The comparative index and transformations of linear Hamiltonian differential systems

In this paper we investigate mutual oscillatory behaviour of two linear differential Hamiltonian systems related via symplectic transformations. The main result extends our previous results in [30], where we presented new explicit relations connecting the multiplicities of proper focal points of conjoined bases Y(t) of the Hamiltonian system and the transformed conjoined bases Y˜(t)=R−1(t)Y(t). In the present paper we omit restrictions on the symplectic transformation matrix R(t) concerning the constant rank of its components. As consequences of the main result we prove generalized reciprocity principles which formulate new sufficient conditions for R(t) concerning preservation of (non)oscillation of the abnormal Hamiltonian systems as t → ∞. The main tool of the paper is the comparative index theory for discrete symplectic systems implemented into the continuous case.