  We investigate how the phase space structure of a 3D autonomous Hamiltonian
system evolves across a series of successive 2D and 3D pitchfork and
period-doubling bifurcations, as the transition of the parent families of
periodic orbits (POs) from stability to simple instability leads to the
creation of new stable POs. Our research illustrates the consecutive
alterations in the phase space structure near POs as the stability of the main
family of POs changes. This process gives rise to new families of POs within
the system, either maintaining the same or exhibiting higher multiplicity
compared to their parent families. Tracking such a phase space transformation
is challenging in a 3D system. By utilizing the color and rotation technique to
visualize the 4D Poincar'e surfaces of section of the system, i.e. projecting
them onto a 3D subspace and employing color to represent the fourth dimension,
we can identify distinct structural patterns. Perturbations of parent and
bifurcating stable POs result in the creation of tori characterized by a smooth
color variation on their surface. Furthermore, perturbations of simple unstable
parent POs beyond the bifurcation point which lead to the birth of new stable
families of POs, result in the formation of figure-8 structures of smooth color
variations. These figure-8 formations surround well-shaped tori around the
bifurcated stable POs, losing their well-defined forms for energies further
away from the bifurcation point. We also observe that even slight perturbations
of highly unstable POs create a cloud of mixed color points, which rapidly move
away from the location of the PO. Our study introduces, for the first time, a
systematic visualization of 4D surfaces of section within the vicinity of
higher multiplicity POs. It elucidates how, in these cases, the coexistence of
regular and chaotic orbits contributes to shaping the phase space landscape.
