  A codimension-three bifurcation, characterized by a pair of purely imaginary
eigenvalues and a nonsemisimple double zero eigenvalue, arises in the study of
a pair of weakly coupled nonlinear oscillators with Z_2 + Z_2 symmetry. The
methodology is based on Arnold's ideas of versal deformations of matrices for
the linear analysis, and Poincar\'e normal forms for the nonlinear analysis of
the system. The stratified subvariety of primary bifurcations of codimensions
one and two is identified in the parameter space. The analysis reveals
different types of solutions in the state space, including equilibria, limit
cycles, invariant tori and the possibility of homoclinic chaos. A mechanism is
identified for energy transfer without strong resonance between two oscillation
modes with widely separated frequencies.
