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The box dimension of degenerate spiral trajectories of a class of ordinary differential equations
Renato Huzak, Domagoj Vlah, Darko Žubrinić, Vesna Županović
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In this paper we initiate the study of 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. 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 spiral 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.
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