O sombreamento de trajetórias no mapa padrão

Abstract

Numerical solutions of a mathematical system presents noise due to the truncation and roundoff errors. If chaos cannot ruled out then these errors are amplified. The Hamiltonian dynamical systems may present chaos and periodicity in the same phase space for a given range of values of the control parameter. In particular the standard map is a Hamiltonian system widely investigated to be derived for many physical systems of interest. An answer for question of validity of numerical solutions is the shadowing of physical trajectories that ensures the existence of real orbits that stays near of noisy trajectories for long time. If the system present hyperbolic structure, then all conditions are fulfilled and shadowing can be done for every point of the set where the system is defined. On the other hand, most of systems are nonhyperbolic like standard map. This loss of hyperbolicity can ocurr in two ways: unstable dimension variability and tangencies between manifolds. This study aims the shadowing problem and investigate regions where tangencies can ocurr caracterizing periodic orbits structure in phase space. With the knowledge of unstable periodic orbits is possible to obtain manifolds and verify regions where shadowing is broken by tangencies. For this the Schmelcher-Diakonos method is employed for found periodic orbits. The manifolds are found by taking a ball of initial conditions in linear neighborhood of points of any period and by iteration foward in time of map to represent an aproximation of unstable manifold and by reverse iteration in time to represent stable manifold. As a result we found regions were possible tangencies ocurr and shadowing cannot be done.