Let be a continuous scalar function defined for for which there exists a sequence such that and changes sign at any point . The oscillatory dimension of near is defined as the box dimension of the graph of near .
If is a scalar function of class such that the curve is a spiral converging to the origin, then the phase dimension of is the box dimension of near the origin.
The authors calculate the oscillatory and phase dimensions for some functions of special type. The results are applied to solutions of some Liénard equations and weakly damped oscillators.