A new approach to measure the fractal dimension of a trajectory in the high-dimensional phase space

Abstract

In this paper, we introduce a new approach, which measures the fractal dimension (FD) of a trajectory in the multi-dimensional phase space based on the self-similarity of the sub-trajectories. Actually, we first compute the length of the sub-trajectories extracted from zooming out the trajectory in the phase space and then estimate the average length of the sub-trajectories in these zooms. Finally, we also calculate the fractal dimension of the trajectory based on the exponent of the power-law between the average length and the zoom-out size. For validating this approach, we also use the Weierstrass cosine function, which can generate fractured (fractal) trajectories with different dimensions. A set of the EEG segments recorded under the eyes-open and eyes-closed resting conditions is also employed to validate this new method by the data of a natural system. Generally, the outcomes of this method represent that it can well follow variations create in the dimension of a fractal trajectory. Therefore, since this new dimension can be estimated in every high-dimensional phase space, it is a good choice for investigating the dimension and the behavior of the high-dimensional strange attractors.