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

Autori
Huzak, Renato ; Vlah, Domagoj ; Žubrinić, Darko ; Županović, Vesna

Izvornik
Applied Mathematics and Computation (0096-3003) 438 (2023); 127569, 15

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Box dimension (Minkowski dimension) ; Degenerate spiral trajectories ; Geometric chirps ; Turning points

Sažetak
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 $\alpha$-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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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