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Títol: Dynamics close to a non semi-simple 1: -1 resonant periodic orbit.
Autor: Ollé Torner, Mercè
Pacha Andújar, Juan Ramón
Villanueva Castelltort, Jordi
Altres autors/autores: Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I
Matèries: Bifurcation theory
Hamiltonian systems
Bifurcation
Complex instability
Invariant tori
Bifurcació, Teoria de la
Hamilton, Sistemes de
/Classificació AMS/37 Dynamical systems and ergodic theory/37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
/Classificació AMS/37 Dynamical systems and ergodic theory/37G Local and nonlocal bifurcation theory
Tipus de document: Article
Descripció: In this work, our target is to analyze the dynamics around the $1:-1$ resonance which appears when a family of periodic orbits of a real analytic three-degree of freedom Hamiltonian system changes its stability from elliptic to a complex hyperbolic saddle passing through degenerate elliptic. Our analytical approach consists of computing, in a constructive way and up to some given arbitrary order, the normal form around that resonant (or \emph{critical}) periodic orbit. Hence, dealing with the normal form itself and the differential equations related to it, we derive the generic existence of a two-parameter family of invariant 2D tori which bifurcate from the critical periodic orbit. Moreover, the coefficient of the normal form that determines the stability of the bifurcated tori is identified. This allows us to show the Hopf-like character of the unfolding: elliptic tori unfold ``around'' hyperbolic periodic orbits (case of \emph{direct} bifurcation) while normal hyperbolic tori appear ``around'' elliptic periodic orbits (case of \emph{inverse} bifurcation). Further, a global description of the dynamics of the normal form is also given.
Altres identificadors i accés: http://hdl.handle.net/2117/840
Disponible al dipòsit:E-prints UPC
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