Interval methods for rigorous investigations of periodic orbits

In this paper, we investigate the possibility of using interval arithmetic for rigorous investigations of periodic orbits in discrete-time dynamical s ystems with special emphasis on chaotic systems. We show that methods based on interval arithmetic when implemented properly are capable of finding al l period-n cycles for considerable large n. We compare several interval met hods for finding periodic orbits. We consider the interval Newton method an d methods based on the Krawczyk operator and the Hansen-Sengupta operator. We also test the global versions of these three methods. We propose algorit hms for computation of the invariant part and nonwandering part of a given set and for computation of the basin of attraction of stable periodic orbit s, which allow reducing greatly the search space for periodic orbits. As examples we consider two-dimensional chaotic discrete-time dynamical sys tems, defined by the Henon map and the Ikeda map, with the "standard" param eter values for which the chaotic behavior is observed. For both maps using the algorithms presented in this paper, we find very good approximation of the invariant part and the nonwandering part of the region enclosing the c haotic attractor observed numerically. For the Henon map we find all cycles with period n less than or equal to 30 belonging to the trapping region. F or the Ikeda map we find the basin of attraction of the stable fixed point and all periodic orbits with period n less than or equal to 15. For both sy stems using the number of short cycles, we estimate its topological entropy .