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

Renato Huzak, Domagoj Vlah, Darko Žubrinić and Vesna Županović

Applied Mathematics and Computation, 2023, vol. 438, issue C

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.

Keywords: Box dimension (Minkowski dimension); Degenerate spiral trajectories; Geometric chirps; Turning points (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:438:y:2023:i:c:s0096300322006439

DOI: 10.1016/j.amc.2022.127569

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