2012/11/05 by Luka Korkut, Korkut, Luka, Domagoj Vlah +4
Computer Science · Mathematics · Physics and Astronomy · #28A80 #34C15 #37C45 #37G10 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems #math.CA #math.DS #msc:28A80 #msc:34C15 #msc:37C45 #msc:37G10 #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1211.0918
openalex publication_date 2012/11/05 · arxiv created 2014/04/21 · arxiv updated 2014/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Here we study a class of second-order nonautonomous differential equations, and the corresponding planar and spatial systems, from the point of view of fractal geometry. The fractal oscillatority of solutions at infinity is measured by oscillatory and phase dimensions. The oscillatory dimension is defined as the box dimension of the reflected solution near the origin, while the phase dimension is defined as the box dimension of a trajectory of the corresponding planar system in the phase plane. Using the phase dimension of the second-order equation we compute the box dimension of a spiral trajectory of the spatial system, lying in Lipschitzian or H" olderian surfaces. This phase dimension of the second-order equation is connected to the asymptotics of the associated Poincaré map. Also, the box dimension of a trajectory of the reduced normal form with one eigenvalue equals to zero, and a pair of pure imaginary eigenvalues has been computed when limit cycles bifurcate from the origin.