On the central stability zone for linear discrete-time Hamiltonian systems

Abstract

In this paper we start from the discrete version of linear Hamiltonian systems with periodic coefficients $_y_(k+1) - _y(_k) = \lambda B(__k)_y(_k) + \lambda D_(k^z_(k+1))$ $_z_(k+1) - _z(_k) = -\lambda A(__k)_y(_k) - \lambda B*_(k^z_(k+1))$ where $A_k$ and $D_k$ are Hermitian matrices, $A_k$, $B_k$, $D_k$ define $N$-periodic sequences, and $\lambda$ is a complex parameter. For this system a Krein-type theory of the $\lambda$-zones of strong (robust) stability may be constructed. Within this theory the side $\lambda$-zones’ width may be estimated using the multipliers’ “traffic rules” of Krein while the central stability zone (centered around $\lambda$ = 0) is estimated using the eigenvalues of a certain boundary value problem which is self-adjoint. In the discrete-time there occur some specific differences with respect to the continuous time case due to the fact that the transition matrix (hence the monodromy matrix also) is not entire with respect to $\lambda$ but rational. During the paper we consider some specific cases (the matrix analogue of the discretized Hill equation, the J-unitary and symplectic systems, real scalar systems) for which the results on the eigenvalues are complete and obtain some simplified estimates of the central stability zones.