INSTABILITY AND BIFURCATION NEAR THE SYMPLECTIC 1 4 RESONANCE/

In coupled mechanical systems with an underlying Hamiltonian structure , the stability analysis-of periodic solutions or periodically forced equilibria-leads via a Poincare' section to an interated symplectic ma p. In this paper, a two-parameter family of symplectic maps on R-4 is considered when the linearization has a loss of stability through a co llision of Floquet multipliers at +/-i. Two approaches to the problem are considered: first, a bifurcation analysis on configuration space w hich leads to a complete local theory for bifurcation and stability of period-4 points in the two-parameter family. The second approach is t o define a model vector field whose time-1 map approximates the dynami cs near the instability. The model vector field derived here has indep endent interest as the truncated normal form for a Z(4)-equivariant Ha miltonian vector field on R-4 with a double-zero eigenvalue. A spinnin g double-orthogonal pendulum with this instability is also presented.