# American Institute of Mathematical Sciences

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

## On the renormalization of Hamiltonian flows, and critical invariant tori

 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.
Citation: Hans Koch. On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete & Continuous Dynamical Systems - A, 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 - A, 2014, 34 (5) : 2091-2104. doi: 10.3934/dcds.2014.34.2091 [2] Gernot Greschonig. Real cocycles of point-distal minimal flows. Conference Publications, 2015, 2015 (special) : 540-548. doi: 10.3934/proc.2015.0540 [3] Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807 [4] Denis G. Gaidashev. Renormalization of isoenergetically degenerate hamiltonian flows and associated bifurcations of invariant tori. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 63-102. doi: 10.3934/dcds.2005.13.63 [5] César J. Niche. Non-contractible periodic orbits of Hamiltonian flows on twisted cotangent bundles. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 617-630. doi: 10.3934/dcds.2006.14.617 [6] 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 - A, 2016, 36 (5) : 2405-2417. doi: 10.3934/dcds.2016.36.2405 [7] Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273 [8] 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 [9] 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 - A, 2016, 36 (6) : 3153-3225. doi: 10.3934/dcds.2016.36.3153 [10] 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 [11] Calin Iulian Martin. A Hamiltonian approach for nonlinear rotational capillary-gravity water waves in stratified flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 387-404. doi: 10.3934/dcds.2017016 [12] Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201 [13] Hassan Najafi Alishah, Pedro Duarte. Hamiltonian evolutionary games. Journal of Dynamics & Games, 2015, 2 (1) : 33-49. doi: 10.3934/jdg.2015.2.33 [14] Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855 [15] G. A. Swarup. On the cut point conjecture. Electronic Research Announcements, 1996, 2: 98-100. [16] Marek Rychlik. The Equichordal Point Problem. Electronic Research Announcements, 1996, 2: 108-123. [17] 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 [18] David Damanik, Anton Gorodetski. The spectrum of the weakly coupled Fibonacci Hamiltonian. Electronic Research Announcements, 2009, 16: 23-29. doi: 10.3934/era.2009.16.23 [19] Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789 [20] Ely Kerman. On primes and period growth for Hamiltonian diffeomorphisms. Journal of Modern Dynamics, 2012, 6 (1) : 41-58. doi: 10.3934/jmd.2012.6.41

2016 Impact Factor: 1.099