Abstract
We investigate symplectic difference systems with periodic coefficients. We show that the known traffic rules for eigenvalues of the monodromy matrix of linear Hamiltonian differential and difference systems can be extended also to discrete symplectic systems.
Highlights
A symplectic difference system is the discrete first-order system zk+ = Skzk, k ∈ Z, where the matrices Sk ∈ C n× n are symplectic, i.e., Sk*J Sk = J, J= –I I. ( )In particular, if Sk ∈ R n× n, we have SkT J Sk = J
In our paper we suppose that the matrices Sk depend on a parameter λ, i.e., we consider the system zk+ = Sk(λ)zk
The stability theory of periodic Hamiltonian differential systems is deeply developed since the fifties of the last century
Summary
The terminology for symplectic matrices is not unique. Symplectic difference systems are discrete counterparts of linear Hamiltonian differential systems z = J H(t)z with the Hermitean matrix H, i.e., H* = H, in the sense that they are the most general firstorder systems whose fundamental matrix is symplectic. Concerning the continuous counterpart of symplectic difference systems, linear Hamiltonian differential systems z = λJ H(t)z, the stability theory of periodic Hamiltonian differential systems is deeply developed since the fifties of the last century. Concerning expression ( ), consider the matrices X [n], . It remains to prove that A(λ, λ ) is Hermitean This matrix is given by an infinite series, and the typical summand in this series is given by the expression in brackets in ( ).
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