Box dimension of Neimark–Sacker bifurcation

Abstract

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.

Box dimension of unit-time map near nilpotent singularity of planar vector field

“…The semihyperbolic cases can be reduced to the center manifold, see [10]. To complete the study about box dimension of the whole unfolding we also use result about discrete Hopf bifurcation called Neimark-Sacker appearing in 2-dimensional systems, see [9]. Discrete spiral orbit at the bifurcation parameter has box dimension equal to 4 3 .…”

“…See Figure 3a. For other cases we use the results from [4], [8], and [9]. At region 1 there are no singularities.…”

“…The article [8] proved the connection between the value of that box dimension and appropriate one and two-parameter bifurcations. Also in the article [9] the box dimension result for the Neimark-Sacker bifurcation is proved. Recent work [22] shows interesting connection beetween box dimension and Minkowski content with formal classification of parabolic diffeomorphisms.…”

“…See for example [13], [14], [18], [19], [29]. For our study of discrete systems it is particularly interesting fractal analysis of bifurcations of discrete dynamical systems (see [4], [8], [9]), [27], [31], [10], also applied to continuous systems. Direct connection between the box dimension of trajectories of dynamical systems and the bifurcation of that system has been proved in the articles.…”

The box dimension of degenerate spiral trajectories of a class of ordinary differential equations

“…Furthermore, an explicit relation between the box dimension and the leading power in the asymptotic expansion of the Poincaré map of the weak focus has been obtained (for more details see [29,31]). The box dimension of spiral trajectories changes from trivial to nontrivial for parameter values at which some bifurcations occur (Hopf-Takens bifurcations [29], Bogdanov-Takens bifurcations [13,14], discrete saddle-node and period doubling bifurcations [6,12], etc.) Our paper is a natural continuation of [29].…”