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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

Denis G. Gaidashev 1,

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.
Keywords: Renormalization, isoenergetically degenerate Hamiltonian flows, and invariant tori..
Mathematics Subject Classification: 37F25, 37E45, 70H08, 37J2.
Citation: Denis G. Gaidashev. Renormalization of isoenergetically degenerate hamiltonian flows and associated bifurcations of invariant tori. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 63-102. doi: 10.3934/dcds.2005.13.63
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