The diffuse vibrational bands, observed in the ultraviolet photodissociation spectrum of nitrous oxide by exciting the molecule in the first (1)A' state, have recently been attributed to resonances localized mainly in the NN stretch and bend degrees of freedom. To further investigate the origin of this localization, fundamental families of periodic orbits emanating from several stationary points of the (1)A' potential energy surface and bifurcations of them are computed. We demonstrate that center-saddle bifurcations of periodic orbits are the main mechanism for creating stable regions in phase space that can support the partial trapping of the wave packet, and thus they explain the observed spectra. A non-linear mechanical methodology, which involves the calculation of equilibria, periodic orbits, and transition states in normal form coordinates, is applied for an in detail exploration of phase space. The fingerprints of the phase space structures in the quantum world are identified by solving the time dependent Schrödinger equation and calculating autocorrelation functions. This demonstrates that different reaction channels could be controlled if exact knowledge of the phase space structure is available to guide the initial excitation of the molecule.