The propagation of a soliton in a nonlinear optical fiber with a periodically modulated but sign-preserving dispersion coefficient is analyzed by means of the variational approximation. The dynamics are reduced to a second-order evolution equation for the width of the soliton that oscillates in an effective potential well in the presence of a periodic forcing induced by the inhomogeneity. This equation of motion is considered analytically and numerically. Resonances between the oscillations in the potential well and the external forcing are analyzed in detail. It is demonstrated that regular forced oscillations take place only at very small values of the amplitude of the inhomogeneity; the oscillations become chaotic as the inhomogeneity becomes stronger and, when the dimensionless amplitude attains a threshold value which is typically less than 1/4, the soliton is completely destroyed by the periodic inhomogeneity.