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In this paper we show how a change of box dimension of the orbits of two-dimensional discrete dynamical systems is connected to their bifurcations in a nonhyperbolic fixed point. This connection is already shown in the case of one-dimensional discrete dynamical systems (see [12],[8]). Namely, at the bifurcation point the box dimension changes from zero to a certain positive value which is connected with 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 is shown that box dimension depends on the order of the nondegeneracy at the nonhyperbolic fixed point and on the angle-displacement map. As it was expected, we prove that the box dimension is different in rational and irrational case.

The purpose of this paper is to derive sufficient conditions for the oscillation of all solutions of a difference equation with several non-monotone deviating arguments and nonnegative coefficients. Corresponding difference equations of both retarded and advanced type are studied. Examples illustrating the significance of the results are also given.

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