Box dimension and bifurcations of one-dimensional discrete dynamical systems

Abstract

This paper is devoted to study the box dimension of the orbits of one-dimensional discrete dynamical systems and their bifurcations in nonhyperbolic fixed points. It is already known that there is a connection between some bifurcations in a nonhyperbolic fixed point of one-dimensional maps, and the box dimension of the orbits near that point. The main purpose of this paper is to generalize that result to the one-dimensional maps of class $C^{k}$ and apply it to one and two-parameter bifurcations of maps with the generalized nondegeneracy conditions. These results show that the value of the box dimension changes at the bifurcation point, and depends only on the order of the nondegeneracy condition. Furthermore, we obtain the reverse result, that is, the order of the nondegeneracy of a map in a nonhyperbolic fixed point can be obtained from the box dimension of a orbit near that point. This reverse result can be applied to the continuous planar dynamical systems by using the Poincare map, in order to get the multiplicity of a weak focus or nonhyperbolic limit cycle. We also apply the main result to the bifurcations of nonhyperbolic periodic orbits in the plane.