Geometrical properties of a class of systems with spiral trajectories in R^3

TL;DR

This paper investigates the fractal geometry of solutions to certain second-order nonautonomous differential equations, focusing on spiral trajectories in three-dimensional space and their box dimensions, revealing connections to bifurcations and Poincaré maps.

Contribution

It introduces a novel approach to measure fractal oscillatory behavior of solutions using phase and oscillatory dimensions, linking these to the asymptotics of Poincaré maps and bifurcation analysis.

Findings

Computed the box dimension of spiral trajectories in R^3.

Linked phase dimension to the asymptotics of Poincaré maps.

Analyzed box dimension near bifurcation points of limit cycles.

Abstract

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 Poincar\'e 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.