  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.
