Second order asymptotics for Krein indefinite multipliers with multiplicity two

Abstract

We consider linear Hamiltonian equations in R2n of the following typedγdt(t)=J2nA(t)γ(t),γ(0)∈Sp(2n,R), where J=J2n=def[0Idn−Idn0] and A:t↦A(t) is a C1 curve in the space of 2n×2n real matrices which are symmetric. Then, t↦γ(t) is a C2 curve in the space of 2n×2n (real) symplectic matrices. We obtain second order asymptotics for the eigenvalues bifurcated from non-real Krein indefinite eigenvalues with algebraic multiplicity two and geometric multiplicity one. As a corollary, we obtain a simple formula about the derivative of the sum of the bifurcated eigenvalues at time t=0. In the end, we discuss possible potential applications for the linear stability of the elliptic Lagrangian solutions of the planar three-body problem.