de Diego. The ubiquity of the symplectic Hamiltonian equations in mechanics. Journal of Geometric Mechanics, 2009, 1 (1) : 1-34. doi: 10.3934/jgm.2009.1.1

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences