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The trajectory intersection: An approach for nonlinear down-sampling

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  • Sohrabi, Faezeh
  • Khodabakhshi, Mohammad Bagher

Abstract

Poincaré section is a method for representing the trajectory of dynamical systems in terms of a set of discrete points in a lower dimension. Herein, inspired by the idea of the Poincaré section, we proposed a method for drastically reducing the size of a data set that describes a dynamical system, principally a dynamical system operating in the chaotic regime, while retaining its essential information. The main idea of the proposed method is finding an optimal plane intersecting the trajectory of the system based on information entropy. First, a novel formulation of the trajectory intersection as a differentiable closed form transformation is proposed. Then, optimal parameters of the intersection plane are found through numerical optimization. Finally, the resulting points are arranged in unique coordination by use of principal component analysis (PCA). Our findings indicate that this method is able to preserve the behavior of dynamical systems in both of the state space and time domain. Moreover, transitions of the system among dynamical regimes are detectable. Therefore, our methodology characterizes the dynamical system adequately while the number of samples is noticeably reduced.

Suggested Citation

  • Sohrabi, Faezeh & Khodabakhshi, Mohammad Bagher, 2019. "The trajectory intersection: An approach for nonlinear down-sampling," Chaos, Solitons & Fractals, Elsevier, vol. 124(C), pages 10-17.
    • Handle: RePEc:eee:chsofr:v:124:y:2019:i:c:p:10-17
    DOI: 10.1016/j.chaos.2019.04.034
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    References listed on IDEAS

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    1. Mukherjee, Sayan & Palit, Sanjay Kumar & Banerjee, Santo & Wahab, A.W.A. & Ariffin, MRK & Bhattacharya, D.K., 2017. "Computing two dimensional Poincaré maps for hyperchaotic dynamics," Applied Mathematics and Computation, Elsevier, vol. 301(C), pages 140-154.
    2. Goshvarpour, Atefeh & Goshvarpour, Ateke, 2018. "Poincaré's section analysis for PPG-based automatic emotion recognition," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 400-407.
    3. Amiri, Mahmood & Davoodi-Bojd, Esmaeil & Bahrami, Fariba & Raza, Mohsin, 2011. "Bifurcation analysis of the Poincaré map function of intracranial EEG signals in temporal lobe epilepsy patients," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(11), pages 2471-2491.
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