engleski

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

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.

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

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