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Fractal analysis of degenerate spiral trajectories of a class of ordinary differential equations

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  • Huzak, Renato
  • Vlah, Domagoj
  • Žubrinić, Darko
  • Županović, Vesna

Abstract

In this paper we initiate the study of the Minkowski dimension, also called the box dimension, of degenerate spiral trajectories of a class of ordinary differential equations. A class of singularities of focus type with two zero eigenvalues (nilpotent or more degenerate) has been studied. We find the box dimension of a polynomial degenerate focus of type (n,n) by exploiting the well-known fractal results for α-power spirals. In the general (m,n) case, we formulate a conjecture about the box dimension of a degenerate focus using numerical experiments. Further, we reduce the fractal analysis of planar nilpotent contact points to the study of the box dimension of a slow-fast spiral generated by their “entry-exit” function. There exists a bijective correspondence between the box dimension of the slow-fast spirals and the codimension of contact points. We also construct a three-dimensional vector field that contains a degenerate spiral, called an elliptical power spiral, as a trajectory.

Suggested Citation

  • Huzak, Renato & Vlah, Domagoj & Žubrinić, Darko & Županović, Vesna, 2023. "Fractal analysis of degenerate spiral trajectories of a class of ordinary differential equations," Applied Mathematics and Computation, Elsevier, vol. 438(C).
    • Handle: RePEc:eee:apmaco:v:438:y:2023:i:c:s0096300322006439
    DOI: 10.1016/j.amc.2022.127569
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    References listed on IDEAS

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    1. Resman, Maja, 2013. "Invariance of the normalized Minkowski content with respect to the ambient space," Chaos, Solitons & Fractals, Elsevier, vol. 57(C), pages 123-128.
    2. Korkut, L. & Vlah, D. & Županović, V., 2016. "Fractal properties of Bessel functions," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 55-69.
    3. Maoan Han & Valery G. Romanovski, 2012. "Limit Cycle Bifurcations from a Nilpotent Focus or Center of Planar Systems," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-28, November.
    4. Elezović, Neven & Županović, Vesna & Žubrinić, Darko, 2007. "Box dimension of trajectories of some discrete dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 244-252.
    5. Panigrahy, Chinmaya & Seal, Ayan & Mahato, Nihar Kumar & Bhattacharjee, Debotosh, 2019. "Differential box counting methods for estimating fractal dimension of gray-scale images: A survey," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 178-202.
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    Cited by:

    1. Khalili Golmankhaneh, Alireza & Tejado, Inés & Sevli, Hamdullah & Valdés, Juan E. Nápoles, 2023. "On initial value problems of fractal delay equations," Applied Mathematics and Computation, Elsevier, vol. 449(C).

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