Dan Offin - Variational structure of the domains of stability

The classical approach to studying parametric resonance in conservative scalar second order differential equations uses Floquet theory. In modern textbooks this is used to prove the celebrated result of Liapunov which states that strong stability corresponds exactly to the condition . The transition from stability to instability then occurs when where R(t) denotes the fundamental matrix solution normalized so that R(0)=I. In this talk we present a different approach, using symplectic geometry. We obtain a completely equivalent criterion to Liapunov's, in terms of the indices of certain variational problems. These indices are interpreted as the rotation and intersection number of certain Lagrangian planes. The main theorem applies to planar quadratic Hamiltonian systems which are periodic in the time variable and which satisfy the classical condition of Legendre. This result has applications to nonlinear problems, as well as special results assuming symmetry of the coefficient matrix, in the case of higher dimensional systems.