This paper reports on a classical phase space bifurcation analysis of the 2345 Fermi resonance of acetylene. The 2345 Fermi resonance is a multimode nonlinear, resonance coupling that is important to the vibrational dynamics and energy flow of highly excited acetylene. The bifurcation analysis is performed on an integrable Hamiltonian that represents a planar five-mode model of acetylene in which the nu(2), nu(3), nu(4), and nu(5) vibrational modes are nonlinearly coupled through the 2345 Fermi resonance. The phase space structures of the 2345 Fermi resonance are shown to be analogous to but more complicated than phase space structures of the two-mode, 1:1 and 2:1 Fermi resonance. The results are presented in terms of bifurcation diagrams and molecular catastrophe maps. The bifurcation analysis of this multidimensional system with a complicated multimode resonance is a step beyond the simple integrable, resonantly coupled two-mode systems that are now well understood. Analysis of this integrable system also represents a necessary step toward using a multiresonance, i.e., "chaotic" model to decipher the vibrational spectra of highly excited acetylene, based on knowledge of the anharmonic modes born from bifurcations of the low-energy normal modes.