engleski

Fractal analysis of Hopf bifurcation for maps

In this paper we show how a change of a box dimension of 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 and Hopf bifurcation for continuous systems. Namely, at the bifurcation point the box dimension changes from zero to a certain positive value which is connected to the appropriate bifurcation. We study a two-dimensional discrete dynamical system with only one multiplier on the unit circle, and show a result for the box dimension of an orbit on the center manifold. We also consider a planar discrete system undergoing a Neimark-Sacker bifurcation. It is shown that box dimension depends on the order of 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 the rational and irrational case.

box dimension; nonhyperbolic fixed point; bifurcation; center manifold; Neimark-Sacker bifurcation

nije evidentirano