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Discrete linear Hamiltonian systems: A survey. (English) Zbl 0942.39009

The authors present a survey on recent results connected to the discrete linear vector Hamiltonian system

Δx(t)=A(t)x(t)+B(t)u(t),Δu(t)=C(t)x(t)-A * (t)u(t),

where B(t) and C(t) are n×n Hermitian matrix functions on the discrete interval [a,b]:={a,a+1,,b},a,b, and it is assumed that I-A(t) is nonsingular on [a,b].

The paper is organized as follows. Section 2 introduces linear Hamiltonian difference systems and explains how they are special cases of symplectic difference systems. Section 3 gives an introduction to symplectic systems and introduces important notions such as conjoined bases, generalized zeros, focal points, and disconjugacy.

In Section 4 are introduced the concept and provided the motivation of trigonometric systems. An explicit transformation is also offered which transforms any symplectic system into a trigonometric system and preserves oscillatory behavior. Section 6 presents an overview of the discrete calculus of variations. In Section 7 an introduction is given to the theory of Stefan Higler in order to unify discrete and continuous analysis.

MSC:
39A12Discrete version of topics in analysis
37J05Relations of dynamical systems with symplectic geometry and topology