The Evolution of the Phase Space Structure Along Pitchfork and Period-Doubling Bifurcations in a 3D-Galactic Bar Potential

We investigate how the phase space structure of a Three-Dimensional (3D) autonomous Hamiltonian system evolves across a series of successive Two-Dimensional (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 Four-Dimensional (4D) Poincar & eacute; 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.