A numerical investigation of the nonlinear dynamics of a passively mode-locked fiber laser containing a long period fiber grating was conducted. The model was based on the complex Ginzburg-Landau equation and the nonlinear coupled mode equations of the grating. The numerical results indicated the existence of passive mode-locking and autosoliton generation in the cavity of the laser. Both single and bound soliton pulse trains were found to exhibit period doubling bifurcations and a route to chaos as the saturated gain was increased. Furthermore, the presence of long period pulsation, soliton sidebands and possible coexisting attractors excited by multisoliton formation and soliton energy quantization was observed. The laser dynamics were presented through plotting the pulse energy against the linearly increasing gain, so obtaining bifurcation diagrams. Higher order nonlinear and dispersive effects, such as Raman self-frequency shift, self-steepening and third order dispersion, were found to change these bifurcation diagrams.