Geometrical properties of systems with spiral trajectories in R^3

Abstract

We study a class of second-order nonautonomous differential equations, and the corresponding planar and spatial systems, from the geometrical point of view. The oscillatory behavior 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 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. This phase dimension of the second-order equation is connected to the asymptotic of the associated Poincare map. Also, the box dimension of a trajectory of the reduced normal form with one eigenvalue equals zero, and a pair of pure imaginary eigenvalues is computed when limit cycles bifurcate from the origin.

Keywords

• Spiral
• chirp
• box dimension
• rectifiability
• oscillatory dimension
• phase dimension
• limit cycle