# Fractal analysis of hyperbolic and nonhyperbolic fixed points and singularities of dynamical systems in $\mathbb{R}^{n}$

## TL;DR

This paper investigates the fractal box dimension of orbits near fixed points in higher-dimensional discrete and continuous dynamical systems, extending known one-dimensional results and introducing new concepts like projective box dimension.

## Contribution

It generalizes box dimension results from one-dimensional to higher-dimensional systems, introduces projective box dimension, and applies findings to continuous systems via the unit time map.

## Key findings

- Box dimension is zero near hyperbolic fixed points on stable/unstable manifolds.
- Box dimension on the center manifold determines the dimension near nonhyperbolic points.
- Projective box dimension serves as a criterion for nonhyperbolicity.

## Abstract

The main purpose of this article is to study box dimension of orbits near hyperbolic and nonhyperbolic fixed points of discrete dynamical systems in higher dimensions. We generalize the known results for one-dimensional systems, that is, the orbits near the hyperbolic fixed point in one-dimensional discrete dynamical system has the box dimension equal to zero and the orbits near nonhyperbolic fixed point has positive box dimension. In the process of studying box dimensions, we use the stable, unstable and center manifolds and appropriate system restrictions. The main result is that the box dimension of orbit equals zero near stable and unstable hyperbolic fixed points and on the stable and unstable manifolds. The main results in the nonhyperbolic case is that box dimension is determined by the box dimension on the center manifold. We also introduce the projective box dimension and use it as a sufficient condition for nonhyperbolicity. At the end, all the results for discrete systems can be applied to continuous systems by using the unit time map so we apply it to the hyperbolic and nonhyperbolic singularities of continuous dynamical systems in higher dimensions.