On the Asymptotic Determination of Invariant Manifolds for Autonomous Ordinary Differential Equations

• Autores: Jesús Francisco Palacián Subiela
• Localización: Revista de la Academia de Ciencias Exactas, Físicas, Químicas y Naturales de Zaragoza, ISSN 0370-3207, Nº. 57, 2002, págs. 7-66
• Idioma: inglés
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• Resumen

  • A methodology to calculate the approximate invariant manifolds of dynamical systems defined through an m-dimensional autonomous vector field is presented. The technique is based on the calculation of formal symmetries and generalized normal forms associated to the vector field making use of Lie transformations for ordinary differential equations. Once a symmetry is determined up to a certain order, a reduction map allows us to pass from the equation in normal form to the orbit space,leading to the so-called reduced system of dimension s < m. Next, a non-degenerate p-dimensional invariant set of the reduced system is transformed, asymptotically, into a (p+m-s)-dimensional invariant set of the departure equation. We put three examples of normal forms computations and reduction process for Hamiltonian and dissipative systems. The procedure is illustrated by three applications: i) we characterize the set of all periodic orbits sufficiently close to the origin of the Hamiltonian vector field defined by the H¿enon and Heiles family when the main frequencies do not satisfy a resonance condition; ii) we calculate the normally hyperbolic invariant manifold together with its stable and unstable manifold of an equilibrium point of type centre×centre×saddle for the three-degrees-of-freedom (3DOF) Hamilton function of the Rydberg atom, explaining the relevance of these invariant structures in the Transition State Theory; and iii) we apply our technique to the reduction process of the Lorenz equations, obtaining periodic orbits and some one-dimensional (1D) and 2D invariant sets.