STABILITY OF PERIODIC-SOLUTIONS NEAR A COLLISION OF EIGENVALUES OF OPPOSITE SIGNATURE

Some general observations about stability of periodic solutions of Hamiltonian systems are presented as well as stability results for the periodic solutions that exist near a collision of pure imaginary eigenvalues. Let I = closed-intergral pdq be the action functional for a periodic orbit. The stability theory is based on the surprising result that changes in stability are associated with changes in the sign of dI/d-omega, where omega is the frequency of the periodic orbit. A stability index based on dI/d-omega is defined and rigorously justified using Floquet theory and complete results for the stability (and instability) of periodic solutions near a collision of pure imaginary eigenvalues of opposite signature (the 1: -1 resonance) are obtained.