Box dimension of trajectories of some discrete dynamical systems

Abstract

Abstract. We study the asymptotics, box dimension, and Minkowski content of trajectories of some discrete dynamical systems. We show that under very general conditions, trajectories corresponding to parameters where saddle-node bifurcation appears have box dimension equal to 1/2, while those corresponding to period-doubling bifurcation parameter have box dimension equal to 2/3. Furthermore, all these trajectories are Minkowski nondegenerate. The results are illustrated in the case of logistic map.

Multiplicity of fixed points and growth of ε-neighborhoods of orbits

“…The article is organized as follows. First, in Subsection 1.1 we recall the connection from [5] between the box dimension of the orbit and the multiplicity of the fixed point in the differentiable case, see Theorem 1. In Subsection 1.2 we recall and introduce definitions and notions we need in non-differentiable cases.…”

Invariance of the normalized Minkowski content with respect to the ambient space

“…For any s ∈ [0, N] and for any bounded set U ⊂ R N , it holds that . The set can be constructed using fractal strings, as in [7], or as a discrete orbit generated by function g(x) = x − x α , α ∈ R, α > 1, as in [2]. It is easy to prove that, if U ⊂ R N is Minkowski measurable in R N , with box dimension d, then U ×[0, 1] is Minkowski measurable in R N +1 , with box dimension d + 1.…”

Box dimension of Neimark–Sacker bifurcation

“…On the other hand, the map G depends on r, which complicates the study of the orbit structure on the invariant curve, and consequently, the calculation of box dimension. All the previously studied bifurcations (see [8], [28], [12]) showed that the box dimension of a orbit around the nonhyperbolic fixed point is connected with the box dimension of the invariant set which emerge at the bifurcation point. For instance, in the bifurcation of one-dimensional discrete dynamical systems when only fixed point can bifurcate, the box dimension is between 0 and 1.…”

“…In the case of dynamical systems, fractal analysis consists of studying the box dimension and Minkowski content of trajectories or orbits. Several articles with fractal analysis of bifurcations of dynamical systems (see [8], [31], [28], [29]) showed that there is a direct connection between the change in box dimension of trajectories of dynamical systems and the bifurcation of that system. Around the hyperbolic singularities the box dimension is trivial (0), while around the nonhyperbolic singularities the box dimension is positive and connected to the appropriate bifurcation.…”