


Next: Ernesto Perez-Chavela - Heteroclinic Up: Algebraic Geometric Methods in Previous: R. Moechel - To
Dan Offin - Variational structure of the domains of stability
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.



Next: Ernesto Perez-Chavela - Heteroclinic Up: Algebraic Geometric Methods in Previous: R. Moechel - To