# American Institute of Mathematical Sciences

April  2005, 13(1): 63-102. doi: 10.3934/dcds.2005.13.63

## Renormalization of isoenergetically degenerate hamiltonian flows and associated bifurcations of invariant tori

 1 Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3, Canada

Received  December 2003 Revised  December 2004 Published  March 2005

The paper presents a study of a renormalization group transformation acting on an appropriate space of Hamiltonian functions in two angle and two action variables. In particular, we study the existence of real invariant tori, on which the flow is conjugate to a rotation with a rotation number equal to a quadratic irrational ($\omega$-tori). We demonstrate that the stable manifold of the renormalization operator at the "simple" fixed point contains isoenergetically degenerate Hamiltonians possessing shearless $\omega$-tori. We also show that one-parameter families of Hamiltonians transverse to the stable manifold undergo a bifurcation: for a certain range of the parameter values the members of these families posses two distinct $\omega$-tori, the members of such families lying on the stable manifold posses one shearless $\omega$-torus, while the members corresponding to other parameter values do not posses any.
Citation: 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
 [1] 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 [2] C. Chandre. Renormalization for cubic frequency invariant tori in Hamiltonian systems with two degrees of freedom. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 457-465. doi: 10.3934/dcdsb.2002.2.457 [3] Hans Koch. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 881-909. doi: 10.3934/dcds.2004.11.881 [4] Fuzhong Cong, Yong Li. Invariant hyperbolic tori for Hamiltonian systems with degeneracy. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 371-382. doi: 10.3934/dcds.1997.3.371 [5] Shengqing Hu, Bin Liu. Degenerate lower dimensional invariant tori in reversible system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3735-3763. doi: 10.3934/dcds.2018162 [6] Tingting Zhang, Àngel Jorba, Jianguo Si. Weakly hyperbolic invariant tori for two dimensional quasiperiodically forced maps in a degenerate case. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6599-6622. doi: 10.3934/dcds.2016086 [7] João Lopes Dias. Brjuno condition and renormalization for Poincaré flows. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 641-656. doi: 10.3934/dcds.2006.15.641 [8] Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 437-446. doi: 10.3934/dcdsb.2016.21.437 [9] Hans Koch, João Lopes Dias. Renormalization of diophantine skew flows, with applications to the reducibility problem. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 477-500. doi: 10.3934/dcds.2008.21.477 [10] Hsuan-Wen Su. Finding invariant tori with Poincare's map. Communications on Pure & Applied Analysis, 2008, 7 (2) : 433-443. doi: 10.3934/cpaa.2008.7.433 [11] Ugo Locatelli, Antonio Giorgilli. Invariant tori in the Sun--Jupiter--Saturn system. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 377-398. doi: 10.3934/dcdsb.2007.7.377 [12] Xiaocai Wang. Non-floquet invariant tori in reversible systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3439-3457. doi: 10.3934/dcds.2018147 [13] Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431 [14] Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098 [15] Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123 [16] Gemma Huguet, Rafael de la Llave, Yannick Sire. Computation of whiskered invariant tori and their associated manifolds: New fast algorithms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1309-1353. doi: 10.3934/dcds.2012.32.1309 [17] 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 [18] Oliver Butterley, Carlangelo Liverani. Robustly invariant sets in fiber contracting bundle flows. Journal of Modern Dynamics, 2013, 7 (2) : 255-267. doi: 10.3934/jmd.2013.7.255 [19] Livio Flaminio, Giovanni Forni, Federico Rodriguez Hertz. Invariant distributions for homogeneous flows and affine transformations. Journal of Modern Dynamics, 2016, 10: 33-79. doi: 10.3934/jmd.2016.10.33 [20] Amadeu Delshams, Pere Gutiérrez, Tere M. Seara. Exponentially small splitting for whiskered tori in Hamiltonian systems: flow-box coordinates and upper bounds. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 785-826. doi: 10.3934/dcds.2004.11.785

2018 Impact Factor: 1.143