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

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  • Karimui, Reza Yaghoobi

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.

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  • Karimui, Reza Yaghoobi, 2021. "A new approach to measure the fractal dimension of a trajectory in the high-dimensional phase space," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    • Handle: RePEc:eee:chsofr:v:151:y:2021:i:c:s0960077921005932
    DOI: 10.1016/j.chaos.2021.111239
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    References listed on IDEAS

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    1. Gallos, Lazaros K. & Song, Chaoming & Makse, Hernán A., 2007. "A review of fractality and self-similarity in complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 386(2), pages 686-691.
    2. Yao, Kui & Chen, Haotian & Peng, W.L. & Wang, Zekun & Yao, Jia & Wu, Yipeng, 2021. "A new method on Box dimension of Weyl-Marchaud fractional derivative of Weierstrass function," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    3. Spasić, Sladjana, 2014. "On 2D generalization of Higuchi’s fractal dimension," Chaos, Solitons & Fractals, Elsevier, vol. 69(C), pages 179-187.
    4. Yaghoobi Karimui, Reza & Azadi, Sassan & Keshavarzi, Parviz, 2019. "The ADHD effect on the high-dimensional phase space trajectories of EEG signals," Chaos, Solitons & Fractals, Elsevier, vol. 121(C), pages 39-49.
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