Fractal analysis of Neimark-Sacker bifurcation

TL;DR

This paper explores how the box dimension of orbits in two-dimensional discrete dynamical systems changes at bifurcation points, especially during Neimark-Sacker bifurcations, revealing differences between rational and irrational cases.

Contribution

It establishes a connection between box dimension changes and bifurcations in 2D systems, extending previous 1D results, and analyzes the dependence on nondegeneracy and angle-displacement.

Findings

Box dimension changes at bifurcation points from zero to positive values.

Box dimension differs between rational and irrational cases.

Dependence of box dimension on nondegeneracy order and angle-displacement.

Abstract

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.