  In this paper we show how a change of box dimension of the orbits of
two-dimensional discrete dynamical systems is connected to bifurcations in a
nonhyperbolic fixed point. This connection is already shown in the case of
one-dimensional discrete dynamical systems. Namely, at the bifurcation point
the box dimension changes from zero to a certain positive value which is
connected to the type of bifurcation. First, we study a two-dimensional
discrete dynamical system with only one multiplier on the unit circle, and get
the result for the box dimension of the orbit on the center manifold. Then we
consider the planar discrete system undergoing a Neimark-Sacker bifurcation. It
shows that the box dimension depends on the order of the nondegeneracy at the
nonhyperbolic fixed point and on the angle-displacement map. We prove that the
box dimension is different in rational and irrational case as we initially
expected.
