2021/09/30 by Renato Huzak, Domagoj Vlah, Huzak, Renato +5
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2109.15167
In this paper we initiate the study of the box dimension of degenerate spiral\ntrajectories of a class of ordinary differential equations. A class of\nsingularities of focus type with two zero eigenvalues (nilpotent or more\ndegenerate) has been studied. We find the box dimension of a polynomial\ndegenerate focus of type (n,n) by exploiting the well-known fractal results\nfor \α-power spirals. In the general (m,n) case, we formulate a\nconjecture about the box dimension of a degenerate focus. Further, we reduce\nthe fractal analysis of planar nilpotent contact points to the study of the box\ndimension of a slow-fast spiral generated by their "entry-exit" function. There\nexists a bijective correspondence between the box dimension of the slow-fast\nspiral and the codimension of contact points. We also construct a\nthree-dimensional vector field that contains a degenerate spiral, called an\nelliptical power spiral, as a trajectory.\n